Finding the probability of an observation V: The backward procedure
Let ?i(t) = P(ot…oT | Xt = i, ?) be the backward variables.
Backward variables can be calculated working backward through the treillis as follows:
Initialization: ?i(T+1) = 1, 1? i ? N
Induction: ?j=1N aijbijot?j(t+1), 1? t ?T, 1? i ? N
Total: P(O|?)=?i=1N?i?i(1)
Backward variables can also be combined with forward variables:
P(O|?) = ?i=1N ?i(t)?i(t), 1? t ? T+1