From 548fd8f794a1c1bdb3187d29c81d3b98a0f44bfd Mon Sep 17 00:00:00 2001 From: Bob Carpenter Date: Sat, 27 Jun 2026 14:54:23 -0400 Subject: [PATCH 1/7] beginning intro --- src/quarto-config | 2 +- src/stan-users-guide/_quarto.yml | 6 +- src/stan-users-guide/intro.qmd | 143 +++++++++++++++++++++++++++++++ 3 files changed, 149 insertions(+), 2 deletions(-) create mode 100644 src/stan-users-guide/intro.qmd diff --git a/src/quarto-config b/src/quarto-config index a7b1139f1..65904751c 160000 --- a/src/quarto-config +++ b/src/quarto-config @@ -1 +1 @@ -Subproject commit a7b1139f10c2049385b110507e29383a31a3d002 +Subproject commit 65904751c830555ec8d81d9270e609ec190c425f diff --git a/src/stan-users-guide/_quarto.yml b/src/stan-users-guide/_quarto.yml index 951012f3d..a4d8a1630 100644 --- a/src/stan-users-guide/_quarto.yml +++ b/src/stan-users-guide/_quarto.yml @@ -29,11 +29,15 @@ format: book: title: "Stan User's Guide" - author: "Stan Development Team" + author: "Stan Developme +nt Team" search: true chapters: - index.qmd + - part: "Introduction" + chapters: + - intro.qmd - part: "Example Models" chapters: - regression.qmd diff --git a/src/stan-users-guide/intro.qmd b/src/stan-users-guide/intro.qmd new file mode 100644 index 000000000..60993a6e0 --- /dev/null +++ b/src/stan-users-guide/intro.qmd @@ -0,0 +1,143 @@ +--- +pagetitle: The Stan Ecosystem +--- + +# The Stan Ecosystem {#ecosystem.chapter} + +Stan is a domain-specific tool for specifying probability models, +performing statistical inference and analyzing model fits. Stan is +largely used for Bayesian models with Monte Carlo-based inference, but +it may also be used to fit penalized maximum likelihood estimates and +estimate confidence intervals with the bootstrap with optimization. + + +## Stan community + +Stan has an active community of users and developers. + +* [Stan forums](https://discourse.mc-stan.org), where users and + developers discuss issues with Stan and its applications. + +* [Stan GitHub](https://github.com/stan-dev), which tracks issues and + contains the complete source code for Stan (under the BSD-3 + license). + +* [StanCon](https://mc-stan.org/learn-stan/stancon-talks.html), for + which talks and executable notebooks are available. + + +## Stan documentation + +Stan's documentation is organized through a top-level web page: + +* [Stan documentation](https://mc-stan.org/docs/) + +This is the *User's Guide*, which is complemented by the *Reference +Manual* and *Functions Reference*. These are all interface-agnostic +guides to the Stan language and its execution. They are available at + +* [*Stan Users' Guide*](https://mc-stan.org/docs/stan-users-guide/) +documentation. + +* [*Stan Reference + Manual*](https://mc-stan.org/docs/reference-manual/) + +* [*Stan Functions Reference*](https://mc-stan.org/docs/functions-reference/) + +Documentation for the specific Stan interfaces are here + +* [*CmdStan User's Guide*](https://mc-stan.org/docs/cmdstan-guide/) + +* [*CmdstanPy Users's Guide*](https://mc-stan.org/cmdstanpy/) + +* [*CmdStanR Users' Guide*](https://mc-stan.org/cmdstanr/) + +* [*Stan.jl User's Guide*]( + +* [*PyStan User's Guide*](https://pystan.readthedocs.io/en/latest/) + +* [*RStan User's Guide*](https://mc-stan.org/rstan/) + +* [*Julia Interface User's Guide*](http://stanjulia.github.io/Stan.jl/stable/INTRO/) + + +## Stan programming language + +Stan provides a domain-specific language for specifying smooth, +data-dependent probability functions on the logarithmic scale. For +Bayesian models, this will be the posterior log density function (up +to an additive constant); in a frequentist setting, it will be a +penalized (marginal) likelihood function. + +Stan is a differentiable programming language, meaning that Stan can +automatically take first, second, and even higher-order derivatives of +any log density it defines. Stan is also a probabilistic programming +language in that its variables represent random variables and +constants. + +## Stan inference + +Stan supports inference for probability models through general, +model-agnostic sampling techniques including Markov chain Monte Carlo +(MCMC), variational inference (VI), and Laplace approximation. Stan +can also be used to calculate a (penalized) maximum likelihood +estimate (MLE) using optimization and estimate confidence intervals +through the bootstrap. + + +## Stan math library + +A program written in the Stan language is translated (aka transpiled) +to a C++ class defining a statistical model. This C++ class depends +on the Stan Math Library, which implements differentiable arithmetic +operations, matrix and linear algebra operations, special mathematical +functions, and special statistical functions. The Stan math library +itself has limited dependencies: Intel Thread Building Blocks for +synchronizing parallelism, Boost for general C++ tooling and special +functions, Eigen for matrix operations and linear algebra solvers, and +Sundials for differential equation solvers. + + +## Stan interfaces + +### BridgeStan language interface + +BridgeStan provides a complete interface to a Stan program through +Python, R, Julia, and Rust. BridgeStan supports evaluation of the log +density and its first-order derivatives (gradients) and second-order +derivatives (Hessian). It also provides I/O interface code for +constraining and unconstraining transforms and the simulation required +for posterior predictive inference. + +BridgeStan is implemented through low-level foreign function +interfaces that link to the compiled C++ class for a Stan program. + +### CmdStan reference interface + +Several Stan interfaces provide posterior (predictive) inference +through Monte Carlo methods, variational methods, and Laplace +approximation. The reference implementation is CmdStan, which provides +a command-line interface (CLI) for Stan. + +### CmdStan wrappers + +Several of Stan's interfaces wrap CmdStan and call it as an external +process, including CmdStanPy (Python), CmdStanR (R), Stan.jl (Julia), +and StataStan (Stata). These interfaces are straightforward to update +with new versions of Stan as only CmdStan is required. + +### In-memory interfaces + +RStan (R) and PyStan (Python), were the first two interfaces developed +after CmdStan, and both provide in-memory and in-process calls to the +underlying C++. Unlike BridgeStan, these in-memory interfaces are +coded using the language's high-level C++ interfaces (Rcpp and Cython, +respectively). Like CmdStan and its wrappers, RStan and PyStan +require a C++ toolchain at runtime to compile models. + +One advantage of RStan is that it encapsulates much of the +functionality of BridgeStan, allows the execution of Stan functions in +R through Rcpp, and supports the development of packages that can be +distributed with precomputed models and thus not depend on a C++ +toolchain at runtime. + From f7a13688775965beef53c8ab01a5349ce32150f9 Mon Sep 17 00:00:00 2001 From: Bob Carpenter Date: Mon, 29 Jun 2026 00:25:52 -0400 Subject: [PATCH 2/7] intro chapters --- src/.gitignore | 2 + src/_quarto.yml | 4 + src/stan-users-guide/_quarto.yml | 4 +- src/stan-users-guide/how-stan-works.qmd | 386 ++++++++++++++++++++++++ src/stan-users-guide/index.qmd | 12 +- src/stan-users-guide/intro.qmd | 296 +++++++++++++----- 6 files changed, 617 insertions(+), 87 deletions(-) create mode 100644 src/stan-users-guide/how-stan-works.qmd diff --git a/src/.gitignore b/src/.gitignore index 8ccecbdb7..c9a4fdaeb 100644 --- a/src/.gitignore +++ b/src/.gitignore @@ -7,3 +7,5 @@ *.toc /.quarto/ + +**/*.quarto_ipynb diff --git a/src/_quarto.yml b/src/_quarto.yml index 0a641af0b..6bd988551 100644 --- a/src/_quarto.yml +++ b/src/_quarto.yml @@ -113,6 +113,10 @@ website: contents: - stan-users-guide/index.qmd - section: "Version {{< env STAN_DOCS_VERSION >}}" + - section: "Introduction" + contents: + - stan-users-guide/intro.qmd + - stan-users-guide/how-stan-works.qmd - section: "Example Models" contents: - stan-users-guide/regression.qmd diff --git a/src/stan-users-guide/_quarto.yml b/src/stan-users-guide/_quarto.yml index a4d8a1630..ebc0a4dcd 100644 --- a/src/stan-users-guide/_quarto.yml +++ b/src/stan-users-guide/_quarto.yml @@ -29,8 +29,7 @@ format: book: title: "Stan User's Guide" - author: "Stan Developme -nt Team" + author: "Stan Development Team" search: true chapters: @@ -38,6 +37,7 @@ nt Team" - part: "Introduction" chapters: - intro.qmd + - how-stan-works.qmd - part: "Example Models" chapters: - regression.qmd diff --git a/src/stan-users-guide/how-stan-works.qmd b/src/stan-users-guide/how-stan-works.qmd new file mode 100644 index 000000000..7fbb2b1a1 --- /dev/null +++ b/src/stan-users-guide/how-stan-works.qmd @@ -0,0 +1,386 @@ +--- +pagetitle: How Stan Works +--- + +Stan is a probabilistic programming language whose variables represent random variables +and constants. + +# How Stan Works + +## A quick Description of Stan + +Stan programs have a similar syntax to C and C++ programs. Stan is +strongly statically typed, which means that every variable's type is +declared (e.g., `int` for integer, `real` for real, or `matrix` for +matrices) and the type of a variable never changes. The three +primitive types are `int`, `real`, and `complex`, but integer values +cannot be parameters and `complex` is treated as a pair of `real` +values. Parameters may be declared with constrained types (e.g., +`lower=0, upper=1` for probabilities, `cov_matrix` for symmetric +positive-definite matrices, and `simplex` for simplexes). These +constraints are maintained implicitly by Stan through unconstraining +and constraining transforms, which adjust for non-linear changes of +variables without user intervention. + +A Stan program is organized into a sequence of blocks, which are +optional, but must appear in the following order. + +1. `functions`: Define functions that can be called in the Stan +program. Functions are executed when called. + +2. `data`: Declare variables that are read in from lists in R, +dictionaries in Python, JSON files externally, etc., depending on the +interface. Executed once when data is ingested. + +3. `transformed data`: Declare variables defined as functions of the +data variables, or generated randomly. Executed once when data is +ingested. + +4. `parameters`: Declare parameters whose unknown values are +estimated through sampling, variational inference, or optimization. +This implicitly adds terms to the log density through +change-of-variables adjustment for any constrained parameters. +Executed every log density and gradient evaluation. + +5. `transformed parameters`: Declare variables defined as functions of +parameters, and add to the log Jacobian adjustment for changes of variables. +Executed every log density and gradient evaluation, with top-level variables +saved for output. + +6. `model`: Define local variables and the kernel of the log density +defined by the Stan program. Executed every log density and gradient +evaluation. + +7. `generated quantities`: Define variabels of interested that are not +needed for the log density, such as posterior predictive quantities; +may use randomization. Executed once per algorithm iteration using, +which may be far fewer times than the log density and gradient is +evaluated. Evaluated with primitive values and no automatic +differentiation, which is *much* faster (3--4 times or more) and lower +memory overhead. + +When compiled a Stan program provides a means to read in data and +transform it. It then provides methods for evaluating the log density +and gradient (and optionally Hessian) at any input value of the +parameters on the unconstrained scale. It can also transform and +inverse transform parameters. Finally, it can take a parameter value +and random number generator, and generate the variables defined in +the `generated quantities` block. + + +## An example Stan program + +To begin with the simplest possible Stan program for fitting data, consider +a program with the three core blocks, `data`, `parameters`, and `model`. + +```stan +data { + int N; // number of observations + array[N] int y; // observations +} +parameters { + real theta; // probability of success +} +model { + theta ~ beta(2.3, 8.1); // prior + y ~ bernoulli(theta); // data generating distribution +} +``` + +As can be seen in the example, a Stan program is organized into blocks. +The `data` block specifies the data that must be provided to fit the +model. Here, there are two data variables, `N` and `y`. The variable +`N` is declared to be an integer with an inclusive lower bound of 0. +The variable `y` is declared to be an array of `N` integers with +a lower bound of 0 and upper bound of `1`, that is, a binary variable. + +The `parameters` block declares the parameters for the model, whose +values are unknown. Here there is a single parameter `theta`, which +is declared to be a real number between 0 and 1. Given data, a Stan +program will infer values for the parameters (and other values). + +The `model` block defines the model's posterior log density up to an +additive constant that doesn't depend on the parameters. +Alternatively, the model block can be thought of as defining the joint +log density, as the two are equivalent up to a constant by Bayes's +rule, +$$ +\log p(y, \theta) = \log p(\theta \mid y) + \textrm{const}. +$$ + +The distribution statements with `~` are syntactic sugar for incrementing +the target log density, which is initialized implicitly at 0. For +example, the model block of the previous program could be equivalently +written as follows. + +```stan +model { + target += beta_lupdf(theta | 2.3, 8.1); + target += bernoulli_lupmf(y | theta); +} +``` + +Note the use of a vertical bar (`|`) rather than comma to separate the +variate from the distribution parameters. The suffix `_lupdf` is an +acronym for "log unnormalized probability density function," and +`_lupmf` for "log unnormalized probability mass function." + + +## Models define log density functions + +The Stan model defines a log density function, which is the sum of the +contributions of its sampling and target increment statements. Here, +$$ +\log p(\theta \mid y, N) += \log \textrm{beta}(\theta \mid 8, 2) + + \sum_{n=1}^N \log \textrm{bernoulli}(y \mid \theta) + + \textrm{const}. +$$ +The statement in the Stan program is vectorized, which means it +automatically broadcasts all of its arguments to repeat as many times +as necessary to match the largest argument. All arguments must be +either scalars or arrays/vectors of the same size, and the scalars +will be reused as many times as necessary (often with more efficient +evaluation, such as only evaluating $\log \sigma$ once for multiple +observations with the same error scale in a normal distribution). + +## Models define unconstrained log density function + +Any parameters declared with constraints, such as `theta` in the +example, are transformed to unconstrained behind the scenes by Stan. +For example, declaring `lower=0, upper=1` specifies a log odds +transform, which is written as $\textrm{logit}$, +$$ +\theta^\textrm{unc} += \textrm{logit}(\theta) += \log \frac{\theta}{1 - \theta}. +$$ + +Stan will account for the change-of-variables adjustment that is +required by the non-linear transform. The *Stan Reference Manual* +specifies all of the transforms and lays out change-of-variable +mathematics and goes through derivations for all of Stan's built-in +transforms. When the chang-of-variables dust settles on the log +scale, Stan defines a density with support (finite value) for all +$\theta^\textrm{unc} \in (-\infty, \infty)$. It does this by +inverting the transform and applying the change of variables formula +on the log scale, which yields +$$ +\log p(\theta^\textrm{unc} \mid y, N) += +\log p(\textrm{logit}^{-1}(\theta^\textrm{unc}) \mid y, N) ++ \log \textrm{logit}^{-1}(\theta^\textrm{unc}) ++ \log (1 - \textrm{logit}^{-1}(\theta^\textrm{unc})) ++ \textrm{const}, +$$ +where +$$\ +textrm{logit}^{-1}(\theta^\textrm{unc}) += \frac{\exp(\theta^\textrm{unc})} + {1 + \exp(\theta^\textrm{unc}}. +$$ + +This unconstrained log density is what Stan is actually sampling. +Values are then automatically transformed back to the constrained +scale using the inverse transforms, +$$ +\theta += \textrm{logit}^{-1}(\theta^\textrm{unc}). +$ + +If there is more than one constrained variable, each one's log +change-of-variables is included and all of them are transformed +back. + + +## Transformed parameters and Jacobians + +The unconstrained parameter