Bayesian network Problem

Question

I am confused in obtaining a term in Bayesian Formula. I have attached my partial solution in the image. How can I calculate the term P(X1/X3)??. Image attached

Answer 1

Hint: The most important part is to solve $p(A|x_1)$ and $p(\neg A|x_1)$.

$$p(A|x_1) = \frac{p(x_1|A)p(A)}{p(x_1)} = \frac{p(x_1|A)p(A)}{p(x_1|A)p(A) + p(x_1|\neg A)p(\neg A)} = 0.25$$

Then, $p(\neg A|x_1) = 0.75$.

(PS: Based on the Chapman-Kolmogorov equation, the result of $p(x_3|x_1)$ is obvious from $p(A|x_1)$ and $p(\neg A|x_1)$. Here, we still explain it in details.)

Note that $p(x_3|x_1) = p(x_3, A|x_1) + p(x_3, \neg A|x_1)$ (by the law of total probability). We calculate the following two terms

$$p(x_3,A|x_1) = p(x_3|A)p(A|x_1) = 0.05,$$ $$p(x_3,\neg A|x_1) = p(x_3|A)p(\neg A|x_1) = 0.45,$$

where $p(x_3|A,x_1) = p(x_3|A)$ and $p(x_3|\neg A,x_1) = p(x_3|\neg A)$.

Now, $p(x_3|x_1) = p(x_3, A|x_1) + p(x_3, \neg A|x_1) = 0.05 + 0.45 = 0.5$.