I am given this Bayesian Network:
I need to calculate $p(+d|+a)$
What I have done up to now is:
$$ p(+d|+a) := \frac{p(+d,+a)}{p(+a)} $$ where \begin{align} p(+d,+a) & = \sum_X p(+d,+a,X) \\ & = p(+d) \sum_X p(+a | X, +d) p(X|+d) \\ & = 0.1 \times \big[ p(+a | +x,+d) p(+x|+d) + p(+a | -x,+d) p(-x|+d) \big] \\ & = 0.1 \times (0.9 \times 0.7 + 0.8 \times 0.3) \\ & = 0.087 \end{align}
How do I find the denominator $p(+a)$?
Is the following correct?
\begin{align} p(+a) & = \sum_{X,D} p(+a,X,D) \\ & = \sum_{X,D} p(+a|X,D) p(X|D) p(D) \\ & = \text{...} \\ & = 0.537 \end{align}
Thus $$ p(+d|+a) \approxeq 0.16 $$