CS6220 Data Mining Techniques Hidden Markov Models, Exponential Families, and the Forward-backward Algorithm

A hidden Markov model (HMM) defines a joint probability distribution of a series of observations xt and hidden states zt for t = 1, . . . , T . We use x1:T and z1:T to refer to the full sequence of observations and states respectively. In a HMM the prior on the state sequence is assumed to satisfy the Markov property, which is to say that the probability of each state zt depends only on the previous state zt−1. In the broadest definition of a HMM, the states zt can be either discrete or continuous valued. Here we will discuss the more commonly considered case where zt takes on a discrete values zt ∈ [K] where [K] := {1, . . . , K}. We will use a matrix A ∈ RK×K and a vector π ∈ R to define the transition probabilities and the probability for the first state z1 respectively p(zt = l | zt−1 = k,A) := Akl, (1) p(z1 = k |π) := πk. (2)