fixed #5 #4

Author: WKaiZ

Gemfile

@@ -3,3 +3,5 @@ source "https://rubygems.org"
 gem "github-pages", group: :jekyll_plugins
 
 gem "webrick", "~> 1.8"
+
+gem 'faraday', '~> 1.10'

Gemfile.lock

@@ -27,9 +27,31 @@ GEM
       ffi (>= 1.15.0)
     eventmachine (1.2.7)
     execjs (2.10.0)
-    faraday (2.0.0)
+    faraday (1.10.4)
+      faraday-em_http (~> 1.0)
+      faraday-em_synchrony (~> 1.0)
+      faraday-excon (~> 1.1)
+      faraday-httpclient (~> 1.0)
+      faraday-multipart (~> 1.0)
+      faraday-net_http (~> 1.0)
+      faraday-net_http_persistent (~> 1.0)
+      faraday-patron (~> 1.0)
+      faraday-rack (~> 1.0)
+      faraday-retry (~> 1.0)
       ruby2_keywords (>= 0.0.4)
-    ffi (1.17.1)
+    faraday-em_http (1.0.0)
+    faraday-em_synchrony (1.0.1)
+    faraday-excon (1.1.0)
+    faraday-httpclient (1.0.1)
+    faraday-multipart (1.1.1)
+      multipart-post (~> 2.0)
+    faraday-net_http (1.0.2)
+    faraday-net_http_persistent (1.2.0)
+    faraday-patron (1.0.0)
+    faraday-rack (1.0.0)
+    faraday-retry (1.0.3)
+    ffi (1.17.1-arm64-darwin)
+    ffi (1.17.1-x86_64-linux-gnu)
     forwardable-extended (2.6.0)
     gemoji (4.1.0)
     github-pages (231)
@@ -211,6 +233,7 @@ GEM
       jekyll-feed (~> 0.9)
       jekyll-seo-tag (~> 2.1)
     minitest (5.25.4)
+    multipart-post (2.4.1)
     nokogiri (1.13.10-arm64-darwin)
       racc (~> 1.4)
     nokogiri (1.13.10-x86_64-linux)
@@ -256,6 +279,7 @@ PLATFORMS
   x86_64-linux
 
 DEPENDENCIES
+  faraday (~> 1.10)
   github-pages
   webrick (~> 1.8)
 

bayes-nets/d-separation.md

@@ -56,7 +56,7 @@ $$P(X | Z, Y) = P(X | Y)$$
 We can prove this statement as follows:
 
 $$P(X | Z, y) = \frac{P(X, Z, y)}{P(Z, y)}
-= \frac{P(Z|y) P(y|X) P(X)}{\sum_{x} P(X, y, Z)}
+= \frac{P(Z|y) P(y|X) P(X)}{\sum_{x} P(x, y, Z)}
 = \frac{P(Z|y) P(y|X) P(X)}{P(Z|y) \sum_{x} P(y|x)P(x)}
 = \frac{P(y|X) P(X)}{\sum_{x} P(y|x)P(x)}
 = P(X|y)$$
@@ -173,9 +173,14 @@ Any path in a graph from $$X$$ to $$Y$$ can be decomposed into a set of 3 consec
 
 ---
 
-**Active triples**: We can enumerate all possibilities of active and inactive triples using the three canonical graphs we presented below in the figures.
+We can enumerate all possibilities of active and inactive triples using the three canonical graphs we presented below in the figures.
+
+**Active triples**: 
 
 <img src="{{ site.baseurl }}/assets/images/active.PNG" alt="Active triples" />
+
+**Inactive triples**:
+
 <img src="{{ site.baseurl }}/assets/images/inactive.PNG" alt="Inactive triples" />
 
 ## 6.5.5 Examples