Bayesian Probability

Question

I have probability P(A|B=T,C=F,D=F,G=T) is this same as computing P(A|B=T)*P(A|C=F)*P(A|D=F) *P(A|G=T) ? P(A|B=T,C=F,D=F,G=T)=P(A|B=T)*P(A|C=F)*P(A|D=F) *P(A|G=T) ? where A is the child of B, C, D, G

thanks!

Answer 1

Conditional independence does not always imply independence: for example, independent random variables are rarely conditionally independent on their total or maximum.

Conditional independence does not imply independence; conditionally independent random variables uniform on (0,u) where u is uniform on (0,1), for example, are not independent.

Furthermore, If (X,Y,Z) are (mutually) independent, then X is conditionally independent of Y. Proof:

fx,y|z=fx,y,z/fz=fxfyfz/fz=fxfy=fx|zfy|z.

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