/ˈhɪd.ən ˈmɑːrkɒv ˈmɒd.əl/
noun … “seeing the invisible through observable clues.”
Hidden Markov Model (HMM) is a statistical model that represents systems where the true state is not directly observable but can be inferred through a sequence of observed emissions. It extends the concept of a Markov Process by introducing hidden states and probabilistic observation models, making it a cornerstone in temporal pattern recognition tasks such as speech recognition, bioinformatics, natural language processing, and gesture modeling.
Formally, an HMM is defined by:
A finite set of hidden states S = {s₁, s₂, ..., s_N}
A transition probability matrix A = [a_ij], where a_ij = P(s_j | s_i)
An observation probability distribution B = [b_j(k)], where b_j(k) = P(o_k | s_j)
An initial state distribution π = [π_i], where π_i = P(s_i at t=0)
The model generates a sequence of observed variables O = {o₁, o₂, ..., o_T} while the underlying state sequence S = {s₁, s₂, ..., s_T} remains hidden. Standard HMM algorithms include the Forward-Backward algorithm for evaluating sequence likelihoods, the Viterbi algorithm for decoding the most probable state path, and the Baum-Welch algorithm for parameter estimation via Maximum Likelihood Estimation.
Hidden Markov Models are closely connected to multiple concepts in statistics and machine learning. They rely on Markov Processes for state dynamics, Probability Distributions for modeling observations, and Expectation Values and Variance for understanding state uncertainty. HMMs also serve as the foundation for sequence models in natural language processing, biosequence alignment, and temporal pattern recognition, often interfacing with machine learning techniques such as Gradient Descent when extended to differentiable architectures.
Example conceptual workflow for applying an HMM:
define the set of hidden states and observation symbols
initialize transition, observation, and initial state probabilities
use training data to estimate parameters via Baum-Welch algorithm
compute sequence likelihoods using Forward-Backward algorithm
decode the most probable hidden state sequence using Viterbi algorithm
analyze results for prediction, classification, or temporal pattern recognitionIntuitively, a Hidden Markov Model is like trying to understand a play behind a curtain: you cannot see the actors directly, but by watching their shadows and hearing the lines (observations), you infer who is on stage and what actions are taking place. It converts hidden dynamics into structured, probabilistic insights, revealing patterns that are otherwise invisible.