model that Stan defines explicitly +can also be defined directly in Stan. In the following program, +the model is reparameterized in terms of `logit_theta`, the +log odds of success, which is unconstrained. + +```stan +data { + int N; + array[N] int y; +} +parameters { + real logit_theta; // log odds of success +} +transformed parameters { + real theta = inv_logit(logit_theta); + jacobian += log_inv_logit(logit_theta) + log1m_inv_logit(logit_theta); +} +model { + theta ~ beta(2.3, 8.1); + y ~ bernoulli(theta); +} +``` + +A new block, `transformed parameters`, is used to define the +probability of success `theta` as the inverse logit of `logit_theta`, +which maps it back to satisfy the `lower=0, upper=1` constraints. +Constraints in the transformed parameters` block are evaluated at the +end of the block and and if they fail, the current algorithm iteration +will be rejected. Because this is a non-linear transform and we wish +to put a prior directly on `theta`, we need to apply a log-scale +change-of-variables correction, which is done by incrementing the +variabler `jacobian`, which acts like `target`, but accumulates the +log Jacobian of the transform. + +In most circumstances, the value of `jacobian` will simply be added to +the `target`. But it can be dropped with settings in the optimizer +and Laplace approximation code so that the optimization result is a +maximum likelihood estimate (or a penalized maximum likelihood +estimate). Maximum a posteriori estimates, i.e., posterior modes, are +computed with the Jacobian included. User-defined Jacobians and +implicit Jacobians from constrained parameters have the same effect on +the final log density and gradient value returned by the Stan program. + +Although this shows the base way of writing this code, Stan's built-in +variable transforms are also available as special functions in the +Stan language, so that the transformed parameter block could be +written as follows. + +```stan +transformed parameters { + real theta + = lower_upper_bound_jacobian(logit_theta, 0, 1); +} +``` + +A suffix `_jacobian` on a function indicates that it has access to the +Jacobian to increment, so such functions are restricted to the +`transformed parameter` block where the `jacobian +=` statement is +available. This particular function applies the inverse logit +transform, then increments the Jacobian with the log +change-of-variables adjustment. + + +## Generated quantities for posterior predictions + +Suppose we want to observe $N$ trials of a boolean process like the positive +or negative outcome of a clinical trial or positive or negative rating of +a business, and we want to predict what the next $\tilde{N}$ trials might +look like. We can do that with our simple example program by +declaring a data variable for $\tilde{N}$ and then defining our predictive +quantity in the `generated quantities` block. + +```stan +data { + ... + int N_tilde; +} +... +generated quantities { + array[N_tilde] int y_tilde; + for (n in 1:N_tilde) { + y_tilde[n] = bernoulli_rng(theta); + } +} +``` + +Rather than sampling notation, this program explicitly calls a random +number generator for the Bernoulli distribution. Thus +rather than just sampling from the posterior $p(\theta \mid y)$, +we also sample from the posterior predictive, $p(\tilde{y} \mid y)$. +The Stan program simply samples from the joint posterior, +$$ +p(\tilde{y}, \theta \mid y) += p(\tilde{y} \mid \theta) \cdot p(\theta \mid y), +$$ +by first sampling $\theta$ from the posterior $p(\theta \mid y)$, then +sampling $\tilde{y}$ from the data generating distribution +$p(\tilde{y} \mid \theta)$ given the sample for $\theta$. This matches +the two sources of uncertainty in the posterior predictive, the +uncertainty in the parameter value $\theta$ given the "training" data +$y$, and the uncertainty in generating new "test" data $\tilde{y}$ +given an estimate of $\theta$. + + +## Fitting a Stan model with Markov chain Monte Carlo + +To fit a model, data must be provided. Stan accepts JavaScript Object +Notation (JSON) formatted data (or lists or dictionaries in the R and +Python interfaces), so suppose we have data matching matching the +declarations in our Stan program, such as the following. + + +```json +{ + 'N': 10; + 'y': [1, 0, 0, 1, 0, 1, 0, 0, 0, 0] +} +``` + +The Stan program must be converted to C++ and then the C++ must be +compiled. Then, it must be given data matching the `data` block +declaration in shape and satisfying all of the declared constraints. + +When a Stan program is executed, the first thing that happens is that +it will read the data file in. All of the constraints declared +are checked and any violations will cause the program to terminate +with an informative error message. + +Once the data is loaded, a Stan program is able to evaluate log +densities and gradients. One way to think about the Stan program is +that the data block defines the signature for the function that reads +in data. Then once the data block is read in, the program defines +a function whose arguments are given by the parameters block and +whose value is a log density and gradient evaluated at the argument. + +That is, we can provide a value such as $\theta^\textrm{unc} = -1.3786352$ and the +Stan program will return the posterior log density up to a normalizing +constant, $\log p(\theta \mid y, N) + +\textrm{const}$, and its gradient, +$$ +\nabla \left( \log p(\theta^\textrm{unc} \mid y, N) + \textrm{const} \right) += \frac{\partial}{\partial \theta^\textrm{unc}} \log p(\theta^\textrm{unc} \mid y, N), +$$ +where the constant drops out because it does not involve $\theta$. + +When fitting a model, the data is only read in once, whereas the log +density is evaluated multiple times, some times as many as 1024 times +per iteration with the no-U-turn sampler. + +Stan's samplers will try to sample values of $\theta$ from the posterior +distribution. That is, it will try to generate several Markov chains +in parallel, each of which has the form +$$ +\theta^{(1)}, \ldots, \theta^{(M)}, +$$ +where if everything is functioning correctly, marginally, +$$ +\theta^{(m)} \sim p(\theta \mid y, N). +$$ +We say ``marginally'' because the draws $\theta^{(m)}$ in the sample +Markov chain may be autocorrelated. + +Sampling will automatically generate all the generated quantities. + +## Summarizing a model fit + +When fitting a model using sampling, either Stan or an external +package (e.g., `ArviZ` in Python) can be used to return a +summary. Such summaries typically include the posterior mean, which +acts as a Bayesian parameter estimate, and posterior standard +deviation. They also typically include posterior quantiles such as +the posterior median (50\% quantile), and by default in Stan, the 5\% +and 95\% quantiles (with others being available). Stan also reports +diagnostics on sampling, such as $\widehat{R}$, which measures +convergence, and effective sample size, which estimates the strength +of Stan's sample in units of independent draws. Finally, it reports +standard error on the mean estimates, which works out to be standard +deviation divided by the square root of the effective sample size. + +Stan's interfaces also allow the posterior draws to be extracted +directly so that they may be used for plotting or to plumb Bayesian +uncertainty through externally generated predictive quantities the same +way Stan does this internally with generated quantities. + +Even Stan's variational inference and Laplace approximations allow +sampling, which is necessary to get the inferences back on the +appropriate scale after fitting on the unconstrained scale (because +averages of functions are not equivalent to functions of averages when +the function is non-linear, as most of the transforms are). diff --git a/src/stan-users-guide/index.qmd b/src/stan-users-guide/index.qmd index decc0dd81..35366e660 100644 --- a/src/stan-users-guide/index.qmd +++ b/src/stan-users-guide/index.qmd @@ -16,7 +16,8 @@ format: This is the official user's guide for [Stan](https://mc-stan.org/). It provides example -models and programming techniques for coding statistical models in Stan. +models and programming techniques for coding statistical models in Stan. It does *not* +provide a guide to installing or using any of the interfaces required to use Stan. - Part 1 gives Stan code and discussions for several important classes of models. @@ -27,17 +28,10 @@ not tied to any particular model. - Part 3 introduces algorithms for calibration and model checking that require multiple runs of Stan. -- The appendices provide an introduction to the stanc3 compiler used in the +- The appendices provide an introduction to the `stanc3` compiler used in the various interfaces to Stan, a style guide, and advice for users of BUGS and JAGS. -We recommend working through this guide using the textbooks _Bayesian -Data Analysis_ and _Statistical Rethinking: A Bayesian Course with -Examples in R and Stan_ as references on the concepts, and using the -[*Stan Reference Manual*](https://mc-stan.org/docs/reference-manual/index.html) -when necessary to clarify programming issues. - - ::: {.content-visible when-format="html"} [Download the pdf version of this manual](https://mc-stan.org/docs/{{< env STAN_DOCS_VERSION_PATH >}}/stan-users-guide-{{< env STAN_DOCS_VERSION_PATH >}}.pdf). ::: diff --git a/src/stan-users-guide/intro.qmd b/src/stan-users-guide/intro.qmd index 60993a6e0..baea7e427 100644 --- a/src/stan-users-guide/intro.qmd +++ b/src/stan-users-guide/intro.qmd @@ -4,140 +4,284 @@ pagetitle: The Stan Ecosystem # The Stan Ecosystem {#ecosystem.chapter} -Stan is a domain-specific tool for specifying probability models, -performing statistical inference and analyzing model fits. Stan is -largely used for Bayesian models with Monte Carlo-based inference, but -it may also be used to fit penalized maximum likelihood estimates and -estimate confidence intervals with the bootstrap with optimization. +Stan is a domain-specific programming language for specifying +probabilistic models along with a collection of algorithms for +performing statistical inference and algorithms and visualizations for +analyzing model fits. +## What is the *Stan User's Guide*? -## Stan community +This document, the *Stan User's Guide*, is intended to explain +techniques for statistical modeling using the Stan modeling language. +As such, it concentrates on the Stan language itself and does *not* +discuss any of the Stan interfaces that are required to actually +use Stan (see below). -Stan has an active community of users and developers. +## Who uses Stan? + +You can find Stan users and example code online from almost every +subfield of science and most branches of statistics other than highly +non-parametric modeling (e.g., neural networks, high-dimensional +Dirichlet processes) and highly coupled discrete models (e.g., Ising +models). Stan applications have been drawn from pretty much every +branch of biological science (from cellular to population ecology and +zoology), physical sciences (from quantum to astro), social sciences +(psychometrics, surveys, economics, and anthropology), engineering +(from civil to mechanical), educational sciences (testing), medical +sciences (from epidemiology to clinical trials), sports statistics +(pretty much every major sport, plus bookmaking), finance (asset +pricing, risk assessment, forecasting), business (advertising +attribution, demand and pricing), and actuarial sciences. + + +## Stan user and developer communities + +Stan's strongest feature is its active community of users. Stan's +users hail from a wide range of mathematical and statistical +backgrounds and work in a broad range of application areas. + +The primary Stan discussion board is hosted by Discourse. * [Stan forums](https://discourse.mc-stan.org), where users and developers discuss issues with Stan and its applications. +The Stan developers discuss designs, plans, and communicate with each +other through the forums and also on GitHub. + * [Stan GitHub](https://github.com/stan-dev), which tracks issues and - contains the complete source code for Stan (under the BSD-3 - license). + contains the complete source code for Stan (the core is licensed + under BSD-3). + +Both developers and users attend Stan conferences, which are a mix +of tutorials, talks about applications, and talks from the Stan +developers about where Stan is going. + +* [StanCon](https://mc-stan.org/learn-stan/stancon-talks.html), the + pages for which host video talks, slides, and executable notebooks. + + +## How is the Stan Project organized? + +Stan is free open source software released under permissive licenses +(BSD-3 for the core code). There are no institutional owners and no +institutional oversight of the project. -* [StanCon](https://mc-stan.org/learn-stan/stancon-talks.html), for - which talks and executable notebooks are available. +The Stan project as a whole is managed through the Stan Governing Body +(SGB), who have final say on all things Stan related. They also +organize StanCon. The members are elected by the developers and +rotate through short (1--2 year) terms. + +Software development is managed through Apache-style voting among the +developers. Lack of consensus is so rare there have only been a few +votes in the history of the project aside from electing SGB members. + +Since development of Stan began in 2011 (the first release was in +2012), funding has been provided through governmental grants across +multiple agencies in several countries, through non-profit +foundations, through donations from corporations, and through in-kind +donations of time by our developer's employers. Thank you! ## Stan documentation -Stan's documentation is organized through a top-level web page: +### Core Stan documentation + +Stan's internal documentation is organized through a top-level web +page. * [Stan documentation](https://mc-stan.org/docs/) -This is the *User's Guide*, which is complemented by the *Reference -Manual* and *Functions Reference*. These are all interface-agnostic -guides to the Stan language and its execution. They are available at +You are currently reading the *User's Guide*, which is complemented by +the *Reference Manual* and *Functions Reference*. These are all +interface-agnostic guides to the Stan language and its execution. +They are available online with integrated navigation and also as pdfs. * [*Stan Users' Guide*](https://mc-stan.org/docs/stan-users-guide/) -documentation. * [*Stan Reference Manual*](https://mc-stan.org/docs/reference-manual/) * [*Stan Functions Reference*](https://mc-stan.org/docs/functions-reference/) -Documentation for the specific Stan interfaces are here +### Stan case studies + +Stan case studies cover specific models or techniques with reproducible +code (usually in Python or R). They can be found in the devoted page of +case studies or through StanCon presentations. + +* [*Stan Case Studies*](https://mc-stan.org/learn-stan/case-studies.html) + +* [*StanCon Case Studies*](https://mc-stan.org/learn-stan/stancon-talks.html) -* [*CmdStan User's Guide*](https://mc-stan.org/docs/cmdstan-guide/) +### Stan interface documentation -* [*CmdstanPy Users's Guide*](https://mc-stan.org/cmdstanpy/) +Documentation for the officially supported Stan interfaces for the language and +inference are here. CmdStan is the reference implementation. -* [*CmdStanR Users' Guide*](https://mc-stan.org/cmdstanr/) +* Command line: [*CmdStan User's Guide*](https://mc-stan.org/docs/cmdstan-guide/) -* [*Stan.jl User's Guide*]( +CmdStanPy, CmdStanR, and Stan.jl make system calls to CmdStan out of +process. -* [*PyStan User's Guide*](https://pystan.readthedocs.io/en/latest/) +* Python: [*CmdstanPy Users's Guide*](https://mc-stan.org/cmdstanpy/) -* [*RStan User's Guide*](https://mc-stan.org/rstan/) +* R: [*CmdStanR Users' Guide*](https://mc-stan.org/cmdstanr/) -* [*Julia Interface User's Guide*](http://stanjulia.github.io/Stan.jl/stable/INTRO/) +* Julia: [*Stan.jl Interface User's Guide*](http://stanjulia.github.io/Stan.jl/stable/INTRO/) +PyStan and RStan call Stan code directly through C++. -## Stan programming language +* Python: [*PyStan User's Guide*](https://pystan.readthedocs.io/en/latest/) + +* R: [*RStan User's Guide*](https://mc-stan.org/rstan/) + +There is also an interface for accessing Stan language components, +including (unconstrained) log densities, gradients, and Hessians, as +well as parameter transformations and the posterior predictive +simulations required for generated quantities. + +* Python, R, C, Rust: [*BridgeStan User's Guide*](https://roualdes.us/bridgestan/latest/) + +BridgeStan is implemented through low-level foreign function +interfaces that link to the compiled C++ class for a Stan program. + +In addition to the documentation maintained by the development team, there +are many resources available online from the community including books, +video courses, talks, papers, and case studies. + + +## What is Stan? + +Stan is made up of a programming language for statistical models, +several integrated statistical inference algorithms, and interfaces in +various higher-level analysis languages (R, Python, Julia, Stata, +MATLAB). The Stan project also maintains higher-level interfaces to +Stan such as the `rstanarm` and `brms`, both of which are based on R's +expression language for regressions. The project also develops and +maintains visualization and model comparison tools like `posterior` +and `loo` (in R; for Python, similar functionality is available through +`ArviZ`). + +Many other individuals and companies have built tools that depend on +the core Stan language or packages, many of which are available open +source. These handle everything from compartment +pharmacometric/pharmacodynamic models (`torsten`) to structural +equation models (`blavaan`) to general additive models for time series +(`prophet`). + +### Stan programming language Stan provides a domain-specific language for specifying smooth, -data-dependent probability functions on the logarithmic scale. For -Bayesian models, this will be the posterior log density function (up -to an additive constant); in a frequentist setting, it will be a -penalized (marginal) likelihood function. +data-dependent target density functions on the logarithmic scale. For +Bayesian applications, this will be the posterior log density function +(up to an additive constant). For frequentist applications, it will +be a penalized (marginal) likelihood function. Stan also provides a +method for coding posterior predictive quantities and generating them +through simulation. Stan is a differentiable programming language, meaning that Stan can automatically take first, second, and even higher-order derivatives of any log density it defines. Stan is also a probabilistic programming language in that its variables represent random variables and -constants. +constants (at least when viewed from the Bayesian perspective). + +Stan provide syntax highlighting through an `atom` specification. +Downstream tools using syntax highlighting for Stan include GitHub, +RStudio, VS Code, Jupyter notebooks, Emacs, Neovim, etc. -## Stan inference +### Stan inference engines Stan supports inference for probability models through general, -model-agnostic sampling techniques including Markov chain Monte Carlo -(MCMC), variational inference (VI), and Laplace approximation. Stan -can also be used to calculate a (penalized) maximum likelihood -estimate (MLE) using optimization and estimate confidence intervals -through the bootstrap. +model-agnostic sampling techniques including the following two Markov +chain Monte Carlo (MCMC) methods. + +* Hamiltonian Monte Carlo (HMC) +* No-U-Turn Sampler (NUTS) + +Stan also includes the following two approximate inference +techniques based on variational inference, + +* Automatic Differentiation Variational Inference (ADVI) +* Pathfinder variational inference +as well the following approximate method based on optimization and curvature. -## Stan math library +* Laplace approximation -A program written in the Stan language is translated (aka transpiled) -to a C++ class defining a statistical model. This C++ class depends -on the Stan Math Library, which implements differentiable arithmetic +Finally, Stan supplies industry-standard optimization methods, which +can be used for maximum likelihood estimates and as the basis for +Laplace approximations. + +* Newton, BFGS, and L-BFGS optimization + +Stan can estimate confidence intervals through the bootstrap as +explained in the posterior inference and model checking part of this +document. + +### Stanc3: Stan-to-C++ transpilation + +Stan uses a standard programming language stack of parser, +intermediate representation layer, and code generator. These are all +written in the OCaml programming language. The code generator +produces a C++ class implementing the model defined by the Stan +program. It is this latter feature, targeting an intermediate +language rather than assembly/machine language, which makes it a +transpiler rather than a compiler. + +### Stan math library + +The C++ class produced by the `stanc3` transpiler depends on the Stan +Math Library. The math library implements differentiable arithmetic operations, matrix and linear algebra operations, special mathematical functions, and special statistical functions. The Stan math library -itself has limited dependencies: Intel Thread Building Blocks for -synchronizing parallelism, Boost for general C++ tooling and special -functions, Eigen for matrix operations and linear algebra solvers, and -Sundials for differential equation solvers. +itself has four dependencies in addition to C++ itself, +* `tbb`: Intel Thread Building Blocks for thread management including +pools and synchronization, -## Stan interfaces +* `boost`: Boost for general C++ tooling, special functions, random +number generators, and densities. -### BridgeStan language interface +* `Eigen`: Eigen for templated general matrix operations and linear + algebra solvers, and -BridgeStan provides a complete interface to a Stan program through -Python, R, Julia, and Rust. BridgeStan supports evaluation of the log -density and its first-order derivatives (gradients) and second-order -derivatives (Hessian). It also provides I/O interface code for -constraining and unconstraining transforms and the simulation required -for posterior predictive inference. +* `sundials`: Sundials for differential equation solvers. -BridgeStan is implemented through low-level foreign function -interfaces that link to the compiled C++ class for a Stan program. +The math library can call out to GPU for some operations (e.g., +Cholesky factorization or matrix-matrix multiplication), but overall, +it has been much more heavily optimzed for CPUs. + + +## Background reading for getting started -### CmdStan reference interface +This section links some resources for getting started with Bayesian +statistics using Stan. The first two recommendations are for +introductions to Bayesian statistics using Stan, a complete hands-on +introductory textbook and shorter introductory article. -Several Stan interfaces provide posterior (predictive) inference -through Monte Carlo methods, variational methods, and Laplace -approximation. The reference implementation is CmdStan, which provides -a command-line interface (CLI) for Stan. +* [_Statistical Rethinking: A Bayesian Course with Examples in R and +Stan_, Second Edition](https://xcelab.net/rm/). 2020. Richard +McElreath. CRC Press. -### CmdStan wrappers +* [Getting started with Bayesian statistics using Stan and + Python](https://bob-carpenter.github.io/stan-getting-started/stan-getting-started.html). 2023. + Bob Carpenter. -Several of Stan's interfaces wrap CmdStan and call it as an external -process, including CmdStanPy (Python), CmdStanR (R), Stan.jl (Julia), -and StataStan (Stata). These interfaces are straightforward to update -with new versions of Stan as only CmdStan is required. +The next step to understanding Bayesian statistics more deeply after +these introductory texts is *BDA3*, a free pdf for which is available +from the linked web site. -### In-memory interfaces +* [_Bayesian Data Analysis, Third +Edition_](https://sites.stat.columbia.edu/gelman/book/). 2013. Andrew +Gelman, John Carlin, Hal Stern, David Dunson, Aki Vehtari, and Donald +Rubin. CRC Press. -RStan (R) and PyStan (Python), were the first two interfaces developed -after CmdStan, and both provide in-memory and in-process calls to the -underlying C++. Unlike BridgeStan, these in-memory interfaces are -coded using the language's high-level C++ interfaces (Rcpp and Cython, -respectively). Like CmdStan and its wrappers, RStan and PyStan -require a C++ toolchain at runtime to compile models. +There has been more interest in statistics lately to consider the +whole process of using statistics from exploratory data analysis to +results presentation. The following short article and longer book +provide advice on statistical workflow from a Bayesian perspective +with worked examples in Stan. -One advantage of RStan is that it encapsulates much of the -functionality of BridgeStan, allows the execution of Stan functions in -R through Rcpp, and supports the development of packages that can be -distributed with precomputed models and thus not depend on a C++ -toolchain at runtime. +* [Statistical workflow](https://sites.stat.columbia.edu/gelman/research/published/Statistical_Workflow_article.pdf). 2026. Andrew Gelman, Aki Vehtari, Richard McElreath. _Philosophical Transactions of the Royal Society A_. +* [_Bayesian Workflow_](https://avehtari.github.io/Bayesian-Workflow/). 2026. Andrew Gelman, Aki Vehtari, Richard McElreath, Daniel Simpson, Charles C. Margossian, Yuling Yao, Lauren Kennedy, Jonah Gabry, Paul-Christian Bürkner, Martin Modrák, Vianey Leos Barajas. Cambridge University Press. From 9775319b1b047b6487a8db8f36e59a6dbc84ad8b Mon Sep 17 00:00:00 2001 From: Bob Carpenter Date: Wed, 1 Jul 2026 02:52:17 -0400 Subject: [PATCH 3/7] revised how-stan-works chapter --- src/stan-users-guide/how-stan-works.qmd | 497 ++++++++++++------------ 1 file changed, 252 insertions(+), 245 deletions(-) diff --git a/src/stan-users-guide/how-stan-works.qmd b/src/stan-users-guide/how-stan-works.qmd index 7fbb2b1a1..0d9bbffee 100644 --- a/src/stan-users-guide/how-stan-works.qmd +++ b/src/stan-users-guide/how-stan-works.qmd @@ -2,25 +2,33 @@ pagetitle: How Stan Works --- -Stan is a probabilistic programming language whose variables represent random variables -and constants. # How Stan Works -## A quick Description of Stan - -Stan programs have a similar syntax to C and C++ programs. Stan is -strongly statically typed, which means that every variable's type is -declared (e.g., `int` for integer, `real` for real, or `matrix` for -matrices) and the type of a variable never changes. The three -primitive types are `int`, `real`, and `complex`, but integer values -cannot be parameters and `complex` is treated as a pair of `real` -values. Parameters may be declared with constrained types (e.g., -`lower=0, upper=1` for probabilities, `cov_matrix` for symmetric -positive-definite matrices, and `simplex` for simplexes). These -constraints are maintained implicitly by Stan through unconstraining -and constraining transforms, which adjust for non-linear changes of -variables without user intervention. +Stan is a differentiable imperative programming language with some +functional constructs. It is a probabilistic programming language in +the sense that its parameters and modeled data are treated as random +variables in Bayesian statistics. + +The primary goal of a Stan program is to define three things, (1) a +differentiable target log density function up to a constant, (2) a way +to generate predictions based on parameter values, and (3) +unconstraining transforms and their inverses. + +Stan's syntax is similar to C's. In Stan, every variable, constant, +and larger expression has a type that does not change. The three +primitive types are `int`, `real`, and `complex`. Integer types may +not be used as parameters. The primitive container types are +`vector`, `row_vector`, and `matrix`. Array types are available to +hold any single type of value. Tuples hold a sequence of values +of fixed types. + +Variables can be declared with constraints such as lower and/or upper +bounds for scalars, and simplex, positive definiteness, and many other +constraints for vectors and matrices. Constraints on parameters are +maintained implicitly by Stan through unconstraining and constraining +transforms. Constraints on other variables are validated at the end +of their respective blocks. A Stan program is organized into a sequence of blocks, which are optional, but must appear in the following order. @@ -33,45 +41,46 @@ dictionaries in Python, JSON files externally, etc., depending on the interface. Executed once when data is ingested. 3. `transformed data`: Declare variables defined as functions of the -data variables, or generated randomly. Executed once when data is -ingested. +data variables, or generated randomly. This is where constant +variables should be defined. Executed once when data is ingested. 4. `parameters`: Declare parameters whose unknown values are estimated through sampling, variational inference, or optimization. -This implicitly adds terms to the log density through -change-of-variables adjustment for any constrained parameters. -Executed every log density and gradient evaluation. - -5. `transformed parameters`: Declare variables defined as functions of -parameters, and add to the log Jacobian adjustment for changes of variables. -Executed every log density and gradient evaluation, with top-level variables -saved for output. - -6. `model`: Define local variables and the kernel of the log density -defined by the Stan program. Executed every log density and gradient -evaluation. - -7. `generated quantities`: Define variabels of interested that are not -needed for the log density, such as posterior predictive quantities; -may use randomization. Executed once per algorithm iteration using, -which may be far fewer times than the log density and gradient is -evaluated. Evaluated with primitive values and no automatic -differentiation, which is *much* faster (3--4 times or more) and lower -memory overhead. - -When compiled a Stan program provides a means to read in data and -transform it. It then provides methods for evaluating the log density -and gradient (and optionally Hessian) at any input value of the -parameters on the unconstrained scale. It can also transform and -inverse transform parameters. Finally, it can take a parameter value -and random number generator, and generate the variables defined in -the `generated quantities` block. - +When this block is executed, unconstrained parameters are inverse +transformed so that they satisfy their declared constraints. For each +non-linear transform, a change-of-variables adjustment is added to the +target log density (unless these adjustments are turned off for +maximum likelihood estimation). Executed every log density and +gradient evaluation. + +5. `transformed parameters`: Declare and define variables that depend +on variables from the previous blocks. Statements in this block may +increment the log Jacobian for changes of variables. Executed every +log density and gradient evaluation. + +6. `model`: Define the target log density using the variables from +previous blocks. Executed every log density and gradient evaluation. + +7. `generated quantities`: Define posterior predictive quantities that +depend on variables in previous blocks. This is where to define +quantities that are not needed for the log density evaluation. +Statements in this block may use random number generators. Executed +once per iteration using primitives. Execution with primitives means +no automatic differentiation and three or more times faster than +similar code in the model or transformed parameter blocks. + +Stan programs are transpiled to C++ by an OCaml program (`stanc3`). +The resulting C++ class definition is compiled and linked to the Stan +math library for special functions and automatic differentiation. At +this point, an object can be constructed from data. This object can +be used to evaluate log densities and derivatives as well as +unconstraining transforms and their inverses. ## An example Stan program -To begin with the simplest possible Stan program for fitting data, consider -a program with the three core blocks, `data`, `parameters`, and `model`. +Consider the following very simple Stan program, which can be used to +estimate a proportion $\theta \in (0, 1)$ from binary observations +$y_n \in \{ 0, 1 \}$. ```stan data { @@ -87,31 +96,41 @@ model { } ``` -As can be seen in the example, a Stan program is organized into blocks. -The `data` block specifies the data that must be provided to fit the -model. Here, there are two data variables, `N` and `y`. The variable -`N` is declared to be an integer with an inclusive lower bound of 0. -The variable `y` is declared to be an array of `N` integers with -a lower bound of 0 and upper bound of `1`, that is, a binary variable. - -The `parameters` block declares the parameters for the model, whose -values are unknown. Here there is a single parameter `theta`, which -is declared to be a real number between 0 and 1. Given data, a Stan -program will infer values for the parameters (and other values). - -The `model` block defines the model's posterior log density up to an -additive constant that doesn't depend on the parameters. -Alternatively, the model block can be thought of as defining the joint -log density, as the two are equivalent up to a constant by Bayes's -rule, +### Data + +The `data` block specifies that a non-negative integer `N` and an +array `y` of size `N`, which contains binary observations. If the +data supplied to the Stan program does not satisfy the defined +constraints (e.g., $y$ has values other than 0 or 1), the algorithm +terminates and the model will not be constructed. + +### Parameters + +There is a single scalar parameter `theta` representing the unknown +probability of success. It is declared to be a real number between 0 +and 1. + +### Target log density + +The `model` block defines the model's target log density $\log +p(\theta \mid y)$ up to an additive constant that doesn't depend on +the parameters. Alternatively, the model block can be thought of as +defining the joint log density or something in between, +because the two are equivalent up to a constant by Bayes's rule, $$ \log p(y, \theta) = \log p(\theta \mid y) + \textrm{const}. $$ -The distribution statements with `~` are syntactic sugar for incrementing -the target log density, which is initialized implicitly at 0. For -example, the model block of the previous program could be equivalently -written as follows. +In frequentist applications, the target is not treated as a density, +but a (possibly penalized) log likelihood function +$\mathcal{L}(\theta) = \log p(y \mid \theta)$ (with optional extra +penalty terms). + +### Distribution statements are syntactic sugar + +The distribution statements with `~` are syntactic sugar for +incrementing the target log density. The model block of the previous +program could be equivalently written as follows. ```stan model { @@ -125,31 +144,149 @@ variate from the distribution parameters. The suffix `_lupdf` is an acronym for "log unnormalized probability density function," and `_lupmf` for "log unnormalized probability mass function." +The target log density is initialized at 0 and implicitly incremented +for change-of-variables adjustments for constrained parameters and for +sampling statements. -## Models define log density functions +## Model blocks define log density functions -The Stan model defines a log density function, which is the sum of the -contributions of its sampling and target increment statements. Here, +In the running example, the Stan program defines the following +target log density function $\log p$ up to a constant, $$ \log p(\theta \mid y, N) = \log \textrm{beta}(\theta \mid 8, 2) + \sum_{n=1}^N \log \textrm{bernoulli}(y \mid \theta) + \textrm{const}. $$ -The statement in the Stan program is vectorized, which means it -automatically broadcasts all of its arguments to repeat as many times -as necessary to match the largest argument. All arguments must be -either scalars or arrays/vectors of the same size, and the scalars -will be reused as many times as necessary (often with more efficient -evaluation, such as only evaluating $\log \sigma$ once for multiple -observations with the same error scale in a normal distribution). +The distribution statement for `y` in the Stan program is vectorized. +The argument `y` is an array of integers, but `theta` is a scalar. +When Stan sees this argument pattern, it reuses scalar arguments like +`theta` for each entry of containers like `y`. This can lead to much +more efficient evaluation (e.g., $\log \theta$ is only computed once +in this example). + -## Models define unconstrained log density function + +## Generated quantities for posterior predictions + +Suppose we want to observe $N$ trials of a binary process like the +positive or negative outcome of a clinical trial or positive or +negative rating of a business and conditioned on those observations, +predict what the next $\tilde{N}$ trials might look like. This is +called posterior predictive inference, and in Stan, we can define the +quantities of interest in the `generated quantity` block. We extned +our example program by declaring a data variable for the number of new +observations $\tilde{N}$ and then defining our predictive quantity in +the `generated quantities` block. + +```stan +data { + ... + int N_tilde; +} +... +generated quantities { + array[N_tilde] int y_tilde; + for (n in 1:N_tilde) { + y_tilde[n] = bernoulli_rng(theta); + } +} +``` + +This program explicitly calls a random number generator for the +Bernoulli distribution which is based on a parameter. That means the +value of $\tilde{y}$ is sampled from the posterior predictive +distribution $p(\tilde{y} \mid y)$ by first sampling $\theta$ from the +posterior, then sampling $\tilde{y}$ given $\theta$. This accounts for +the estimation uncertainty in $\theta$ as well as the randomness of +the data generating process, which here takes Bernoulli draws conditional +on $\theta$. + +## Fitting a Stan model with Markov chain Monte Carlo + +To fit a model, data must be provided. Stan accepts file-based data +in JavaScript Object Notation (JSON), and can also directly accept +data formatted as lists in R or dictionaries in Python. For the +running example, consider the data defined by the following JSON. + + +```json +{ + 'N': 10; + 'y': [1, 0, 0, 1, 0, 1, 0, 0, 0, 0] +} +``` + +Once the data is loaded, a Stan program is able to evaluate log +densities and gradients. For example, we could provide a value such as +$\theta^\textrm{unc} = -1.3786352$ and the Stan program will return +the posterior log density up to a normalizing constant, $\log p(\theta +\mid y, N) + \textrm{const}$, and its gradient, $$ \nabla \left( \log +p(\theta^\textrm{unc} \mid y, N) + \textrm{const} \right) = +\frac{\partial}{\partial \theta^\textrm{unc}} \log +p(\theta^\textrm{unc} \mid y, N), $$ where the constant drops out +because it does not involve $\theta$. + +When fitting a model, the data is only read in once, whereas the log +density is evaluated multiple times, some times as many as 1024 times +per iteration with the no-U-turn sampler. + +Stan's samplers will try to sample values of $\theta$ from the +posterior distribution. That is, it will try to generate several +Markov chains in parallel, each of which has the form $$ \theta^{(1)}, +\ldots, \theta^{(M)}, $$ where if everything is functioning correctly, +marginally, $$ \theta^{(m)} \sim p(\theta \mid y, N). $$ While these +draws are marginally distributed approximately according to the target +distribituion, the Markov chain from which they arose may be +autocorrelated. In these cases, the number of draws needs to be +discounted to figure out the effective number of independent draws +(i.e., the effective sample size) that would have the same +informativeess. + +Stan's inference engines will also genreate all generated quantities +based on the data and the current draw of the parameters. + +## Summarizing a model fit + +When fitting a model using sampling, either Stan or an external +package (e.g., `ArviZ` in Python) can be used to return a +summary. Such summaries typically include the posterior mean, which +acts as a Bayesian parameter estimate, and posterior standard +deviation. They also typically include posterior quantiles such as +the posterior median (50\% quantile), and by default in Stan, the 5\% +and 95\% quantiles (with others being available). Stan also reports +diagnostics on sampling, such as $\widehat{R}$, which measures +convergence, and effective sample size, which estimates the +informativeness of the potentially autocorrelated Markov chain Monte +Carlo sample in units of independent draws. Finally, summaries report +standard error on the mean estimates, which are calculated as +standard deviation divided by the square root of the effective sample size. + +Stan's interfaces also allow the posterior draws to be extracted +directly so that they may be used for plotting or to plumb Bayesian +uncertainty through externally generated predictive quantities the same +way Stan does this internally with generated quantities. + +Even Stan's variational inference and Laplace approximations allow +sampling, which is necessary to get the inferences back on the +appropriate scale after fitting on the unconstrained scale. This +needs to be done before averaging for estimating expectations because +averages of functions are not equivalent to functions of averages when +the function is non-linear, as most of the transforms are. + + + +## Technical detail: Models define unconstrained log density functions + +This section describe how Stan works under the hood by transforming +constrained log density functions. Understranding the details of this +section is not necessary to write Stan code. Any parameters declared with constraints, such as `theta` in the example, are transformed to unconstrained behind the scenes by Stan. -For example, declaring `lower=0, upper=1` specifies a log odds -transform, which is written as $\textrm{logit}$, +For example, declaring `theta` with the type `real` +specifies a log odds transform on $\theta$, $\textrm{logit}:(0, 1) +\rightarrow (-\infty, \infty)$, defined by $$ \theta^\textrm{unc} = \textrm{logit}(\theta) @@ -158,13 +295,15 @@ $$ Stan will account for the change-of-variables adjustment that is required by the non-linear transform. The *Stan Reference Manual* -specifies all of the transforms and lays out change-of-variable -mathematics and goes through derivations for all of Stan's built-in -transforms. When the chang-of-variables dust settles on the log -scale, Stan defines a density with support (finite value) for all +specifies all of the constrained parameters and corresponding +transforms and change-of-variables adjustments. + +When the change-of-variables dust settles on the log scale, Stan +defines a density with support (finite value) for all $\theta^\textrm{unc} \in (-\infty, \infty)$. It does this by inverting the transform and applying the change of variables formula -on the log scale, which yields +on the log scale, which yields the unconstrained unnormalized log density +function $$ \log p(\theta^\textrm{unc} \mid y, N) = @@ -174,31 +313,28 @@ $$ + \textrm{const}, $$ where -$$\ -textrm{logit}^{-1}(\theta^\textrm{unc}) -= \frac{\exp(\theta^\textrm{unc})} - {1 + \exp(\theta^\textrm{unc}}. -$$ - -This unconstrained log density is what Stan is actually sampling. -Values are then automatically transformed back to the constrained -scale using the inverse transforms, $$ \theta -= \textrm{logit}^{-1}(\theta^\textrm{unc}). -$ - -If there is more than one constrained variable, each one's log -change-of-variables is included and all of them are transformed -back. +\ = \ +\textrm{logit}^{-1}(\theta^\textrm{unc}) +\ = \ +\frac{\exp(\theta^\textrm{unc})} + {1 + \exp(\theta^\textrm{unc})}. +$$ +This unconstrained log density is what Stan is actually sampling or +approximating with variational inference or Laplace approximations. +After inference, parameter values are automatically transformed back +to the constrained scale using the inverse transforms (here, +$\textrm{logit}^{-1}$). -## Transformed parameters and Jacobians +### Transformed parameters and Jacobians -The unconstrained parameter model that Stan defines explicitly -can also be defined directly in Stan. In the following program, -the model is reparameterized in terms of `logit_theta`, the -log odds of success, which is unconstrained. +Although this is rarely something a user will need to do, the +unconstrained parameter model that Stan defines explicitly can also be +defined directly in Stan. In the following program, the model is +reparameterized in terms of `logit_theta`, the log odds of success, +which is unconstrained. ```stan data { @@ -210,6 +346,7 @@ parameters { } transformed parameters { real theta = inv_logit(logit_theta); + // change-of-variables adjustment jacobian += log_inv_logit(logit_theta) + log1m_inv_logit(logit_theta); } model { @@ -217,7 +354,6 @@ model { y ~ bernoulli(theta); } ``` - A new block, `transformed parameters`, is used to define the probability of success `theta` as the inverse logit of `logit_theta`, which maps it back to satisfy the `lower=0, upper=1` constraints. @@ -227,21 +363,20 @@ will be rejected. Because this is a non-linear transform and we wish to put a prior directly on `theta`, we need to apply a log-scale change-of-variables correction, which is done by incrementing the variabler `jacobian`, which acts like `target`, but accumulates the -log Jacobian of the transform. +log Jacobian of the transform (cf. the *Reference Manual* chapter on +constraining transforms for a derivation). In most circumstances, the value of `jacobian` will simply be added to the `target`. But it can be dropped with settings in the optimizer and Laplace approximation code so that the optimization result is a -maximum likelihood estimate (or a penalized maximum likelihood -estimate). Maximum a posteriori estimates, i.e., posterior modes, are -computed with the Jacobian included. User-defined Jacobians and -implicit Jacobians from constrained parameters have the same effect on -the final log density and gradient value returned by the Stan program. +(penalized) maximum likelihood estimate rather than a maximum a posteriori +(MAP) estimate computed at posterior modes after adjusting for any +change of variables.. Although this shows the base way of writing this code, Stan's built-in variable transforms are also available as special functions in the Stan language, so that the transformed parameter block could be -written as follows. +written as followsm. ```stan transformed parameters { @@ -255,132 +390,4 @@ Jacobian to increment, so such functions are restricted to the `transformed parameter` block where the `jacobian +=` statement is available. This particular function applies the inverse logit transform, then increments the Jacobian with the log -change-of-variables adjustment. - - -## Generated quantities for posterior predictions - -Suppose we want to observe $N$ trials of a boolean process like the positive -or negative outcome of a clinical trial or positive or negative rating of -a business, and we want to predict what the next $\tilde{N}$ trials might -look like. We can do that with our simple example program by -declaring a data variable for $\tilde{N}$ and then defining our predictive -quantity in the `generated quantities` block. - -```stan -data { - ... - int N_tilde; -} -... -generated quantities { - array[N_tilde] int y_tilde; - for (n in 1:N_tilde) { - y_tilde[n] = bernoulli_rng(theta); - } -} -``` - -Rather than sampling notation, this program explicitly calls a random -number generator for the Bernoulli distribution. Thus -rather than just sampling from the posterior $p(\theta \mid y)$, -we also sample from the posterior predictive, $p(\tilde{y} \mid y)$. -The Stan program simply samples from the joint posterior, -$$ -p(\tilde{y}, \theta \mid y) -= p(\tilde{y} \mid \theta) \cdot p(\theta \mid y), -$$ -by first sampling $\theta$ from the posterior $p(\theta \mid y)$, then -sampling $\tilde{y}$ from the data generating distribution -$p(\tilde{y} \mid \theta)$ given the sample for $\theta$. This matches -the two sources of uncertainty in the posterior predictive, the -uncertainty in the parameter value $\theta$ given the "training" data -$y$, and the uncertainty in generating new "test" data $\tilde{y}$ -given an estimate of $\theta$. - - -## Fitting a Stan model with Markov chain Monte Carlo - -To fit a model, data must be provided. Stan accepts JavaScript Object -Notation (JSON) formatted data (or lists or dictionaries in the R and -Python interfaces), so suppose we have data matching matching the -declarations in our Stan program, such as the following. - - -```json -{ - 'N': 10; - 'y': [1, 0, 0, 1, 0, 1, 0, 0, 0, 0] -} -``` - -The Stan program must be converted to C++ and then the C++ must be -compiled. Then, it must be given data matching the `data` block -declaration