Statistical comparison of Hidden Markov Models via Fragment Analysis

Standard practice in Hidden Markov Model (HMM) selection favors the candidate with the highest full-sequence likelihood, although this is equivalent to making a decision based on a single realization. We introduce a \emph{fragment-based} framework that redefines model selection as a formal statistical comparison. For an unknown true model ${\mathrm{H}\mathrm{M}\mathrm{M}}_0$ and a candidate ${\mathrm{H}\mathrm{M}\mathrm{M}}_j$, let ${\mu }_j(r)$ denote the probability that ${\mathrm{H}\mathrm{M}\mathrm{M}}_j$ and ${\mathrm{H}\mathrm{M}\mathrm{M}}_0$ generate the same sequence of length~$r$. We show that if ${\mathrm{H}\mathrm{M}\mathrm{M}}_i$ is closer to ${\mathrm{H}\mathrm{M}\mathrm{M}}_0$ than ${\mathrm{H}\mathrm{M}\mathrm{M}}_j$, there exists a threshold $r^{\ast }$ – often small – such that ${\mu }_i(r)>{\mu }_j(r)$ for all $r\ge r^{\ast }$. Sampling $k$ independent fragments yields unbiased estimators ${\widehat{\mu }}_j(r)$ whose differences are asymptotically normal, enabling a straightforward $Z$-test for the hypothesis $H_0\text{\hspace{0.17em}⁣}:{\mu }_i(r)={\mu }_j(r)$. By evaluating only short subsequences, the procedure circumvents full-sequence likelihood computation and provides valid $p$-values for model comparison.