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Improved order selection method for hidden Markov models: a case study with movement data

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A double-penalized likelihood method recovers hidden Markov states when AIC and BIC overfit.

desk verdict A genuinely new extension of double-penalized order selection to non-stationary HMMs, backed by a solid simulation study; the main caveats are a heuristic penalty with no consistency proof and an abstract that overstates the results. read the letter →

arxiv 2411.18826 v2 pith:7YHVSHLU submitted 2024-11-28 stat.ME stat.AP

Fanny Dupont , Marianne Marcoux , Nigel Hussey , Marie Auger-Méthé This is my paper · ORCID
classification stat.MEstat.AP MSC 62M0562F1262P12
keywords AnimalmovementDoublepenalizedmaximumlikelihoodestimate(DPMLE)HiddenMarkovmodelOrderselectionNon-stationarymisspecificationSCADInformationcriteria
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes a double penalized maximum likelihood estimate (DPMLE) that estimates the number of hidden states and all model parameters of a non-stationary hidden Markov model in a single pass, rather than fitting many candidate models and comparing information criteria. The method adds two penalty terms to the log-likelihood: one discourages states the process almost never occupies, and the other uses the SCAD penalty to fuse states whose emission distributions are nearly identical. Using forward and backward probabilities to approximate stationary probabilities, the DPMLE extends an existing stationary-HMM technique to models with time-varying covariates. In a simulation study spanning six forms of misspecification common in movement data, the DPMLE identified the true three states more often than AIC and BIC, and on narwhal tracking data it selected two interpretable behavioural states where AIC and BIC chose four or more poorly separated ones. The result is a one-stage order selection tool for ecologists that is more robust to misspecified models than standard criteria.

What carries the argument

The load-bearing object is the double penalized log-likelihood $\tilde l_M(\Psi|y) = l_M(\Psi|y) + C_N \sum_{j=1}^N \log \pi_j - \sum_{j=1}^{N-1} p_{\lambda_M}(\eta_j)$, with $\eta_j = \mu_{j+1} - \mu_j$ under an ordering $\mu_1 \le \mu_2 \le \cdots \le \mu_N$. The first penalty term suppresses states with tiny stationary probabilities (type I overfitting), while the second, the smoothly clipped absolute deviation (SCAD) penalty, shrinks nearly equal means together so redundant states merge (type II overfitting). For non-stationary HMMs, the stationary probability $\pi_j$ is replaced by $\hat\pi_j = \frac{1}{T}\sum_t P[S_t=j \mid Y=y]$, computed from forward and backward probabilities, and the SCAD term is maximized via a local linear approximation inside the EM algorithm. The Group-Sort-Fuse procedure extends the same penalties to multivariate emission distributions, as used for step length and turning angle in the narwhal analysis.

What would settle it

Generate non-stationary HMM data from a known covariate model with three states, fit the DPMLE at increasing sample sizes such as T = 5,000, T = 50,000, and T = 500,000, and check whether the proportion of trials recovering three states converges to 100%. If the success rate plateaus below 100% or the penalty merges true states as T grows, the time-averaged approximation to the stationary probabilities does not preserve consistency.

Watch

Extended reading notes

Core claim

The central claim is that the DPMLE, which maximizes the log-likelihood penalized by a term on the stationary probabilities and a SCAD term on the gaps between state-dependent means, estimates both the number of states and the parameters of non-stationary HMMs at once. For non-stationary chains the stationary probabilities are replaced by time-averaged posterior state probabilities obtained from forward and backward recursions. The paper reports that this double penalized estimator outperformed AIC and BIC in recovering the true number of states under model misspecification, with the non-stationary DPMLE exceeding 85% success in ten of twelve simulation settings. In the narwhal case study it selected a two-state model with stable, interpretable behaviours, while AIC and BIC selected four to eight states with overlapping and unstable components.

