more wip

docs/2. Complex Bayesian Network.md

@@ -768,25 +768,39 @@ graph TD
 
 ```math
 
-\begin{align}
+\begin{align*}
 
 P(Q_1|\hat{Y}_1,\hat{Y}_2,\hat{Y}_3) 
-    = & \frac{P(\hat{Y}_1,\hat{Y}_2,\hat{Y}_3|Q_1)P(Q_1)}{P(\hat{Y}_1,\hat{Y}_2,\hat{Y}_3)} \\
-    = & \frac{P(\hat{Y}_1,\hat{Y}_2,\hat{Y}_3|Q_1)P(Q_1)}{\sum\limits_{q_1}{P(\hat{Y}_1,\hat{Y}_2,\hat{Y}_3|Q_1)P(Q_1)}} \\
-    = & \frac{a_1(q_1)b_1(q_1)}{\sum\limits_{q_1}{a_1(q_1)b_1(q_1)}} \\
+    = & P(Q_1)\frac{P(\hat{Y}_1,\hat{Y}_2,\hat{Y}_3|Q_1)}{P(\hat{Y}_1,\hat{Y}_2,\hat{Y}_3)} \\
+    = & a_1(q_1)b_1(q_1) \\
 
 P(Q_2|\hat{Y}_1,\hat{Y}_2,\hat{Y}_3) 
-    = & \frac{P(\hat{Y}_1,\hat{Y}_2,\hat{Y}_3|Q_2)P(Q_2)}{P(\hat{Y}_1,\hat{Y}_2,\hat{Y}_3)}  \\
-    = & \frac{P(\hat{Y}_1,\hat{Y}_2,\hat{Y}_3,Q_2)}{\sum\limits_{q_2}{P(\hat{Y}_1,\hat{Y}_2,\hat{Y}_3,Q_2)}}  \\
-    = & \frac{P(\hat{Y}_2,\hat{Y}_3|Q_2)P(\hat{Y}_1, Q_2)}{\sum\limits_{q_2}P(\hat{Y}_2,\hat{Y}_3|Q_2)P(\hat{Y}_1, Q_2)} \\
-    = & \frac{a_5(q_2)b_5(q_2)}{\sum\limits_{q_2}{a_5(q_2)b_5(q_2)}} \\
-    
-\end{align}
+    = & \frac{P(\hat{Y}_2,\hat{Y}_3|Q_2,\hat{Y}_1)P(Q_2|\hat{Y}_1)}{P(\hat{Y}_2,\hat{Y}_3|\hat{Y}_1)}  \\
+    = & \frac{P(\hat{Y}_2,\hat{Y}_3|Q_2)}{P(\hat{Y}_2,\hat{Y}_3|\hat{Y}_1)}P(Q_2|\hat{Y}_1) \\
+    = & a_5(q_2)b_5(q_2)
+
+\end{align*}
 
 ```
 
+The likelihood of the model can be obtained by summing all normalization factors corresponding to the local connections:
+
+```math
+
+\begin{align*}
+
+P(\hat{Y}_1,\hat{Y}_2,\hat{Y}_3)
+    = & \frac{P(\hat{Y}_1,\hat{Y}_2,\hat{Y}_3)}{P(\hat{Y_1}|\hat{Y_2},\hat{Y_3})P(\hat{Y}_2,\hat{Y}_3|\hat{Y}_1)}P(\hat{Y}_1|\hat{Y}_2,\hat{Y}_3)\frac{P(\hat{Y}_2,\hat{Y}_3|\hat{Y}_1)}{P(\hat{Y}_2|\hat{Y}_3,\hat{Y}_1)P(\hat{Y}_3|\hat{Y}_2,\hat{Y}_1)}P(\hat{Y}_2|\hat{Y}_3,\hat{Y}_1)P(\hat{Y}_3|\hat{Y}_2,\hat{Y}_1) \\
+    = & c_1c_4c_5c_8c_9
+
+\end{align*}
+
+``` 
+
 ### Python example
 
+HIER WAS IK GEBLEVEN
+
 ```python
 import numpy as np