in shape and satisfying all of the declared constraints. - -When a Stan program is executed, the first thing that happens is that -it will read the data file in. All of the constraints declared -are checked and any violations will cause the program to terminate -with an informative error message. - -Once the data is loaded, a Stan program is able to evaluate log -densities and gradients. One way to think about the Stan program is -that the data block defines the signature for the function that reads -in data. Then once the data block is read in, the program defines -a function whose arguments are given by the parameters block and -whose value is a log density and gradient evaluated at the argument. - -That is, we can provide a value such as $\theta^\textrm{unc} = -1.3786352$ and the -Stan program will return the posterior log density up to a normalizing -constant, $\log p(\theta \mid y, N) + -\textrm{const}$, and its gradient, -$$ -\nabla \left( \log p(\theta^\textrm{unc} \mid y, N) + \textrm{const} \right) -= \frac{\partial}{\partial \theta^\textrm{unc}} \log p(\theta^\textrm{unc} \mid y, N), -$$ -where the constant drops out because it does not involve $\theta$. - -When fitting a model, the data is only read in once, whereas the log -density is evaluated multiple times, some times as many as 1024 times -per iteration with the no-U-turn sampler. - -Stan's samplers will try to sample values of $\theta$ from the posterior -distribution. That is, it will try to generate several Markov chains -in parallel, each of which has the form -$$ -\theta^{(1)}, \ldots, \theta^{(M)}, -$$ -where if everything is functioning correctly, marginally, -$$ -\theta^{(m)} \sim p(\theta \mid y, N). -$$ -We say ``marginally'' because the draws $\theta^{(m)}$ in the sample -Markov chain may be autocorrelated. - -Sampling will automatically generate all the generated quantities. - -## Summarizing a model fit - -When fitting a model using sampling, either Stan or an external -package (e.g., `ArviZ` in Python) can be used to return a -summary. Such summaries typically include the posterior mean, which -acts as a Bayesian parameter estimate, and posterior standard -deviation. They also typically include posterior quantiles such as -the posterior median (50\% quantile), and by default in Stan, the 5\% -and 95\% quantiles (with others being available). Stan also reports -diagnostics on sampling, such as $\widehat{R}$, which measures -convergence, and effective sample size, which estimates the strength -of Stan's sample in units of independent draws. Finally, it reports -standard error on the mean estimates, which works out to be standard -deviation divided by the square root of the effective sample size. - -Stan's interfaces also allow the posterior draws to be extracted -directly so that they may be used for plotting or to plumb Bayesian -uncertainty through externally generated predictive quantities the same -way Stan does this internally with generated quantities. - -Even Stan's variational inference and Laplace approximations allow -sampling, which is necessary to get the inferences back on the -appropriate scale after fitting on the unconstrained scale (because -averages of functions are not equivalent to functions of averages when -the function is non-linear, as most of the transforms are). +change-of-variables adjustment. \ No newline at end of file From d202696f27678f3b27001e33e69ed4ac055364ee Mon Sep 17 00:00:00 2001 From: Bob Carpenter Date: Sat, 4 Jul 2026 15:59:56 -0400 Subject: [PATCH 4/7] stan ecosystem draft --- src/_quarto.yml | 2 +- src/stan-users-guide/_quarto.yml | 2 +- .../{intro.qmd => ecosystem.qmd} | 151 ++++++++++++------ src/stan-users-guide/img/stan-ecosystem.png | Bin 0 -> 144486 bytes 4 files changed, 101 insertions(+), 54 deletions(-) rename src/stan-users-guide/{intro.qmd => ecosystem.qmd} (63%) create mode 100644 src/stan-users-guide/img/stan-ecosystem.png diff --git a/src/_quarto.yml b/src/_quarto.yml index 6bd988551..4e87749bc 100644 --- a/src/_quarto.yml +++ b/src/_quarto.yml @@ -115,7 +115,7 @@ website: - section: "Version {{< env STAN_DOCS_VERSION >}}" - section: "Introduction" contents: - - stan-users-guide/intro.qmd + - stan-users-guide/ecosystem.qmd - stan-users-guide/how-stan-works.qmd - section: "Example Models" contents: diff --git a/src/stan-users-guide/_quarto.yml b/src/stan-users-guide/_quarto.yml index ebc0a4dcd..54a480ef8 100644 --- a/src/stan-users-guide/_quarto.yml +++ b/src/stan-users-guide/_quarto.yml @@ -36,7 +36,7 @@ book: - index.qmd - part: "Introduction" chapters: - - intro.qmd + - ecosystem.qmd - how-stan-works.qmd - part: "Example Models" chapters: diff --git a/src/stan-users-guide/intro.qmd b/src/stan-users-guide/ecosystem.qmd similarity index 63% rename from src/stan-users-guide/intro.qmd rename to src/stan-users-guide/ecosystem.qmd index baea7e427..3cee221db 100644 --- a/src/stan-users-guide/intro.qmd +++ b/src/stan-users-guide/ecosystem.qmd @@ -6,8 +6,7 @@ pagetitle: The Stan Ecosystem Stan is a domain-specific programming language for specifying probabilistic models along with a collection of algorithms for -performing statistical inference and algorithms and visualizations for -analyzing model fits. +performing statistical inference and analyzing model fits. ## What is the *Stan User's Guide*? @@ -19,13 +18,16 @@ use Stan (see below). ## Who uses Stan? -You can find Stan users and example code online from almost every -subfield of science and most branches of statistics other than highly -non-parametric modeling (e.g., neural networks, high-dimensional -Dirichlet processes) and highly coupled discrete models (e.g., Ising -models). Stan applications have been drawn from pretty much every -branch of biological science (from cellular to population ecology and -zoology), physical sciences (from quantum to astro), social sciences +You can find Stan users and example code online from almost +every subfield of science and most branches of statistics other than +highly non-parametric modeling (e.g., neural networks, +high-dimensional Dirichlet processes), highly coupled discrete models +(e.g., Ising models), huge scale (e.g., e-commerce, global weather +models), and real-time processing (e.g., quad-copter control). + +Stan applications have been drawn from pretty much every branch of +biological science (from cellular to population ecology and zoology), +physical sciences (from quantum to astro), social sciences (psychometrics, surveys, economics, and anthropology), engineering (from civil to mechanical), educational sciences (testing), medical sciences (from epidemiology to clinical trials), sports statistics @@ -33,7 +35,6 @@ sciences (from epidemiology to clinical trials), sports statistics pricing, risk assessment, forecasting), business (advertising attribution, demand and pricing), and actuarial sciences. - ## Stan user and developer communities Stan's strongest feature is its active community of users. Stan's @@ -42,11 +43,12 @@ backgrounds and work in a broad range of application areas. The primary Stan discussion board is hosted by Discourse. -* [Stan forums](https://discourse.mc-stan.org), where users and +* [Stan Forums](https://discourse.mc-stan.org), where users and developers discuss issues with Stan and its applications. The Stan developers discuss designs, plans, and communicate with each -other through the forums and also on GitHub. +other through the forums and also on GitHub. Users can also access it +and make feature requests or report bugs. * [Stan GitHub](https://github.com/stan-dev), which tracks issues and contains the complete source code for Stan (the core is licensed @@ -56,30 +58,34 @@ Both developers and users attend Stan conferences, which are a mix of tutorials, talks about applications, and talks from the Stan developers about where Stan is going. -* [StanCon](https://mc-stan.org/learn-stan/stancon-talks.html), the - pages for which host video talks, slides, and executable notebooks. +* [StanCon](https://mc-stan.org/learn-stan/stancon-talks.html), which is + where you'll find the talks, slides, and executable notebooks. -## How is the Stan Project organized? +## How is the Stan project organized? Stan is free open source software released under permissive licenses (BSD-3 for the core code). There are no institutional owners and no institutional oversight of the project. -The Stan project as a whole is managed through the Stan Governing Body -(SGB), who have final say on all things Stan related. They also -organize StanCon. The members are elected by the developers and +The Stan project as a whole is managed through the Stan Governing +Body (SGB), who have final say on all things Stan related. They +also organize StanCon. The members are elected by the developers and rotate through short (1--2 year) terms. -Software development is managed through Apache-style voting among the -developers. Lack of consensus is so rare there have only been a few -votes in the history of the project aside from electing SGB members. +Software development is managed through Apache-style +voting among the developers. Lack of consensus is so rare there +have only been a few votes in the history of the project aside from +electing SGB members. Since development of Stan began in 2011 (the first release was in 2012), funding has been provided through governmental grants across multiple agencies in several countries, through non-profit -foundations, through donations from corporations, and through in-kind -donations of time by our developer's employers. Thank you! +foundations, through universities, through donations from +corporations, and through in-kind donations of time by our developer's +employers. Thank you! ## Stan documentation @@ -91,7 +97,7 @@ page. * [Stan documentation](https://mc-stan.org/docs/) -You are currently reading the *User's Guide*, which is complemented by +You are currently reading the *Stan User's Guide*, which is complemented by the *Reference Manual* and *Functions Reference*. These are all interface-agnostic guides to the Stan language and its execution. They are available online with integrated navigation and also as pdfs. @@ -113,10 +119,52 @@ case studies or through StanCon presentations. * [*StanCon Case Studies*](https://mc-stan.org/learn-stan/stancon-talks.html) +### Stan GitHub organization + +![The core Stan foundation and interface ecosystem as organized in GitHub repositories. The color coding is based on external interface language.](img/stan-ecosystem.png){#fig-stan-projects width=75% fig-align="center"} + +The projects shown in @fig-stan-projects are just the tip of the iceberg. +Stan is organized into more than fifty code repositories, all of which can +be found in one place. + +* GitHub organization for Stan's source code and documentation: [`stan-dev`](https://github.com/stan-dev) + +The core Stan code repository not included int he diagram is the language itself, +which is at the following location. + +* `stanc3` (OCaml): Stan language transpiler (parser, intermediate representation, C++ code generator) + +The prominent R packages not included in the diagram are the +following. + +* `posterior` (R): posterior analysis tools +* `bayesplot` (R): plotting tools +* `loo` (R): approximate leave-one out cross-validation +* `rstantools` (R): tools for developing R packages interfacing with Stan +* `shinystan` (R): dashboard for exploring Stan model fits + +Stan's core documentation is also maintained on GitHub, with +package-specific documentation maintained in the same repository as +the code. + +* `docs` (Quarto markdown): where this document's source is hosted +* `design-docs` (GitHub markdown): where the developers discuss designs +* `stan-dev.github.io` (Markdown, YAML): mc-stan.org web site + +Stan's syntax highlighters are on GitHub. + +* `atom-language-stan` (Atom): syntax highlighting and autocomplete for the Stan language +* `stan-mode` (emacs lisp): indentation and highlighting in emacs + +The Stan developers also curate a repository of test programs +available as text and data files or through with programmatic +interfaces. + +* `posteriordb` (R, Python, Stan, PyMC): example models with reference draws and R and Python interfaces for access + ### Stan interface documentation -Documentation for the officially supported Stan interfaces for the language and -inference are here. CmdStan is the reference implementation. +CmdStan is the reference implementation of Stan and is coded entirely in C++. * Command line: [*CmdStan User's Guide*](https://mc-stan.org/docs/cmdstan-guide/) @@ -135,15 +183,16 @@ PyStan and RStan call Stan code directly through C++. * R: [*RStan User's Guide*](https://mc-stan.org/rstan/) -There is also an interface for accessing Stan language components, -including (unconstrained) log densities, gradients, and Hessians, as -well as parameter transformations and the posterior predictive -simulations required for generated quantities. +BridgeStan is the cross-platform interface for accessing Stan language +components, including (unconstrained) log densities, gradients, and +Hessians, as well as parameter transformations and the posterior +predictive simulations required for generated quantities. * Python, R, C, Rust: [*BridgeStan User's Guide*](https://roualdes.us/bridgestan/latest/) BridgeStan is implemented through low-level foreign function -interfaces that link to the compiled C++ class for a Stan program. +interfaces that link to the compiled C++ class for a Stan program and +the Stan math library. In addition to the documentation maintained by the development team, there are many resources available online from the community including books, @@ -175,9 +224,8 @@ Stan provides a domain-specific language for specifying smooth, data-dependent target density functions on the logarithmic scale. For Bayesian applications, this will be the posterior log density function (up to an additive constant). For frequentist applications, it will -be a penalized (marginal) likelihood function. Stan also provides a -method for coding posterior predictive quantities and generating them -through simulation. +be a penalized likelihood function. Stan also provides a method for +efficiently coding posterior predictive quantities through simulation. Stan is a differentiable programming language, meaning that Stan can automatically take first, second, and even higher-order derivatives of @@ -204,7 +252,7 @@ techniques based on variational inference, * Automatic Differentiation Variational Inference (ADVI) * Pathfinder variational inference -as well the following approximate method based on optimization and curvature. +It also supplies an approximate method based on optimization and curvature. * Laplace approximation @@ -249,38 +297,37 @@ number generators, and densities. The math library can call out to GPU for some operations (e.g., Cholesky factorization or matrix-matrix multiplication), but overall, -it has been much more heavily optimzed for CPUs. +it has been much more heavily optimized for CPUs. When using GPUs, +there is a further dependency. + +* `OpenCL`: OpenCL for GPU management. -## Background reading for getting started +## Background reading on Bayesian Statistics and Stan This section links some resources for getting started with Bayesian statistics using Stan. The first two recommendations are for introductions to Bayesian statistics using Stan, a complete hands-on introductory textbook and shorter introductory article. -* [_Statistical Rethinking: A Bayesian Course with Examples in R and -Stan_, Second Edition](https://xcelab.net/rm/). 2020. Richard -McElreath. CRC Press. +* [_Statistical Rethinking: A Bayesian Course with Examples in R and Stan_, Second Edition](https://xcelab.net/rm/). 2020. Richard McElreath. CRC Press. -* [Getting started with Bayesian statistics using Stan and - Python](https://bob-carpenter.github.io/stan-getting-started/stan-getting-started.html). 2023. - Bob Carpenter. +* [Getting started with Bayesian statistics using Stan and Python](https://bob-carpenter.github.io/stan-getting-started/stan-getting-started.html). 2023. Bob Carpenter. The next step to understanding Bayesian statistics more deeply after these introductory texts is *BDA3*, a free pdf for which is available from the linked web site. -* [_Bayesian Data Analysis, Third -Edition_](https://sites.stat.columbia.edu/gelman/book/). 2013. Andrew -Gelman, John Carlin, Hal Stern, David Dunson, Aki Vehtari, and Donald -Rubin. CRC Press. +* [_Bayesian Data Analysis, Third Edition_](https://sites.stat.columbia.edu/gelman/book/). 