(in the same series HMM (part I): recurrence equations for filtering and prediction)

Consider a Hidden Markov Model (HMM) with hidden states (for ), initial probability , observed states , transition probability and observation model . This model can be factorized as

. We will use the notation to represent the set .

In this post we will present the details of the method to find the smoothing distribution of a HMM, given a set of observations :

Our starting point is the marginal probability of given all the observations .