Load-bearing premise

The method assumes that the same two-penalty objective that has proven consistency for stationary HMMs remains valid when the stationary probabilities are replaced by time-averaged posterior state probabilities in non-stationary models; the paper states that this large-sample justification is not yet established.

Editorial extensions

If this is right

  • Practitioners can select the number of states and estimate parameters in one optimization, avoiding the two-stage fit-every-order workflow.
  • Under misspecification, the DPMLE gives higher success rates than AIC and BIC, and its performance degrades less as sample size grows.
  • The method supports time-varying covariates in the transition process, providing a one-stage order selection tool for non-stationary HMMs where none existed before.
  • Applied to the narwhal data, the non-stationary DPMLE selects a two-state model with stable states and clear behavioural interpretation, while AIC and BIC select models with more states that are harder to interpret.
  • The Group-Sort-Fuse adaptation extends the approach to multivariate emissions, so it applies to movement models with multiple observation variables.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: if the time-averaged posterior approximation preserves the consistency arguments from the stationary setting, the DPMLE could be applied to any non-stationary HMM with covariates, not just movement data, for example in finance or speech processing.
  • Beyond the paper: because the penalty orders states by mean parameters, the method is most natural when states differ mainly in location; states differing chiefly in variance or shape may need alternative penalty constructions, which the paper only gestures at.
  • Beyond the paper: the hyperparameter selection rule (NIC) is BIC-like and the paper notes it is only rigorously derived for Gaussian models, so a cheaper or more robust selection rule for the two tuning parameters would be a natural next step once the computational cost of the DPMLE drops.
  • Beyond the paper: the narwhal result suggests AIC and BIC may systematically overstate behavioural complexity in telemetry studies; re-analysing published movement datasets with the DPMLE would be a concrete test of how often information criteria select unstable or overlapping states.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a double penalized maximum likelihood estimator (DPMLE) for simultaneous order selection and parameter estimation in non-stationary hidden Markov models (HMMs), extending the stationary HMM method of Hung et al. (2013). The method penalizes small stationary (or time-averaged posterior) probabilities and close state-dependent means, with hyperparameters selected by a BIC-type criterion (NIC). The authors evaluate the method on six simulation scenarios, including misspecified and non-stationary settings, and compare it with AIC and BIC. They also apply the method to narwhal movement data, where DPMLE selects two states while AIC and BIC select more. The core empirical claim is that DPMLE outperforms AIC and BIC in identifying the correct number of states under model misspecification, particularly for non-stationary HMMs.

Significance. If the empirical claims hold, the paper would introduce the first double-penalized likelihood order-selection procedure for non-stationary HMMs, a setting where standard information criteria are known to be unreliable. The simulation study covers several practically relevant misspecification scenarios, and the narwhal case study illustrates a plausible real-data outcome. The authors are also transparent about the lack of asymptotic theory for the non-stationary extension and about the limitations of the NIC hyperparameter selection. However, the abstract overstates the results, the non-stationary penalty is introduced without formal justification, and the reported success rates lack uncertainty quantification. These issues affect the strength of the central claim and need to be addressed before the paper can be recommended for publication.

