Baum–Welch algorithm

In electrical engineering, computer science, statistical computing and bioinformatics, the Baum–Welch algorithm is used to find the unknown parameters of a hidden Markov model (HMM). It makes use of the forward-backward algorithm and is named for Leonard E. Baum and Lloyd R. Welch.

Hidden Markov models and the Baum–Welch algorithm were first described in a series of articles by Leonard E. Baum and his peers at the Institute for Defense Analysis in the late 1960s. One of the first major applications of HMMs was to the field of speech processing. In the 1980s, HMMs were emerging as a useful tool in the analysis of biological systems and information, and in particular genetic information. They have since become an important tool in the probabilistic modeling of genomic sequences.

A hidden Markov model describes the joint probability of a collection of "hidden" and observed discrete random variables. It relies on the assumption that the i-th hidden variable given the (i − 1)-th hidden variable is independent of previous hidden variables, and the current observation variables depend only on the current hidden state.

The Baum–Welch algorithm uses the well known EM algorithm to find the maximum likelihood estimate of the parameters of a hidden Markov model given a set of observed feature vectors.

Let ${\displaystyle X_{t}}$ be a discrete hidden random variable with ${\displaystyle N}$ possible values. We assume the ${\displaystyle P(X_{t}|X_{t-1})}$ is independent of time ${\displaystyle t}$, which leads to the definition of the time-independent stochastic transition matrix

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