2013. Andrew Gelman, John Carlin, Hal Stern, David Dunson, Aki Vehtari, and Donald Rubin. CRC Press. -There has been more interest in statistics lately to consider the -whole process of using statistics from exploratory data analysis to -results presentation. The following short article and longer book -provide advice on statistical workflow from a Bayesian perspective -with worked examples in Stan. +Model building and inference, as supplied by Stan, are only two pieces +in a bigger toolchain needed for effective applied statistical +modeling. The bigger picture extends to exploratory data analysis, +model construction and choice, algorithm calibration checking, prior +and posterior predictive checking, plotting and visualization, and +results presentation. The following short article and book provide +advice on this broader statistical workflow from a Bayesian +perspective with worked examples in Stan. * [Statistical workflow](https://sites.stat.columbia.edu/gelman/research/published/Statistical_Workflow_article.pdf). 2026. Andrew Gelman, Aki Vehtari, Richard McElreath. _Philosophical Transactions of the Royal Society A_. diff --git a/src/stan-users-guide/img/stan-ecosystem.png b/src/stan-users-guide/img/stan-ecosystem.png new file mode 100644 index 0000000000000000000000000000000000000000..ae891d108ea96e5408923e682bfadf6b68d96857 GIT binary patch literal 144486 zcmcG0bySsI^EMzLiUOi2A)yFJE2*?ahtds7iF9{ah|(z_A%cjMbR0@iy1PN?I&{}> zjy@{-y!x#5t@Zt(>u}<}_r3S*nQN}OW_#U~7CCyD^e`G4+EFpl>#}HQ2kp?%4rJnB z!#moIT0v-N=yZnHuH7`cCUVWp*vwMS{I-tX60omeqFqRMt>dNA)4Lvh2lkb@d@)f}wIW%nPJM&WMUKoH*9$ z`*k0JD{=E>WuFCkEK^c7NVA-0xUH>Shn5*qD{)w8hJs4%96ec`Cfc1-=;S!MX6PBM zXJ(FTa}unQzl-pvGN9Tz;9nnxsn6~)^F_k*9hQ*sn_?=eiRXba!EezPiyyyCNO(l+ zI?C*M0`tuE7m8xk9gM+6AOLDyR~zf;B>V29Rq?p zIPL_I)bGoNN`pN-M)9xK#fN#^_P{%EU{C(x^N2GC9=Hn}C@;BOop3|%rVeqqHwiJ$ z0{SsU@mqHvY6im0UD8t))0dJ$V}RES~BS#yNkq>aaESHFGXR(hM!4dylViAS#^Q zpy9=!3hTSM25&9rtAvCYI8^RvfBny01b5wOd4l-#U%&PB8t!L=%){_8{_~4NSQwag z#Y1N=c+9KUa_{zt-SC 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ztSf!ROX8$zvOkN-NDy-N_C%DwwQS5Sz@^(gD-+pq4V1cGptU}*#_9&AZN+g&(kji4 zzm}dW0-hl0vwL2HLiY4^aosyrd^-jF*U69-n62MCy_Uc?qz}oL?%CiAy5nm;$ZU(2 zz{2?JeYswOXkH-#4~GhcJ=4wO09 z@2Y2%0q4aNGJyfvr+MTF;72IM7!%2bMb>?%qIRUQKfU9SM4P>eojcod{6u!ubtw)B2ey`0wNweYB;ISV)qn>Ywb<7h}~LA=T6l>=H(F@ z_7{dQx8mT1Q9pXke%^v0ps|wprOx8QGlP`GtZ%kAwA-?(fExi1WFavQc!y4g4Y>7{ za97zOX_>~Uz1M@txD@(b_NzVM%O>U=SW)H&ey16w|1=K}sRw@~cN$cX%4_d* zU#hZoB$4o#&A;C9mzS*qa02MhBE=9U1#VxCG z)5tw&OL7S8--fl}v-URDAvuQBhNU zKSbA6p#`{%`zPQA$q$&HH&XW<0%6UI`_t1E`w?Js!tO=Mb|B8k-|M>v;uMqXz~G9p zW1-5imuqPlw=hU5medm3?gys(q#_=eA|3buykt#*e$W;iO=2z81iD zUQ6gQPrz!K;yfqL75B9Dd^`&gQqE~%Wbn8;`6QLDJ)K=$YCYN4Q-E!I)YR_p8=a_r zZorPY`Z1`l*mK+lxI{HxzWlGar2qf` literal 0 HcmV?d00001 From e9f93e167d48f3b4e4f4909a5875b3df1431952d Mon Sep 17 00:00:00 2001 From: Bob Carpenter Date: Sat, 4 Jul 2026 20:48:34 -0400 Subject: [PATCH 5/7] how stan works draft --- src/stan-users-guide/how-stan-works.qmd | 254 +++++++++++++++--------- 1 file changed, 159 insertions(+), 95 deletions(-) diff --git a/src/stan-users-guide/how-stan-works.qmd b/src/stan-users-guide/how-stan-works.qmd index 0d9bbffee..8393ffd06 100644 --- a/src/stan-users-guide/how-stan-works.qmd +++ b/src/stan-users-guide/how-stan-works.qmd @@ -17,21 +17,26 @@ unconstraining transforms and their inverses. Stan's syntax is similar to C's. In Stan, every variable, constant, and larger expression has a type that does not change. The three -primitive types are `int`, `real`, and `complex`. Integer types may -not be used as parameters. The primitive container types are -`vector`, `row_vector`, and `matrix`. Array types are available to -hold any single type of value. Tuples hold a sequence of values -of fixed types. +primitive types are `int`, `real`, and `complex`. Integers may not be +used as parameters. The primitive container types are `vector`, +`row_vector`, and `matrix`. Arrays are homogeneous and array types +are available for any single type of value. Tuples are heterogeneous +and hold a sequence of values of possibly different fixed types. Variables can be declared with constraints such as lower and/or upper bounds for scalars, and simplex, positive definiteness, and many other constraints for vectors and matrices. Constraints on parameters are maintained implicitly by Stan through unconstraining and constraining transforms. Constraints on other variables are validated at the end -of their respective blocks. +of their respective blocks. Local variables and function argument +types must be declared with unconstrained types (though they may hold +constrained values). Function argument types do not include sizes. -A Stan program is organized into a sequence of blocks, which are -optional, but must appear in the following order. +A Stan program is organized into a sequence of blocks, all of which +are optional, but must appear in the following order. Stan code +is executed sequentially, and in each block, functions or variables +declared in previous blocks remain available for use (but not for +reassignment). 1. `functions`: Define functions that can be called in the Stan program. Functions are executed when called. @@ -54,32 +59,40 @@ maximum likelihood estimation). Executed every log density and gradient evaluation. 5. `transformed parameters`: Declare and define variables that depend -on variables from the previous blocks. Statements in this block may +on parameters, are needed in the `model` block or `generated +quantities` block, and should be saved for output. Variables that +don't need to be printed, but depend on parameters and are needed in +the model block should be declared as local variables in the `model` +and/or `generated quantities` block. Statements in this block may increment the log Jacobian for changes of variables. Executed every log density and gradient evaluation. -6. `model`: Define the target log density using the variables from -previous blocks. Executed every log density and gradient evaluation. +6. `model`: Define the target log density using the constrained +variables from previous blocks. May define local variables. Executed +every log density and gradient evaluation. 7. `generated quantities`: Define posterior predictive quantities that depend on variables in previous blocks. This is where to define -quantities that are not needed for the log density evaluation. -Statements in this block may use random number generators. Executed -once per iteration using primitives. Execution with primitives means -no automatic differentiation and three or more times faster than -similar code in the model or transformed parameter blocks. +variables that should be saved, but are not needed for the log density +evaluation. Statements in this block may use random number +generators. using primitives. The `generated quantities` block does +not need to calculate derivatives, making it at least three times +faster than variables defined in the transformed parameter block in +other blocks and is much much less memory intensive. Executed once +per algorithm iteration (*not* every log density evaluation). Stan programs are transpiled to C++ by an OCaml program (`stanc3`). The resulting C++ class definition is compiled and linked to the Stan math library for special functions and automatic differentiation. At -this point, an object can be constructed from data. This object can -be used to evaluate log densities and derivatives as well as -unconstraining transforms and their inverses. +this point, an object can be constructed from data. This object +implements all of (1)--(3) above, differentiable log densities, +posterior predictive quantities, and variable transforms and their +inverses (sometimes pseudoinverses for many-to-one inverse transforms). ## An example Stan program Consider the following very simple Stan program, which can be used to -estimate a proportion $\theta \in (0, 1)$ from binary observations +estimate a proportion $\theta \in (0, 1)$ from $N$ binary observations $y_n \in \{ 0, 1 \}$. ```stan @@ -91,7 +104,7 @@ parameters { real theta; // probability of success } model { - theta ~ beta(2.3, 8.1); // prior + theta ~ beta(23.1, 85.7); // prior y ~ bernoulli(theta); // data generating distribution } ``` @@ -129,12 +142,12 @@ penalty terms). ### Distribution statements are syntactic sugar The distribution statements with `~` are syntactic sugar for -incrementing the target log density. The model block of the previous -program could be equivalently written as follows. +incrementing the target log density. An equivalent way to write the +previous model block without distribution statements is as follows. ```stan model { - target += beta_lupdf(theta | 2.3, 8.1); + target += beta_lupdf(theta | 23.1, 85.7); target += bernoulli_lupmf(y | theta); } ``` @@ -142,7 +155,11 @@ model { Note the use of a vertical bar (`|`) rather than comma to separate the variate from the distribution parameters. The suffix `_lupdf` is an acronym for "log unnormalized probability density function," and -`_lupmf` for "log unnormalized probability mass function." +`_lupmf` for "log unnormalized probability mass function." The +normalizing constants are not needed for Stan's inference algorithms. +There are corresponding `_lpdf` and `_lpmf` versions that maintain +normalizing constants where necessary, for instance in heterogeneous +mixture models or to calculate log likelihoods for model comparison. The target log density is initialized at 0 and implicitly incremented for change-of-variables adjustments for constrained parameters and for @@ -151,10 +168,11 @@ sampling statements. ## Model blocks define log density functions In the running example, the Stan program defines the following -target log density function $\log p$ up to a constant, +target log density function $\log p$ up to a constant for a parameter +$\theta \in (0, 1)$, $$ \log p(\theta \mid y, N) -= \log \textrm{beta}(\theta \mid 8, 2) += \log \textrm{beta}(\theta \mid 23.1, 85.7) + \sum_{n=1}^N \log \textrm{bernoulli}(y \mid \theta) + \textrm{const}. $$ @@ -162,22 +180,20 @@ The distribution statement for `y` in the Stan program is vectorized. The argument `y` is an array of integers, but `theta` is a scalar. When Stan sees this argument pattern, it reuses scalar arguments like `theta` for each entry of containers like `y`. This can lead to much -more efficient evaluation (e.g., $\log \theta$ is only computed once -in this example). - - +more efficient evaluation (e.g., $\log \theta$ and $\log(1 - \theta)$ +are only computed once in this example). ## Generated quantities for posterior predictions Suppose we want to observe $N$ trials of a binary process like the positive or negative outcome of a clinical trial or positive or negative rating of a business and conditioned on those observations, -predict what the next $\tilde{N}$ trials might look like. This is -called posterior predictive inference, and in Stan, we can define the -quantities of interest in the `generated quantity` block. We extned -our example program by declaring a data variable for the number of new -observations $\tilde{N}$ and then defining our predictive quantity in -the `generated quantities` block. +and then predict what the next $\tilde{N}$ trials might look like. +This is called _posterior predictive inference_, and in Stan, we can +define the quantities of interest in the `generated quantities` block. +We extend our example program by declaring a data variable for the +number of new observations $\tilde{N}$ and then defining our +predictive quantity in the `generated quantities` block. ```stan data { @@ -206,8 +222,8 @@ on $\theta$. To fit a model, data must be provided. Stan accepts file-based data in JavaScript Object Notation (JSON), and can also directly accept -data formatted as lists in R or dictionaries in Python. For the -running example, consider the data defined by the following JSON. +lists in R or dictionaries in Python. For the running example, +consider the data defined by the following JSON. ```json @@ -218,18 +234,28 @@ running example, consider the data defined by the following JSON. ``` Once the data is loaded, a Stan program is able to evaluate log -densities and gradients. For example, we could provide a value such as +densities and gradients. This is done on transformed parameters, +which here means $\theta^\text{unc} = \textrm{logit}(\theta)$; +see the last section in this chapter, @sec-hsw-transformed-parameters, for +details. + +For example, we could provide a value such as $\theta^\textrm{unc} = -1.3786352$ and the Stan program will return -the posterior log density up to a normalizing constant, $\log p(\theta -\mid y, N) + \textrm{const}$, and its gradient, $$ \nabla \left( \log -p(\theta^\textrm{unc} \mid y, N) + \textrm{const} \right) = -\frac{\partial}{\partial \theta^\textrm{unc}} \log -p(\theta^\textrm{unc} \mid y, N), $$ where the constant drops out -because it does not involve $\theta$. +the posterior log density up to a normalizing constant, $\log p(\theta^\textrm{unc} +\mid y, N) + \textrm{const}$, and its gradient, + +$$ +\nabla \left( + \log p(\theta^\textrm{unc} \mid y, N) + \textrm{const} + \right) += \frac{\partial} + {\partial \theta^\textrm{unc}} + \, \log p(\theta^\textrm{unc} \mid y, N). +$$ When fitting a model, the data is only read in once, whereas the log -density is evaluated multiple times, some times as many as 1024 times -per iteration with the no-U-turn sampler. +density is evaluated multiple times, up to 1024 times per iteration +with default settings for the the no-U-turn sampler. Stan's samplers will try to sample values of $\theta$ from the posterior distribution. That is, it will try to generate several @@ -237,13 +263,13 @@ Markov chains in parallel, each of which has the form $$ \theta^{(1)}, \ldots, \theta^{(M)}, $$ where if everything is functioning correctly, marginally, $$ \theta^{(m)} \sim p(\theta \mid y, N). $$ While these draws are marginally distributed approximately according to the target -distribituion, the Markov chain from which they arose may be +distribution, the Markov chain from which they arose may be autocorrelated. In these cases, the number of draws needs to be discounted to figure out the effective number of independent draws (i.e., the effective sample size) that would have the same -informativeess. +informativeness. -Stan's inference engines will also genreate all generated quantities +Stan's inference engines will also simulate all generated quantities based on the data and the current draw of the parameters. ## Summarizing a model fit @@ -255,32 +281,62 @@ acts as a Bayesian parameter estimate, and posterior standard deviation. They also typically include posterior quantiles such as the posterior median (50\% quantile), and by default in Stan, the 5\% and 95\% quantiles (with others being available). Stan also reports -diagnostics on sampling, such as $\widehat{R}$, which measures -convergence, and effective sample size, which estimates the -informativeness of the potentially autocorrelated Markov chain Monte -Carlo sample in units of independent draws. Finally, summaries report -standard error on the mean estimates, which are calculated as -standard deviation divided by the square root of the effective sample size. +diagnostics on sampling. The primary statistic of interest is +$\widehat{R}$, which approaches 1 asymptotically if multiple Markov +chains are sampling from the same target distribution. The secondary +statistic of interest givne that $\widehat{R}$ is reasonable, is +effective sample size (ESS). ESS is the number of equivalent +independent draws required to get the same standard error in estimates +as the draws from the correlated Markov chain. ESS can be higher than +the number of draws when the chains are anticorrelated, as they can be +for parameter estimates in simple models. Finally, summaries report +standard error on the mean estimates, which are calculated as standard +deviation divided by the square root of the effective sample size. Stan's interfaces also allow the posterior draws to be extracted -directly so that they may be used for plotting or to plumb Bayesian -uncertainty through externally generated predictive quantities the same -way Stan does this internally with generated quantities. - -Even Stan's variational inference and Laplace approximations allow -sampling, which is necessary to get the inferences back on the -appropriate