Training Hidden Markov Models with Baum-Welch

This article explains how the Baum-Welch forward-backward algorithm trains Hidden Markov Models (HMMs) in artificial intelligence systems. We will explore the fundamentals of HMMs, why standard supervised learning fails when states are hidden, and how the Baum-Welch algorithm uses the Expectation-Maximization (EM) framework to iteratively learn both transition and emission probabilities from unlabelled sequential data.

The Challenge of Hidden States

A Hidden Markov Model is a statistical model used in AI to represent systems that change states over time. An HMM consists of two main components: hidden states (which cannot be observed directly) and observations (the visible data generated by those hidden states).

To be useful, an HMM needs two sets of probabilities:

  1. Transition Probabilities: The likelihood of moving from one hidden state to another.
  2. Emission Probabilities: The likelihood of a specific observation being generated from a specific hidden state.

If we could see the hidden states, we could simply count the transitions and emissions to calculate these probabilities. Because the states are hidden, we must learn these probabilities purely from the sequences of observations. This is the exact problem the Baum-Welch algorithm solves.

Step 1: Initialization

The Baum-Welch algorithm begins with guesswork. Before looking at the observation data, the AI system initializes the transition probabilities, emission probabilities, and initial state probabilities with random values (or rough estimates based on prior knowledge). The only constraint is that the probabilities must sum to one.

Step 2: The Expectation Step (Forward-Backward)

Once initialized, the algorithm evaluates how well the current probabilities explain the observed data. It does this using the "Forward-Backward" procedure, which constitutes the Expectation (E) step of the algorithm.

The Forward Pass

The forward pass calculates the "alpha" values. Moving from the beginning of the sequence to the end, the algorithm computes the probability of seeing the partial sequence of observations up to time t and ending up in a specific hidden state at time t. It does this by summing the probabilities of all possible paths that could lead to that state.

The Backward Pass

The backward pass calculates the "beta" values. Moving in reverse from the end of the sequence back to the beginning, the algorithm calculates the probability of seeing the remaining observations from time t+1 to the end, given that the system is in a specific hidden state at time t.

Combining Forward and Backward

By multiplying the forward (alpha) and backward (beta) probabilities together, the algorithm determines two crucial statistical expectations:

  • Gamma: The probability of being in a specific hidden state at time t, given the entire observation sequence.
  • Xi: The probability of transitioning from one specific hidden state at time t to another specific hidden state at time t+1, given the entire observation sequence.

Step 3: The Maximization Step (Updating Probabilities)

With the Gamma and Xi probabilities calculated, the algorithm moves to the Maximization (M) step. Here, the AI updates the model's parameters to better fit the data.

  • Updating Transition Probabilities: The algorithm sums the Xi probabilities over time to find the expected number of transitions from State A to State B. It then divides this by the sum of Gamma probabilities (the expected number of times the system was in State A). This ratio becomes the new transition probability.
  • Updating Emission Probabilities: The algorithm sums the Gamma probabilities for all instances where the system was in State A and a specific observation O occurred. It divides this by the total expected number of times the system was in State A. This ratio becomes the new emission probability for observation O.

Step 4: Iteration to Convergence

After the Maximization step, the HMM has a new, slightly more accurate set of transition and emission probabilities.

The Baum-Welch algorithm then repeats the process. It uses the newly updated probabilities to run the Forward-Backward pass again (E-step), which yields even more accurate Gamma and Xi values. Those values are then used to update the transition and emission probabilities once more (M-step).

Each iteration is mathematically guaranteed to increase the likelihood that the model generates the observed data. The AI repeats this loop until the changes in probabilities between iterations become negligibly small, meaning the algorithm has converged and learned the optimal parameters for the Hidden Markov Model.