Problem description
With the gaussian copula, you are modeling a distribution with the cumulative distribution function $\text{F}(\mathbf{x}) = \Phi_R(\Phi^{-1}(\text{G}_1(x_1)), \dots, \Phi^{-1}(\text{G}_d(x_d)))$, where $\text{G}_i$ is the cumulative distribution function of the $i$-th margin. If I want to call the method probability_density(x), then as far as I can tell, it gives me $p(z) = \phi_R(\Phi^{-1}(\text{G}_1(x_1)), \dots, \Phi^{-1}(\text{G}_d(x_d)))$. But doesn't that function give me the probability of $z$, given by $z = (\Phi^{-1}(\text{G}_1(x_1)), \dots, \Phi^{-1}(\text{G}_d(x_d)))$, instead of $p(x)$? Because to get $f(x)$, you take the derivative of $\text{F}(x)$, so you need to apply the chain rule on the right hand side, to get $p(x) = f(x) = \phi_R(\Phi^{-1}(\text{G}_1(x_1)), \dots, \Phi^{-1}(\text{G}_d(x_d))) \prod _{i=1}^{d} \frac{g_i(x)}{\phi(\Phi^{-1}(\text{G}_i(x_i)))}$. Are you accounting for this anywhere? Because in the documentation, it says the method gives probability density values for points in x.
Problem description
With the gaussian copula, you are modeling a distribution with the cumulative distribution function$\text{F}(\mathbf{x}) = \Phi_R(\Phi^{-1}(\text{G}_1(x_1)), \dots, \Phi^{-1}(\text{G}_d(x_d)))$ , where $\text{G}_i$ is the cumulative distribution function of the $i$ -th margin. If I want to call the method probability_density(x), then as far as I can tell, it gives me $p(z) = \phi_R(\Phi^{-1}(\text{G}_1(x_1)), \dots, \Phi^{-1}(\text{G}_d(x_d)))$ . But doesn't that function give me the probability of $z$ , given by $z = (\Phi^{-1}(\text{G}_1(x_1)), \dots, \Phi^{-1}(\text{G}_d(x_d)))$ , instead of $p(x)$ ? Because to get $f(x)$ , you take the derivative of $\text{F}(x)$ , so you need to apply the chain rule on the right hand side, to get $p(x) = f(x) = \phi_R(\Phi^{-1}(\text{G}_1(x_1)), \dots, \Phi^{-1}(\text{G}_d(x_d))) \prod _{i=1}^{d} \frac{g_i(x)}{\phi(\Phi^{-1}(\text{G}_i(x_i)))}$ . Are you accounting for this anywhere? Because in the documentation, it says the method gives probability density values for points in x.