Entropy Rate of Hidden Markov Processes

Computing the entropy rate of HMMs — the fundamental limit of lossless compression for structured sequences

Controls

HMM View
Entropy
Sequence

Hidden Markov Model

3
0.80
0.10

Entropy Estimates

Empirical H rate—
LZ estimate—
Plugin H(1)—
Sequence length—
Alphabet size—

Theory

For an HMM with transition T and emission B, the entropy rate exists but has no closed form. It's computed as a limit via the transfer matrix.
h = lim_{n→∞} H(X_n | X_1..X_{n-1})
h = -Σ π_i Σ_j T_ij log T_ij
The true entropy rate of the observed (noisy) sequence is strictly LESS than that of the hidden chain, but has no closed form — a fundamental open problem.