Bayes network predecessor relation

Question

I have the following Bayes network.

I know when "SHOES WET" becomes true, the probability of "GROUND WET" will change. But why will the probability of "RAINING" also change? And how can I calculate the probability of it?

Answer 1

If your assessment of the probability of "Ground wet" increases when you have observed "Shoes wet", and your assessment of "Rain" increases when your assessment of the probability of "Ground wet" increases, then these combined is likely to suggest that (given constant conditional probabilities) your assessment of "Rain" also increases when you have observed "Shoes wet".

You could use Bayes' theorem a few times, but perhaps an easier way of illustrating this is with a table of the possible combinations based on the conditional probabilities:

$$\begin{array}{llll} R&G&S&\text{Prob}\\\hline T& T& T& 0.09 \\ F& T& T& 0.06 \\ T& F& T& 0.0025 \\ F& F& T& 0.06 \\ T& T& F& 0.135 \\ F& T& F& 0.09 \\ T& F& F& 0.0225 \\ F& F& F& 0.54 \end{array}$$

By adding up the probabilities, this gives $P(R) =0.25$, $P(G)=0.375$ and $P(S)=0.2125$.

It then gives $P(R \mid S) = \frac{0.09+0.0025}{0.2125} \approx 0.435$ and $P(G \mid S) = \frac{0.09+0.06}{0.2125} \approx 0.706$, showing how observing wet shoes affects the probability there was rain and the the ground was wet.