scale after fitting on the unconstrained scale. This -needs to be done before averaging for estimating expectations because -averages of functions are not equivalent to functions of averages when -the function is non-linear, as most of the transforms are. - - - -## Technical detail: Models define unconstrained log density functions - -This section describe how Stan works under the hood by transforming -constrained log density functions. Understranding the details of this -section is not necessary to write Stan code. +directly so that they may be used for plotting. With the draws, it is +simple to plumb Bayesian uncertainty through externally generated +predictive quantities the same way Stan does this internally with +generated quantities. + +## Variational inference and Laplace approximation + +Stan provides two variational inference algorithms, ADVI and +Pathfinder. These both try to find normal approximations to the +posterior centered at the posterior mean. ADVI can produce dense +or diagonal approximations, and Pathfinder produces low-rank plus +diagonal approximations. + +Laplace approximation works similarly to variational inference, but is +based on straight optimization to a mode, and thus only works when the +mode is well defined or sensible (e.g., not in a hierarchical model). +In cases where it is well defined and in relatively low dimensions, Laplace +approximation will be faster than variational approximation. + +Laplace approximation centers a second-order Taylor approximation +around the posterior mode. This produces a normal distribution located +at the mode with covariance equal to the negative Hessian (matrix of +second-order derivatives). Because Laplace approximation constructs +the Hessian and Cholesky factors it to generate draws, it will be +cubic in cost once and then quadratic in cost per draw. This is the +same cost as the dense variational approximations, but the diagonal +approximations of ADVI and low-rank plus diagonal approximation of +Pathfinder are more efficient in both time and memory in higher dimensions. + +These approximations are performed on the transformed scale, which is +unconstrained. To put the results back on the natural scale where +parameters satisfy their declared constraints, samples from the +approximate posterior can be drawn and inverse transformed back to the +constrained scale by the Stan model. Because the relation is not +linear, it does not make sense to take the point estimates from +variational inference or Laplace approximation and transform those. + +## Technical detail: Transformed parameters {#sec-hsw-transformed-parameters} + +Under the hood, Stan transforms the user-defined parameterization to +an unconstrained form where the model has support over all of real +space. Understranding the details of this section is not necessary to +write Stan code, but it helps to write code that samples efficiently. Any parameters declared with constraints, such as `theta` in the example, are transformed to unconstrained behind the scenes by Stan. @@ -295,8 +351,8 @@ $$ Stan will account for the change-of-variables adjustment that is required by the non-linear transform. The *Stan Reference Manual* -specifies all of the constrained parameters and corresponding -transforms and change-of-variables adjustments. +specifies all of the constrained types and their corresponding +transforms, (pseudo)inverse transforms, and change-of-variables adjustments. When the change-of-variables dust settles on the log scale, Stan defines a density with support (finite value) for all @@ -304,15 +360,18 @@ $\theta^\textrm{unc} \in (-\infty, \infty)$. It does this by inverting the transform and applying the change of variables formula on the log scale, which yields the unconstrained unnormalized log density function -$$ +\begin{align*} \log p(\theta^\textrm{unc} \mid y, N) -= +&= \log p(\textrm{logit}^{-1}(\theta^\textrm{unc}) \mid y, N) + \log \textrm{logit}^{-1}(\theta^\textrm{unc}) + \log (1 - \textrm{logit}^{-1}(\theta^\textrm{unc})) + \textrm{const}, -$$ -where +\\[4pt] +&= +\log p(\theta \mid y, N) + \log \theta + \log (1 - \theta) + \textrm{const}, +\end{align*} +where the inverse transform is applied to $\theta^\textrm{unc}$ to retrieve $$ \theta \ = \ @@ -322,11 +381,12 @@ $$ {1 + \exp(\theta^\textrm{unc})}. $$ -This unconstrained log density is what Stan is actually sampling or +This unconstrained log density is what Stan samples with Hamiltonian Monte +Carlo or approximates with variational inference or Laplace approximation. approximating with variational inference or Laplace approximations. -After inference, parameter values are automatically transformed back +After inference, draws of parameters can be automatically transformed back to the constrained scale using the inverse transforms (here, -$\textrm{logit}^{-1}$). +$\textrm{logit}^{-1}()$). ### Transformed parameters and Jacobians @@ -342,22 +402,25 @@ data { array[N] int y; } parameters { - real logit_theta; // log odds of success + real logit_theta; // log odds of success } transformed parameters { + // inverse transform real theta = inv_logit(logit_theta); + // change-of-variables adjustment - jacobian += log_inv_logit(logit_theta) + log1m_inv_logit(logit_theta); + jacobian += log_inv_logit(logit_theta) + + log1m_inv_logit(logit_theta); } model { - theta ~ beta(2.3, 8.1); + theta ~ beta(23.1, 85.7); y ~ bernoulli(theta); } ``` A new block, `transformed parameters`, is used to define the probability of success `theta` as the inverse logit of `logit_theta`, which maps it back to satisfy the `lower=0, upper=1` constraints. -Constraints in the transformed parameters` block are evaluated at the +Constraints in the `transformed parameters` block are evaluated at the end of the block and and if they fail, the current algorithm iteration will be rejected. Because this is a non-linear transform and we wish to put a prior directly on `theta`, we need to apply a log-scale @@ -375,8 +438,8 @@ change of variables.. Although this shows the base way of writing this code, Stan's built-in variable transforms are also available as special functions in the -Stan language, so that the transformed parameter block could be -written as followsm. +Stan language, so that the transformed parameter block defined +above could be simplified to the following. ```stan transformed parameters { @@ -385,9 +448,10 @@ transformed parameters { } ``` -A suffix `_jacobian` on a function indicates that it has access to the -Jacobian to increment, so such functions are restricted to the -`transformed parameter` block where the `jacobian +=` statement is -available. This particular function applies the inverse logit -transform, then increments the Jacobian with the log +The suffix `_jacobian` on a function indicates that it has access to +the Jacobian to increment. As such, functions with `_jacobian` +suffixes are restricted to the `transformed parameters` block where +the `jacobian +=` statement is also available. Matching the explicit +definition earlier, the `lower_upper_bound_jacobian` function applies +the inverse logit transform, then increments the Jacobian with the log change-of-variables adjustment. \ No newline at end of file From 17e706d3d0052eae96ca4350d9c468eb32fc0586 Mon Sep 17 00:00:00 2001 From: Brian Ward Date: Mon, 6 Jul 2026 09:36:15 -0400 Subject: [PATCH 6/7] Repair quarto config --- src/quarto-config | 2 +- src/stan-users-guide/_quarto.yml | 4 ++-- 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/src/quarto-config b/src/quarto-config index 65904751c..cb46109d1 160000 --- a/src/quarto-config +++ b/src/quarto-config @@ -1 +1 @@ -Subproject commit 65904751c830555ec8d81d9270e609ec190c425f +Subproject commit cb46109d1312380012f1d5d271ad2c50e8630841 diff --git a/src/stan-users-guide/_quarto.yml b/src/stan-users-guide/_quarto.yml index 7fb5c2120..4ee186407 100644 --- a/src/stan-users-guide/_quarto.yml +++ b/src/stan-users-guide/_quarto.yml @@ -36,8 +36,8 @@ book: - index.qmd - part: "Introduction" chapters: - - ecosystem.qmd - - how-stan-works.qmd + - ecosystem.qmd + - how-stan-works.qmd - part: "Example Models" chapters: - regression.qmd From 2b65802eea23ef8fb0c73b370129026d344f33da Mon Sep 17 00:00:00 2001 From: Bob Carpenter Date: Mon, 6 Jul 2026 20:11:29 -0400 Subject: [PATCH 7/7] code review links, etc. --- src/functions-reference/functions_index.qmd | 30 ++- src/stan-users-guide/ecosystem.qmd | 262 ++++++++++++-------- src/stan-users-guide/how-stan-works.qmd | 52 ++-- src/stan-users-guide/img/stan-ecosystem.png | Bin 144486 -> 130516 bytes src/stan-users-guide/index.qmd | 20 +- 5 files changed, 221 insertions(+), 143 deletions(-) diff --git a/src/functions-reference/functions_index.qmd b/src/functions-reference/functions_index.qmd index d1c0a90a3..82a08ba45 100644 --- a/src/functions-reference/functions_index.qmd +++ b/src/functions-reference/functions_index.qmd @@ -1488,8 +1488,28 @@ pagetitle: Alphabetical Index **integrate_1d**: - -
[`(function integrand, real a, real b, array[] real theta, array[] real x_r, array[] int x_i) : real`](higher-order_functions.qmd#index-entry-39f0cbb60146855f8f98ba56502efcbfd04df2aa) (higher-order_functions.html)
- -
[`(function integrand, real a, real b, array[] real theta, array[] real x_r, array[] int x_i, real relative_tolerance) : real`](higher-order_functions.qmd#index-entry-c201769a78980f7f6e28951ac4b8ccf6e750698e) (higher-order_functions.html)
+ -
[`(function integrand, real a, real b, array[] real theta, array[] real x_r, array[] int x_i) : real`](deprecated_functions.qmd#index-entry-39f0cbb60146855f8f98ba56502efcbfd04df2aa) (deprecated_functions.html)
+ -
[`(function integrand, real a, real b, array[] real theta, array[] real x_r, array[] int x_i, real relative_tolerance) : real`](deprecated_functions.qmd#index-entry-c201769a78980f7f6e28951ac4b8ccf6e750698e) (deprecated_functions.html)
+ + +**integrate_1d_double_exponential**: + + -
[`(function integrand, real a, real b, ...) : real`](higher-order_functions.qmd#index-entry-c8cc4abcc68f88de6c338cdd1feb24eb4d60c96e) (higher-order_functions.html)
+ + +**integrate_1d_double_exponential_tol**: + + -
[`(function integrand, real a, real b, data real relative_tolerance, data real absolute_tolerance, data int max_refinements, ...) : real`](higher-order_functions.qmd#index-entry-c83f3890cc67e87e4fb7822aa9f7474d19ee854d) (higher-order_functions.html)
+ + +**integrate_1d_gauss_kronrod**: + + -
[`(function integrand, real a, real b, ...) : real`](higher-order_functions.qmd#index-entry-609db55f251cb21eec33b53ae0f9d2137cf56d76) (higher-order_functions.html)
+ + +**integrate_1d_gauss_kronrod_tol**: + + -
[`(function integrand, real a, real b, data real relative_tolerance, data real absolute_tolerance, data int max_depth, ...) : real`](higher-order_functions.qmd#index-entry-520246de3803e2fe6b309c9071d48973bb187c7b) (higher-order_functions.html)
**integrate_ode**: @@ -2027,6 +2047,9 @@ pagetitle: Alphabetical Index **log_softmax**: + -
[`(array[] row_vector x) : array[] row_vector`](matrix_operations.qmd#index-entry-cf2cd5d419ea6f5e410621d019fb6d857095ea90) (matrix_operations.html)
+ -
[`(array[] vector x) : array[] vector`](matrix_operations.qmd#index-entry-d4a63c4e3656af5cb7c71a1fbe6751c4bc1a3141) (matrix_operations.html)
+ -
[`(row_vector x) : row_vector`](matrix_operations.qmd#index-entry-4b30f160dc58e9df0572d4ef31dc45f2266c515f) (matrix_operations.html)
-
[`(vector x) : vector`](matrix_operations.qmd#index-entry-49224e65079937569f0952ce5740c03bd8dcb98a) (matrix_operations.html)
@@ -3768,6 +3791,9 @@ pagetitle: Alphabetical Index **softmax**: + -
[`(array[] row_vector x) : array[] row_vector`](matrix_operations.qmd#index-entry-a69431a2ac47b5a69504de480eb13928ea1e201d) (matrix_operations.html)
+ -
[`(array[] vector x) : array[] vector`](matrix_operations.qmd#index-entry-4b08e211476579b25d26a5da171b63934b244f83) (matrix_operations.html)
+ -
[`(row_vector x) : row_vector`](matrix_operations.qmd#index-entry-4a9b64c87e2e17a50db554ecf28c217363fad221) (matrix_operations.html)
-
[`(vector x) : vector`](matrix_operations.qmd#index-entry-9267f9986949f0269013001d932bef84c805fa45) (matrix_operations.html)
diff --git a/src/stan-users-guide/ecosystem.qmd b/src/stan-users-guide/ecosystem.qmd index 3cee221db..0dec1bcaa 100644 --- a/src/stan-users-guide/ecosystem.qmd +++ b/src/stan-users-guide/ecosystem.qmd @@ -35,7 +35,7 @@ sciences (from epidemiology to clinical trials), sports statistics pricing, risk assessment, forecasting), business (advertising attribution, demand and pricing), and actuarial sciences. -## Stan user and developer communities +## User and developer communities Stan's strongest feature is its active community of users. Stan's users hail from a wide range of mathematical and statistical @@ -62,7 +62,7 @@ developers about where Stan is going. where you'll find the talks, slides, and executable notebooks. -## How is the Stan project organized? +## How is the project organized? Stan is free open source software released under permissive licenses (BSD-3 for the core code). There are no institutional owners and no @@ -88,12 +88,12 @@ corporations, and through in-kind donations of time by our developer's employers. Thank you! -## Stan documentation +## Documentation -### Core Stan documentation +### Core Stan language documentation -Stan's internal documentation is organized through a top-level web -page. +Stan's core internal documentation for the Stan language is organized +through a top-level web page. * [Stan documentation](https://mc-stan.org/docs/) @@ -109,7 +109,7 @@ They are available online with integrated navigation and also as pdfs. * [*Stan Functions Reference*](https://mc-stan.org/docs/functions-reference/) -### Stan case studies +### Case studies Stan case studies cover specific models or techniques with reproducible code (usually in Python or R). They can be found in the devoted page of @@ -119,55 +119,18 @@ case studies or through StanCon presentations. * [*StanCon Case Studies*](https://mc-stan.org/learn-stan/stancon-talks.html) -### Stan GitHub organization +### Interface documentation -![The core Stan foundation and interface ecosystem as organized in GitHub repositories. The color coding is based on external interface language.](img/stan-ecosystem.png){#fig-stan-projects width=75% fig-align="center"} - -The projects shown in @fig-stan-projects are just the tip of the iceberg. -Stan is organized into more than fifty code repositories, all of which can -be found in one place. - -* GitHub organization for Stan's source code and documentation: [`stan-dev`](https://github.com/stan-dev) - -The core Stan code repository not included int he diagram is the language itself, -which is at the following location. - -* `stanc3` (OCaml): Stan language transpiler (parser, intermediate representation, C++ code generator) - -The prominent R packages not included in the diagram are the -following. - -* `posterior` (R): posterior analysis tools -* `bayesplot` (R): plotting tools -* `loo` (R): approximate leave-one out cross-validation -* `rstantools` (R): tools for developing R packages interfacing with Stan -* `shinystan` (R): dashboard for exploring Stan model fits - -Stan's core documentation is also maintained on GitHub, with -package-specific documentation maintained in the same repository as -the code. +Stan provides several interfaces for compiling Stan models, performing +inference, and analyzing the results. Each supplies its own documentation. -* `docs` (Quarto markdown): where this document's source is hosted -* `design-docs` (GitHub markdown): where the developers discuss designs -* `stan-dev.github.io` (Markdown, YAML): mc-stan.org web site - -Stan's syntax highlighters are on GitHub. - -* `atom-language-stan` (Atom): syntax highlighting and autocomplete for the Stan language -* `stan-mode` (emacs lisp): indentation and highlighting in emacs - -The Stan developers also curate a repository of test programs -available as text and data files or through with programmatic -interfaces. - -* `posteriordb` (R, Python, Stan, PyMC): example models with reference draws and R and Python interfaces for access - -### Stan interface documentation CmdStan is the reference implementation of Stan and is coded entirely in C++. * Command line: [*CmdStan User's Guide*](https://mc-stan.org/docs/cmdstan-guide/) +It includes tools for posterior analysis. + CmdStanPy, CmdStanR, and Stan.jl make system calls to CmdStan out of process. @@ -183,6 +146,7 @@ PyStan and RStan call Stan