Digital Signal Processing (DSP) with Python Programming

3Inferences on HMM

In this chapter, we will give a brief overview of hidden Markov models (HMM) [BAS 93]. These models are widely used in many areas. They have a fundamental property that results in the existence of recursive algorithms, meaning that the number of operations and the size of the memory needed to calculate the required values do not increase with the number of samples. The best-known example of this is the Kalman filter [EMI 60].

Throughout this chapter, for simplicity, the notation (n₁ : n₂) will be used to denote the sequence of integer values from n₁ to n₂ inclusive.

3.1. Hidden Markov models (HMM)

A hidden Markov model (HMM) is a bivariate discrete-time process (X_n, Y_n), where X_n and Y_n are two real random vectors of finite dimension, such that:

• – X_n, n ≥ 0, is a Markov process, i.e. for any function f, the conditional expectation of f(X_n+1) given the σ-algebra generated by {X_s; s ≤ n} (the past until n) coincides with the conditional expectation of f(X_n+1) given the σ-algebra generated by {X_n}. If the conditional distributions have a density, it can be written as:
  [3.1]
• – Y_n, n ≥ 0, is a process such that the conditional distribution of Y₀, …, Y_n−1 given X₀, …, X_n−1 is the product of the distributions of Y_k conditionally on X_k. If the conditional distributions have a density, it can be written as:
  [3.2]
• – the initial r.v. X₀ has a known probability law. If ...