Module variational 

Variational Bayes EM for Hidden Markov Models with Dirichlet priors.

Reference: Beal 2003, “Variational Algorithms for Approximate Bayesian Inference”, §3.4.

Standard Baum-Welch places ML point estimates on π, A, B. VB-EM instead places conjugate Dirichlet priors on those parameters and maintains a factored variational posterior q(π, A, B) = q(π) · ∏_i q(A_i) · ∏_i q(B_i) whose factors are themselves Dirichlet distributions. The sufficient statistics from forward-backward (computed with expected log-parameters derived via the digamma function) update the Dirichlet concentration parameters in the M-step, and the ELBO is tracked for convergence.

Structs

VbHmmConfig

Configuration for Variational Bayes HMM training.

VbHmmResult

Result of Variational Bayes HMM training.

Functions

digamma

Scalar digamma function ψ(x) implemented via upward recursion followed by an asymptotic Stirling expansion.

dirichlet_log_normalizer

Log-normaliser of a Dirichlet distribution: log B(α) = Σ_i ln Γ(α_i) − ln Γ(Σ_i α_i).

log_gamma

Log-Gamma function ln Γ(x) via the Lanczos approximation with g = 7 and 9 pre-computed coefficients (Spouge 1994 / Numerical Recipes form).

variational_hmm

Run Variational Bayes EM for a discrete HMM with Dirichlet priors.