code directly through C++. * R: [*RStan User's Guide*](https://mc-stan.org/rstan/) + BridgeStan is the cross-platform interface for accessing Stan language components, including (unconstrained) log densities, gradients, and Hessians, as well as parameter transformations and the posterior @@ -194,22 +158,55 @@ BridgeStan is implemented through low-level foreign function interfaces that link to the compiled C++ class for a Stan program and the Stan math library. +### Documentation source code web site + +Stan's core documentation is also maintained on GitHub, with +package-specific documentation maintained in the same repository as +the code. + +* [`docs`](https://github.com/stan-dev/docs) (Quarto markdown): where this document's source is hosted +* [`design-docs`](https://github.com/stan-dev/design-docs) (GitHub markdown): where the developers discuss designs + +### Web site source code + +The Stan web site, [`https://mc-stan.org`](https://mc-stan.org), is managed +through GitHub. + +* [`stan-dev.github.io`](https://github.com/stan-dev/stan-dev.github.io) (Markdown, YAML): `mc-stan.org` web site + +### Further documentation resources + In addition to the documentation maintained by the development team, there are many resources available online from the community including books, -video courses, talks, papers, and case studies. +video courses, talks, papers, and case studies. -## What is Stan? -Stan is made up of a programming language for statistical models, -several integrated statistical inference algorithms, and interfaces in -various higher-level analysis languages (R, Python, Julia, Stata, -MATLAB). The Stan project also maintains higher-level interfaces to -Stan such as the `rstanarm` and `brms`, both of which are based on R's -expression language for regressions. The project also develops and -maintains visualization and model comparison tools like `posterior` -and `loo` (in R; for Python, similar functionality is available through -`ArviZ`). + +## Core C++ and interface organization + +![The core Stan foundation and interface ecosystem as organized in GitHub repositories. The color coding is based on external interface language.](img/stan-ecosystem.png){#fig-stan-projects width=75% fig-align="center"} + +The projects shown in @fig-stan-projects are just the tip of the +iceberg. Stan is organized into more than fifty GitHub repositories, +all of which can be found in the following location. + +* GitHub organization for Stan's source code and documentation: [`stan-dev`](https://github.com/stan-dev) + +### Higher-level Stan interfaces: `brms` and `rstanarm` + +The Stan project also maintains two high-level interfaces to Stan, +`brms` and `rstanarm`. These both use R's expression language for +regressions to allow users to write Bayesian models in a natural way +while compiling down to well-implemented Stan programs on the back end. + +* Bayesian regression models using Stan: [`brms`](https://paulbuerkner.com/brms/) +* Bayesian applied regression modeling (arm) via Stan: [`rstanarm`](https://mc-stan.org/rstanarm/) + +Although they cover slightly different model families, the main +difference is that `rstanarm` does not require a C++ toolchain to +compile models, whereas `brms` does. This makes `rstanarm` much easier +to install and distribute. Many other individuals and companies have built tools that depend on the core Stan language or packages, many of which are available open @@ -218,89 +215,134 @@ pharmacometric/pharmacodynamic models (`torsten`) to structural equation models (`blavaan`) to general additive models for time series (`prophet`). -### Stan programming language +### Programming language Stan provides a domain-specific language for specifying smooth, -data-dependent target density functions on the logarithmic scale. For -Bayesian applications, this will be the posterior log density function -(up to an additive constant). For frequentist applications, it will -be a penalized likelihood function. Stan also provides a method for -efficiently coding posterior predictive quantities through simulation. - +data-dependent target density functions on the logarithmic scale. Stan is a differentiable programming language, meaning that Stan can automatically take first, second, and even higher-order derivatives of any log density it defines. Stan is also a probabilistic programming language in that its variables represent random variables and constants (at least when viewed from the Bayesian perspective). -Stan provide syntax highlighting through an `atom` specification. -Downstream tools using syntax highlighting for Stan include GitHub, -RStudio, VS Code, Jupyter notebooks, Emacs, Neovim, etc. - -### Stan inference engines - -Stan supports inference for probability models through general, -model-agnostic sampling techniques including the following two Markov -chain Monte Carlo (MCMC) methods. - -* Hamiltonian Monte Carlo (HMC) -* No-U-Turn Sampler (NUTS) - -Stan also includes the following two approximate inference -techniques based on variational inference, - -* Automatic Differentiation Variational Inference (ADVI) -* Pathfinder variational inference +Under the hood, Stan uses a standard programming language stack of +parser, intermediate representation layer, and code generator. These +are all written in the OCaml programming language. The code generator +produces a C++ class implementing the model defined by the Stan +program. -It also supplies an approximate method based on optimization and curvature. +The code for the parser, intermediate representation, and C++ +code generator is maintained in its own repository. -* Laplace approximation +* [`stanc3`](https://github.com/stan-dev/stanc3) (OCaml): Stan language compiler -Finally, Stan supplies industry-standard optimization methods, which -can be used for maximum likelihood estimates and as the basis for -Laplace approximations. +For Bayesian applications, the target log density will be the +posterior log density function (up to an additive constant). For +frequentist applications, it will be a penalized likelihood +function. Stan also provides a method for efficiently coding posterior +predictive quantities through simulation. -* Newton, BFGS, and L-BFGS optimization +### IDE support -Stan can estimate confidence intervals through the bootstrap as -explained in the posterior inference and model checking part of this -document. +Stan provide syntax highlighting and integrated development environment (IDE) +support as outlined on the following page. -### Stanc3: Stan-to-C++ transpilation +* [Editor and IDE support](https://mc-stan.org/tools/#editor-and-ide-support) -Stan uses a standard programming language stack of parser, -intermediate representation layer, and code generator. These are all -written in the OCaml programming language. The code generator -produces a C++ class implementing the model defined by the Stan -program. It is this latter feature, targeting an intermediate -language rather than assembly/machine language, which makes it a -transpiler rather than a compiler. -### Stan math library +## Stan math library -The C++ class produced by the `stanc3` transpiler depends on the Stan +The C++ class produced by the `stanc3` compiler depends on the Stan Math Library. The math library implements differentiable arithmetic operations, matrix and linear algebra operations, special mathematical functions, and special statistical functions. The Stan math library itself has four dependencies in addition to C++ itself, -* `tbb`: Intel Thread Building Blocks for thread management including -pools and synchronization, +* [`tbb`](https://www.intel.com/content/www/us/en/developer/tools/oneapi/onetbb.html) : Intel Thread Building Blocks for thread management including pools and synchronization, -* `boost`: Boost for general C++ tooling, special functions, random +* [`boost`](https://www.boost.org): Boost for general C++ tooling, special functions, random number generators, and densities. -* `Eigen`: Eigen for templated general matrix operations and linear +* [`Eigen`](https://libeigen.gitlab.io): Eigen for templated general matrix operations and linear algebra solvers, and -* `sundials`: Sundials for differential equation solvers. +* [`sundials`](https://computing.llnl.gov/projects/sundials): Suite of nonlinear and differential/algebraic equation solvers (SUNDIALS). The math library can call out to GPU for some operations (e.g., Cholesky factorization or matrix-matrix multiplication), but overall, it has been much more heavily optimized for CPUs. When using GPUs, there is a further dependency. -* `OpenCL`: OpenCL for GPU management. +* [`OpenCL`](https://www.khronos.org/opencl/): Open Computing Language (OpenCL) for GPU management. + + +## Additional R packages + +Stan provides a dedicated tool for posterior summarization and analysis in R, + +* [`posterior`](https://github.com/stan-dev/posterior) (R): posterior analysis tools + +along with complementary plotting components, + +* [`bayesplot`](https://github.com/stan-dev/bayesplot) (R): plotting tools + +There is a tool for cross-validation. + +* [`loo`](https://github.com/stan-dev/loo) (R): approximate leave-one out cross-validation + +There is a dashboard for visualizing and analyzing posterior results. + +* [`shinystan`](https://github.com/stan-dev/shinystan) (R): dashboard for exploring Stan model fits + +And there is a package + +* [`rstantools`](https://github.com/stan-dev/rstantools) (R): tools for developing R packages interfacing with Stan + + +## Stan inference engines + +Stan supports inference for probability models through general, +model-agnostic sampling techniques. + +### Markov chain Monte Carlo + +Stan's default inference method is Markov chain Monte Carlo sampling +using the no-U-turn sampler (NUTS). supports two Markov chain Monte +Carlo (MCMC) methods. There is more detail in the reference manual. + +* MCMC (_Reference Manual_) + + +### Variational inference + +Stan also includes the following two approximate inference +techniques based on variational inference, + +* [Automatic differentiation variational inference]( + https://mc-stan.org/docs/reference-manual/variational.html) (_Reference Manual_) + +* [Pathfinder variational + inference](https://mc-stan.org/docs/reference-manual/pathfinder.html) (_Reference Manual_) + + +### Laplace approximation + +Stan also supplies an approximate method based on optimization and curvature. + +* [Laplace approximation](https://mc-stan.org/docs/reference-manual/laplace.html) (_Reference Manual_) + +### Optimization for maximum likelihood and maximum a posterior estimation + +Stan supplies industry-standard optimization methods, which +can be used for maximum likelihood estimates and as the basis for +Laplace approximations. Stan supplies standard Newton optimization, +BFGS, and L-BFGS. + +* [Optimization](https://mc-stan.org/docs/reference-manual/optimization.html) + +Stan can estimate confidence intervals through the bootstrap as +explained in the posterior inference and model checking part of this +document. ## Background reading on Bayesian Statistics and Stan diff --git a/src/stan-users-guide/how-stan-works.qmd b/src/stan-users-guide/how-stan-works.qmd index 8393ffd06..8a35d1df0 100644 --- a/src/stan-users-guide/how-stan-works.qmd +++ b/src/stan-users-guide/how-stan-works.qmd @@ -1,5 +1,5 @@ --- -pagetitle: How Stan Works +page-title: How Stan Works --- @@ -16,27 +16,33 @@ to generate predictions based on parameter values, and (3) unconstraining transforms and their inverses. Stan's syntax is similar to C's. In Stan, every variable, constant, -and larger expression has a type that does not change. The three -primitive types are `int`, `real`, and `complex`. Integers may not be -used as parameters. The primitive container types are `vector`, -`row_vector`, and `matrix`. Arrays are homogeneous and array types -are available for any single type of value. Tuples are heterogeneous -and hold a sequence of values of possibly different fixed types. - -Variables can be declared with constraints such as lower and/or upper -bounds for scalars, and simplex, positive definiteness, and many other -constraints for vectors and matrices. Constraints on parameters are -maintained implicitly by Stan through unconstraining and constraining -transforms. Constraints on other variables are validated at the end -of their respective blocks. Local variables and function argument -types must be declared with unconstrained types (though they may hold -constrained values). Function argument types do not include sizes. - -A Stan program is organized into a sequence of blocks, all of which -are optional, but must appear in the following order. Stan code -is executed sequentially, and in each block, functions or variables -declared in previous blocks remain available for use (but not for -reassignment). +and larger expression [has a +type](https://mc-stan.org/docs/reference-manual/types.html) that does +not change. The three primitive types are `int`, `real`, and +`complex`. Integers may not be used as parameters. The primitive +container types are `vector`, `row_vector`, and `matrix`. Arrays are +homogeneous and array types are available for any single type of +value. Tuples are heterogeneous and hold a sequence of values of +possibly different fixed types. + +Variables can be declared with +[constraints](https://mc-stan.org/docs/reference-manual/transforms.html) +such as lower and/or upper bounds for scalars, and simplex, positive +definiteness, and many other constraints for vectors and matrices. +Constraints on parameters are maintained implicitly by Stan through +unconstraining and constraining transforms. Constraints on other +variables are validated at the end of the block in which they are +declared; they may not be modified later. Local variables and +function argument types must be declared with unconstrained types +(though they may hold constrained values). Function argument types do +not include sizes. + +A Stan program is organized into a sequence of +[blocks](https://mc-stan.org/docs/reference-manual/blocks.html), all +of which are optional, but must appear in the following order. Stan +code is executed sequentially, and in each block, functions or +variables declared in previous blocks remain available for use (but +not for reassignment). 1. `functions`: Define functions that can be called in the Stan program. Functions are executed when called. @@ -81,7 +87,7 @@ faster than variables defined in the transformed parameter block in other blocks and is much much less memory intensive. Executed once per algorithm iteration (*not* every log density evaluation). -Stan programs are transpiled to C++ by an OCaml program (`stanc3`). +Stan programs are compiled to C++ by an OCaml program (`stanc3`). The resulting C++ class definition is compiled and linked to the Stan math library for special functions and automatic differentiation. At this point, an object can be constructed from data. 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It provides models and programming techniques for coding statistical models in Stan. It does *not* provide a guide to installing or using any of the interfaces required to use Stan. -- Part 1 gives Stan code and discussions for several important classes +- Introduction: Outline of the Stan project's scope and +documentation, plus an outline of how Stan works. + +- Example Models: Stan code and discussions for several important classes of models. -- Part 2 discusses various general Stan programming techniques that are -not tied to any particular model. +- Programming Techniques: A discussion of general Stan +programming techniques that are not tied to any particular model. -- Part 3 introduces algorithms for calibration and model checking that -require multiple runs of Stan. +- Posterior Inference and Model Checking: An overview of +algorithms for calibration and model checking that require multiple +runs of Stan. -- The appendices provide an introduction to the `stanc3` compiler used in the - various interfaces to Stan, a style guide, and advice for users of BUGS and - JAGS. +- Appendices The appendices provide an introduction to the + `stanc3` compiler used in the various interfaces to Stan, a style + guide, and advice for users of BUGS and JAGS. ::: {.content-visible when-format="html"} [Download the pdf version of this manual](https://mc-stan.org/docs/{{< env STAN_DOCS_VERSION_PATH >}}/stan-users-guide-{{< env STAN_DOCS_VERSION_PATH >}}.pdf).