Hidden Markov Model (HMM)

The Hidden Markov Model (HMM) is a powerful statistical tool used in various AI applications like speech recognition, natural language processing, handwriting recognition, and bioinformatics. It allows us to model systems that follow a sequence of states where the actual state is not directly visible, but outputs (observations) dependent on the state are visible. HMMs are particularly useful when the system has hidden patterns that we want to uncover using observed data.

What is a Markov Chain?

A Markov Chain is a mathematical system that transitions from one state to another within a finite set of possible states. It is called “memoryless” because the next state depends only on the current state, not on the sequence of states that preceded it.

Example: Consider a simple weather model with three states: Sunny, Rainy, and Cloudy. If today is Sunny, there might be a 70% chance that tomorrow is also Sunny, a 20% chance of Cloudy, and a 10% chance of Rainy. The transition depends only on today’s weather, not the past few days.

What is the Markov Assumption?

The Markov Assumption states that the future state depends only on the current state and not on the sequence of past states. This simplifies the computation and modeling, making it practical to work with sequential data.

The Hidden Markov Model (HMM)

A Hidden Markov Model builds on the concept of a Markov Chain by introducing hidden states. In an HMM, the actual state is not observable (hidden), but each state produces an observable output. By analyzing the sequence of observed outputs, we can estimate the most likely sequence of hidden states.

For example, in speech recognition, the actual phoneme being spoken (hidden state) cannot be directly observed, but the sound wave (observation) can. HMM helps in predicting the sequence of phonemes from the observed sound signals.

HMM Components

Set of States

These are the possible hidden states of the system. For instance, in a weather model, states could be Sunny, Rainy, or Cloudy. The actual state is not directly visible.

Transition Probability Matrix

This matrix defines the probability of moving from one state to another. Each element A[i][j] represents the probability of transitioning from state i to state j.

Observation Likelihoods

Also known as emission probabilities, these define the probability of an observation being generated from a state. For example, if a state is “Rainy”, the observation might be “Carrying Umbrella” with a high likelihood.

Initial Probability Distribution

This specifies the probability of the system starting in each state. For example, there might be a 60% chance that the weather starts off as Sunny.

Output Independence

HMM assumes that the output (observation) at a given time depends only on the current state and is independent of previous outputs or states. This is known as the output independence assumption and simplifies modeling significantly.