Some Questions In Reading PRML

Here follows some questions that I met in reading Pattern Recognition and Machine Intelligence. Hope to get through them in the next week:

1. Why the Markov blanket of a node in a Bayesian network must include co-parents, whereas the Markov blanket of a node in a Markov Random Field includes only those direct neighbors. (ref. P383, Figure 8.26 and P385, Figure 8.28)
2. Lagrange multiplier with constraint g(X)>0. "Now, however, the sign of the Lagrange multiplier is crucial, because the function f(x) will only be at a maximum if its gradient is oriented away from the region g(x) > 0, as illustrated in Figure E.3. We therefore have ∇f(x) = −λ∇g(x) for some value of λ > 0."
3. On variational inference (P465, Equation 10.6): how to dissect out the dependence on one of the factors q_j(Z_j).
4. Factor graphs and the sum-product algorithm (also known as belief propagation).
5. Variational inference and expectation propagation.