The forward score for the merged state s at time t is just the sum of the two split scores: while the backward score is a weighted sum of the split scores:β where p 1 and p 2 are the relative (posterior) frequencies of the states s 1 and s 2 .
the likelihood after merging s 1 and s 2 at time t can be computed from these merged forward and backward scores as: where the second sum is over the other substates of x t , i.e.