The standard HMM

Following the notation of Rabiner [57], there are $ T$ observation times. At each time $ 1\leq t \leq T$, there is a discrete state variable $ q_t$ which takes one of $ N$ values $ q_t\in\{S_1,S_2,\cdots, S_N\}$. According to the Markovian assumption, the probability distribution of $ q_{t+1}$ depends only on the value of $ q_t$. This is described compactly as a state transition probability matrix $ A$ whose elements $ a_{ij}$ represent the probability that $ q_{t+1}$ equals $ j$ given that $ q_{t}$ equals $ i$. The initial state probabilities are denoted $ \pi_i$, the probability that $ q_1$ equals $ S_i$.

It is a hidden Markov model because the states $ q_t$ are hidden from view; we cannot observe them. But, we can observe the random data $ O_t$ which is generated according to a PDF dependent on the state at time $ t$. We denote the PDF of $ O_t$ under state $ j$ as $ b_{j}(O_t)$.

The complete set of model parameters that define the HMM are

$\displaystyle \Lambda = \{\pi_j, a_{ij}, b_j \}

The Baum-Welch algorithm calculates new estimates $ \Lambda$ given an observation sequence $ {\bf O}=O_1 O_2\cdots O_T$ and a previous estimate of $ \Lambda$. The algorithm is composed of two parts: the forward/backward procedure, and the reestimation of parameters.

Baggenstoss 2017-05-19