Simple Bayes Net Calculation

Question

I'm reading a paper and I'm trying to understand the Bayes net example they give. Here's the Bayes network in question:

Here's the simple calculation they perform using the net above:

How do they arrive at the conclusion that $P(H = 1|E = 1) = 0.73$? Here's my attempt at using Bayes rule to calculate the value: $P(H = 1|E = 1) = \frac{P(E=1|H=1)P(H=1)}{P(E=1)} = \frac{(0.8)(0.4)}{1} = 0.32$. Obviously, this is quite off. I suspect that my $P(E = 1)$ is wrong, but I have no idea what else it could be other than $1$.

What am I doing wrong here?

Answer 1

The correct value of $P(E = 1)$ is $0.44$. This gives you $P(H = 1|E = 1) \approx 0.73$ using the calculation above.

To calculate $P(E = 1)$, we do $P(E = 1) = P(E = 1|H = 0)P(H = 0) + P(E = 1|H=1)P(H=1) = (0.8)(0.4) + (0.2)(0.6) = 0.44$

Answer 2

First, see this article.

$\displaystyle p(H|E)\times p(E) = p(EH) \implies $

$\displaystyle p(H|E) = \frac{p(EH)}{p(E)} $

$\displaystyle = \frac{(0.4) \times (0.8)}{[(0.4) \times (0.8)] + {(0.6) \times (0.2)}} $

$\displaystyle = \frac{0.32}{0.32 + 0.12}.$