Hidden Markov Models (HMMs)

These are Doubly Stochastic Models in the following sense:

Suppose that our observations in the case of plain Markov Models were not deterministically output by the states. For example, assume S₁ did not always mean ``Rainy'', but rather ``90% chance of rain, 7% chance of cloud and a 3% chance of Sun''. A model that takes this into account is called a HMM.

It is completely specified by $\lambda = (A, B, \pi)$ where A and $\pi$ are the same as before and B is the observation symbol probability distribution given by $B = \{b_j(k): 0 \le j \le N, 0 \le k \le M \}$. We assume that there are M distinct output symbols, $V = \{v_1, v_2, \cdots, v_M\}$. (Again assume that N is implicit in A, M is implicit in V and V is implicit in B).

The probability that the kth symbol is produced by the ith state is b_j(k).

Hereafter, let $O = O_1O_2\cdots O_T$ be an observation sequence, $Q = q_1q_2\cdots q_T$ be a state sequence and $\lambda = (A, B, \pi)$ be a HMM.

Now, discuss how a given O can be generated given $\lambda$.

Three interesting questions about HMMs:

1.

Given O, and $\lambda$, what is $P(O\vert\lambda )$?

2.

Given O and $\lambda$, what is the optimal Q?

3.

How do we adjust the model parameters of $\lambda$ to maximize $P(O\vert\lambda )$?

We will only look at answering questions (1) and (2) in these two lectures. (3) is reserved to serve as illustration of the EM algorithm during lectures 3/4 on Bayesian Learning later.