# Calculating priors of leaf nodes in Bayesian networks

I am studying the book Bayesian Artificial Intelligence. There is an example bayesian network see the figure: For this network the author calculated some numbers. I could not understand how author calculated highlighted probabilities for Xray and Dyspnoea when there is no evidence. See the figure please:

I am trying to reproduce the results represented in the table. If you can at least help me with no evidence column, I guess I can replicate reproduce remaining ones. However, I don't mind seeing the calculation for other reasoning cases (i.e. diagnostic or predictive or intercausal). Bel refers to posterior probability. E.g., Bel(D=T) refers to posterior probability of Dyspnoea being true.

It is a simple calculation using law of total probability: $$P(X=pos) = P(X=pos|C=T)P(C=T)+P(X=pos|C=F)P(C=F) = 0.9\,P(C=T) + 0.2\, P(C=F) = 0.9\cdot 0.011 + 0.2\cdot (1-0.011) = 0.2077 \approx 0.208$$ The other things P(D=T) etc are computed in the same way.
• You need to use Bayes theorem and the network. I.e. for P(S=T|D=T) (2nd row diagnostic) $P(S=T|D=T) = P(D=T|S=T)P(S=T)/P(D=T)$ (Bayes theorem) $P(D=T|S=T)=P(D=T|C=T)P(C=T|S=T)+P(D=T|C=F)P(C=F|S=T)$ – sega_sai Jan 10 '17 at 18:14