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 $\mathrm{HMM}_0$ and a candidate $\mathrm{HMM}_j$, let $\mu_j(r)$ denote the probability that $\mathrm{HMM}_j$ and $\mathrm{HMM}_0$ generate the same sequence of length~$r$. We show that if $\mathrm{HMM}_i$ is closer to $\mathrm{HMM}_0$ than $\mathrm{HMM}_j$, there exists a threshold $r^{*}$ -- often small -- such that $\mu_i(r)>\mu_j(r)$ for all $r\geq r^{*}$. Sampling $k$ independent fragments yields unbiased estimators $\hat{\mu}_j(r)$ whose differences are asymptotically normal, enabling a straightforward $Z$-test for the hypothesis $H_0\!:\,\mu_i(r)=\mu_j(r)$. By evaluating only short subsequences, the procedure circumvents full-sequence likelihood computation and provides valid $p$-values for model comparison.