The following diagram represents a simple, 3-step HMM represented as a 6-node Bayesian Network; the same one as before.
The Zi are the state variables; the Xi are the observable "evidence". The state transition model is the CPD P(Z|Z;-1). The sensor model or emission probabilities are the CPD P(Xi|Z). The initial state is described by the CPD P(Zo). You don't have any numeric probabilities, so just work on the algebra, as in the previous question.
(a) Show that
P(X2|Z1) = 2P(X2|Z2)P(Z2|Z1)
Hint: Use conditional independence given Z2. Don't sum over all nuisance variables!
(b) Show that P(X2, Xi|Zo) = Î P(Xi|Zi)P(Zi|Zo)
Hint: Use conditional independence given Z1. Note: This might be the first tricky derivation so far.
(c) Show that P(X2|Zo)P(X1, Xo|Zo)P(Z1|Xo, X1,X2) = P(Zi|Xo, X1,X2) P(Xo, X1, X2)
Hint: Use conditional independence given Z1. Note: The factor P(X2|Z1) was derived directly above, and the factor P(Zi|X1,Xo) earlier in the assignment. This formula tells us how all the data affects the state at a single time point: Observable evidence X2 affects Z1 from one direction using the backward algorithm; observable evidence Xo, X1 affects Z1 from the other direction using the forward algorithm. To see this more clearly, we have to use a larger HMM; but the derivations are essentially the same. See the next question.
Clarification: As before, the factor P(Xi|Xo) in the numerator is constant relative to the query variable Z1. As a result, we might rewrite the formula for Q5.c as follows: P(Zi|Xo, X1,X2) = aP(X2|Z1)P(Zi|X1, Xo)