Hidden Markov model

Builds on Markov chain Transition matrix Conditional independence Markov chain

Definition

Let $S$ and $O$ be finite sets, the hidden states and the observations. Let $\lambda$ be a probability distribution on $S$ , and let $A = (a_{ij})_{i, j \in S}$ be a transition matrix. Let $E = (e_i(o))_{i \in S, o \in O}$ be a matrix with $e_i(o) \ge 0$ and $\sum_{o \in O} e_i(o) = 1$ for each $i$ . A hidden Markov model with these parameters is a pair of processes $(X_n, Y_n)_{n \ge 0}$ with values in $S \times O$ such that:

$(X_n)$ is a Markov chain with initial law $\lambda$ and transition matrix $A$ ;
given $(X_n)_{n \ge 0}$ , the variables $Y_0, Y_1, \ldots$ are conditionally independent;
$P(Y_n = o \mid (X_k)_{k \ge 0}) = e_{X_n}(o)$ for all $n \ge 0$ and $o \in O$ .

Only $(Y_n)$ is observed.

Remarks

The joint law is $P(X_0 = i_0, Y_0 = o_0, \ldots, X_n = i_n, Y_n = o_n) = \lambda_{i_0} e_{i_0}(o_0) \prod_{k=1}^{n} a_{i_{k-1} i_k} e_{i_k}(o_k)$ . Example: let $S = \set{F, L}$ for a fair and a loaded die, with $a_{FL} = a_{LF} = 0.05$ , and let $O = \set{1, \ldots, 6}$ . Take $e_F(o) = 1/6$ for all $o$ , $e_L(6) = 1/2$ and $e_L(o) = 1/10$ for $o \le 5$ ; each row sums to $1$ , so clauses 1 to 3 define a model. Non-example: if $Y_n$ is allowed to depend on $Y_{n-1}$ as well as on $X_n$ , clause 3 fails. The observed process $(Y_n)$ alone is in general not a Markov chain. The forward algorithm computes the likelihood $P(Y_0 = o_0, \ldots, Y_n = o_n)$ in time of order $n \abs{S}^2$ , and the Viterbi algorithm finds a most likely hidden path.

Reading path

To understand this page from the ground up, read these 54 pages in order. The foundations come first.

Conditional independence concept
Construction of the reals from Cauchy sequences concept
Element definition
Integers object
Markov chain concept
Ordered field definition
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Transition matrix concept
Set definition
Ring of integers statement
Disjunction definition
Implication definition
Predicate definition
Ordered pair definition
Union definition
Sample space definition
Existential quantifier definition
Universal quantifier definition
The rationals form a field statement
Cartesian product definition
Subset definition
Uniqueness quantifier definition
The rationals form an ordered field statement
Binary relation definition
Universal set definition
Power set definition
Order on the reals definition
Function definition
Complement definition
Interval definition
Injective function definition
Preimage of a set definition
Sigma-algebra definition
Countable set definition
Event definition
Borel sigma-algebra definition
Probability axioms concept
Probability measure definition
Probability space definition
Conditional probability definition
Random variable definition
Independence of two events definition
Discrete random variable definition
Stochastic process concept
Random variable with a density definition
Conditional independence definition
Expected value definition
Conditional expectation definition
Markov property definition
Markov chain definition
Hidden Markov model definition

Tag	`CDVYCKZ2XZ`
URI	`m2://CDVYCKZ2XZ`
Kind	definition
Writing	W0 stub
MSC	62M05, 60J10
Defines	Hidden Markov model
Uses	Markov chain, Transition matrix, Conditional independence, Markov chain
Changed	2 November 2026

Hidden Markov model

Definition

Remarks

Reading path

Further reading

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