Exact Inference in Bayesian Networks

I'm doing some exam study and came across a question I'm not really sure on. Consider the Bayesian network below:

Let's denote "Disease" with $D$ and "Symptom" with $S$.

I want to find $P(D \mid S_A, S_B)$ but it's not working for me. I've tried (among other things) to apply this rule:

$\qquad \displaystyle P(x_1, x_2, \dots , x_n) = P(x_i \mid parents(x_i))$

\qquad\begin{align} P(D \mid S_A, S_B) &= \frac{P(D, S_A, S_B)}{P(S_A, S_B)} \\ &= \frac{1}{\alpha} \cdot P(D) \cdot P(S_A \mid D) \cdot P(S_B \mid D, S_A) \end{align}

where $\alpha = P(S_A, S_B)$

However, I'm not really sure how to calculate my $\alpha$ value. I've tried making $D$ a hidden variable and calculating it like that, but my final answer was less than $P(D)$ was, so it can't be right.

Could someone please guide me through this? I'm not really sure what I'm doing wrong.

• I reformatted your post to be more readable; please take care of such things in the future. If you don't receive a good answer over the next few days, you may want to consider reposting on Cross Validated or Mathematics (one at a time). – Raphael Jun 11 '13 at 10:21