I'm taking a course on bayesian statistics and I'm having trouble with one assigment, it goes like this:
Construct an example in which two variables have a common effect, and the presence of one of the variables makes the other more likely. More precisely, consider a v-structure X → Z ← Y over three binary-valued variables. Construct a CPD P(Z|X, Y ) such that:
• X and Y both in crease the probability of the effect, that is P(Z = T|X = T) > P(Z = T) and P(Z = T|Y = T) > P(Z = T).
• each of X and Y increases the probability of the other one, that is P(X = T|Z = T) < P(X = T|Y = T, Z = T) and P(Y = T|Z = T) < P(Y = T|X = T, Z = T).
For what I understand, if I draw this V-structure the like it says, X and Y are independent and therefore if I apply Bayes rule on the first condition:
P(Z = T|X = T) = P(X = T|Z = T)P(Z = T)/P(X=T) > P(Z = T)
and if I cancel terms:
P(X = T|Z = T) > P( X = T )
but the v-structure of the graph tells me that X is independent of Z...
And the other condition also bothers me, as it states that for example
P( X = T ) < P ( X = T | Y = T )
but again the graph states independence between X and Y ( or I'm wrong? ) And introducing this dependence with and edge between X and Y will induce a loop in the graph.
I tried to think with the variables being X = Burglary , Y = Have a broken window, and Z = Alarm. In the sense that having a broken window increase the change of a burglary, and having a burglary increase the chance of having a broken window ( second condition ) and any of those events increase the change of the alarm being triggered.
Well, any help in sorting this out ( finding the CPD given the conditions) is greatly appreciated.
Thanks in advance