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I have been using Sebastian Thrun's course on AI and I have encountered a slightly difficult problem with probability theory.

He poses the following statement:

$$ P(R \mid H,S) = \frac{P(H \mid R,S) \; P(R \mid S)}{P(H \mid S)} $$

I understand he used Bayes' Rule to get the RHS equation, but fail to see how he did this. If somebody could provide a breakdown of the application of the rule in this problem that would be great.

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Taking it one step at a time: $$\begin{align} \mathsf P(R\mid H, S) & = \frac{\mathsf P(R,H,S)}{\mathsf P(H, S)} \\[1ex] & =\frac{\mathsf P(H\mid R,S)\,\mathsf P(R, S)}{\mathsf P(H, S)} \\[1ex] & =\frac{\mathsf P(H\mid R,S)\,\mathsf P(R\mid S)\,\mathsf P(S)}{\mathsf P(H, S)} \\[1ex] & =\frac{\mathsf P(H\mid R,S)\,\mathsf P(R\mid S)}{\mathsf P(H\mid S)}\frac{\mathsf P(S)}{\mathsf P(S)} \\[1ex] & =\frac{\mathsf P(H\mid R,S)\;\mathsf P(R\mid S)}{\mathsf P(H\mid S)} \end{align}$$

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You don't really need Bayes' Theorem. Just apply the definition of conditional probability in two ways. Firstly,

\begin{eqnarray*} P(R\mid H,S) &=& \dfrac{P(R,H\mid S)}{P(H\mid S)} \\ && \\ \therefore\quad P(R,H\mid S) &=& P(R\mid H,S)P(H\mid S). \end{eqnarray*}

Secondly,

\begin{eqnarray*} P(H\mid R,S) &=& \dfrac{P(R,H\mid S)}{P(R\mid S)} \\ && \\ \therefore\quad P(R,H\mid S) &=& P(H\mid R,S)P(R\mid S). \end{eqnarray*}

Combine these two to get the result.

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    $\begingroup$ I would vote up, but alas, I do not have enough reputation to vote up an answer to my own question :/ $\endgroup$ – slyaer May 14 '15 at 9:55
  • $\begingroup$ How to you get from $P(R|H,S)$ to $P(R,H|S)/P(H|S)$? $\endgroup$ – Mo Prog Jul 19 at 16:35
  • $\begingroup$ @MoProg From the definition of conditional probability. Just as you can have $P(A\mid B) = P(A,B)/P(B)$, which is from the standard definition, you can also have $P(A\mid B,C) = P(A,B\mid C)/P(B\mid C)$, for events $A,B,C$. $\endgroup$ – Mick A Jul 19 at 22:50
  • $\begingroup$ Sorry @MickA, but I still don't understand how you use the definition of conditional probablity $P(X|Y)= P(X,Y)/P(Y)$ to get from $P(A|B,C)$ to $P(A,B|C)/P(B|C)$. If I use the definition of conditional probability I would have $P(A|B,C) = P(A,B,C)/P(B,C)$ , because I will substitute $Y = B,C$. $\endgroup$ – Mo Prog Jul 22 at 7:57
  • $\begingroup$ @MoProg We can derive it this way: $\frac{P(A,B|C)}{P(B|C)} = \frac{P(A,B,C)/P(C)}{P(B,C)/P(C)}$ $ = \frac{P(A,B,C)}{P(B,C)} =P(A|B,C)$. $\endgroup$ – Mick A Jul 22 at 13:59

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