adding bayes

doc/src/week36/week36.do.txt

@@ -883,15 +883,61 @@ p(X \cup Y)= p(X)+p(Y)-p(X \cap Y).
 !eblock
 
 
-!bblock The product rule is given by
+!bblock The product rule (aka joint probability) is given by
 !bt
 \[
-p(X \cup Y)= p(X,y)= p(X\vert Y)p(Y)=p(Y\vert X)p(x),
+p(X \cup Y)= p(X,Y)= p(X\vert Y)p(Y)=p(Y\vert X)p(x),
 \]
 !et
 where we read $p(X\vert Y)$ as the likelihood of obtaining $X$ given $Y$.
 !eblock
 
+If we have independent events then $p(X,Y)=p(X)p(Y)$.
+
+
+!split
+===== Marginal Probability =====
+
+The marginal probability is defined in terms of only of the set of variables $X,Y$. For a discrete probability we have
+!bblock Discrete Probability
+!bt
+\[
+p(X)=\sum_{i=0}^{n-1}p(X,Y=y_i)=\sum_{i=0}^{n-1}p(X\vert Y=y_i)p(Y=y_i)=\sum_{i=0}^{n-1}p(X\vert y_i)p(y_i).
+\]
+!et
+!eblock
+
+
+!split
+===== Conditional  Probability =====
+
+The conditional  probability, if $p(Y) > 0$, is 
+!bblock 
+!bt
+\[
+p(X\vert Y)= \frac{p(X,Y)}{p(Y)}=\frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}.
+\]
+!et
+!eblock
+
+
+!split
+===== Bayes' Theorem =====
+
+If we combine the conditional probability with the marginal probability and the standard product rule, we have
+!bt
+\[
+p(X\vert Y)= \frac{p(X,Y)}{p(Y)},
+\]
+!et
+we can rewrite this as
+
+!bt
+\[
+p(X\vert Y)= \frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}=\frac{p(Y\vert X)p(X)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)},
+\]
+!et
+Which is Bayes' theorem. More text to be added here by Friday September 10.
 
 
 !split