ABSTRACT
Three comments are provided below. First, in maximizing Q = Qi + Q2, Qi and Q2 can be maximized independently-Q1 contains only the parameters in the trend functions and in the residuals, while Q2 involves just the parameters in the Markov chain. Second, in maximizing Q, the weights in Eq. 10.9 and Eq. 10.10, P(st = ilof , (1)0) and P(st = st}1
= 4)0 which we denote by ryt(i) and respectively, are treated as known constants due to their conditioning on 4)o. They can be computed efficiently via the use of the forward and backward probabilities. The posterior state transition probabilities are
j) = at(i)A-F1
exP(Nt+ (i)) P(oT14o)
for t = 1, 2, ..., T —1. (Note that .T•(i, j) has no definition.) The posterior state occupancy probabilities can be obtained by summing t (i, j) over all the destination states j
7t(i) = E et (i, (10.12)
(10.6)
(10.8)
for t = 1, 2, ..., T — 1. 7T(i) can be obtained by its very definition:
7T(i) = P(sT = il0f, 4,o) = P(sT = nT(i) P(of1430) P(0D)0) .