major comments (4)
  1. [Abstract, paragraph 4; Results, Section 3; Tables A1–A2] The abstract states that the proposed method 'outperformed AIC and BIC in identifying the correct number of states under model misspecification,' but the simulation results show two settings where BIC with covariates is superior: Scenario 4 (heterogeneity in emission) with T=5,000 (BIC cov 87% vs. DPMLE cov 71%) and Scenario 5 (violation of conditional independence) with T=5,000 (BIC cov 100% vs. DPMLE cov 99%). The results section acknowledges this, but the abstract and the closing claim of the Discussion do not. Please qualify the outperformance claim to reflect the settings where BIC is competitive, or restrict the claim to the majority of scenarios.
  2. [Estimation, Eq. (17); Discussion, paragraph 3] The non-stationary DPMLE replaces the stationary probability π_j with the time-averaged posterior probability \hatπ_j(Ψ) = T^{-1} Σ_t P(S_t = j | Y = y) in the type-I penalty. This is a data-dependent quantity that is not a parameter of the model, and the consistency theorem of Hung et al. (2013) applies to an objective with the stationary distribution, not this modified objective. The paper explicitly defers formal asymptotic analysis to future work. This is a load-bearing gap because the central claim concerns non-stationary HMMs, and the simulations are the only evidence that the approximation behaves as intended. At minimum, please state this as a conjecture, provide diagnostic checks (e.g., convergence of \hatπ_j to the empirical occupancy in the simulated settings), or discuss conditions under which the approximation is expected to be valid.
  3. [Estimation, Eq. (19); Discussion, paragraph 3] The hyperparameters λ_M and C_N are selected by a BIC-type criterion (NIC), which reintroduces an information-criterion step inside the proposed one-stage procedure. The paper acknowledges that NIC 'has only been rigorously derived in Gaussian models' and may share BIC's sample-size dependence, but no sensitivity analysis is provided for the choice of the hyperparameter grid or for the impact of NIC on the final order estimate. Since the method's main selling point is its departure from information criteria, the reliance on NIC should be discussed more explicitly and its influence on the results should be assessed (e.g., by varying the grid or comparing with a small set of fixed hyperparameters).
  4. [Results; Tables A1–A2; Figure 3] Success rates are reported as percentages from 100 simulation trials without any measure of uncertainty. With 100 trials, a reported difference of, say, 14 percentage points (e.g., Scenario 4, T=5,000: BIC cov 87% vs. DPMLE cov 71%) has a standard error of roughly 4–5 percentage points. Several comparisons in the text rely on differences of this magnitude. Please report standard errors or confidence intervals for the success rates, or at least state the number of trials explicitly in the figure and tables, so that readers can assess whether observed differences are meaningful.
minor comments (5)
  1. [Results, Section 3] The sentence 'Both DPMLE methods outperformed BIC and AIC, and had a success rate higher than 99% for more than half of the scenarios' is ambiguous because 'more than half' counts settings, not scenarios; please specify how many of the twelve settings this refers to.
  2. [Estimation, Eq. (17)] The notation \hatβ^{(p+1)} is used both for the regression coefficients in the multinomial logit (Eq. 5–6) and for the backward probabilities; this is confusing. Please use distinct symbols for the two quantities.
  3. [Estimation, Eq. (15)] The expression π = 1(I_N − Γ + U)^{-1} is standard, but the derivation of the stationary distribution from the tpm is not shown; a brief explanation or a reference to Eq. (3) would help readers not familiar with this identity.
  4. [Discussion, paragraph 1] The sentence 'With a higher upper bound, the method is expected to handle greater levels of misspecification' is speculative; the simulation study only considers N=4. Please soften this or provide evidence from a scenario with a larger upper bound.
  5. [Section 2, Simulation study] The description of scenario 5 is incomplete: the text says the mean parameters of the first state are generated with an AR(1) process, but it is not stated whether the fitted models account for this autocorrelation or whether the conditional independence assumption is violated by design; a sentence clarifying the exact misspecification would be useful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the DPMLE is a well-defined penalized estimator, its consistency citations are external, and its non-stationary extension is an openly acknowledged approximation validated by simulation with known truth.

full rationale

The paper's central claim — that the double penalized likelihood estimator (DPMLE) can simultaneously estimate the number of states and parameters of non-stationary HMMs and outperform AIC/BIC under misspecification — is not obtained by a circular reduction. The DPMLE objective is explicitly defined in Eq. (9), and the non-stationary version in Eq. (17) replaces stationary probabilities with time-averaged posterior state probabilities, a data-dependent approximation. This is not a renaming of the target quantity: the order estimate is the number of distinct fitted means after penalized maximization, and the method is tested on simulated datasets with known true order, providing independent empirical grounding. The consistency properties are cited to external work (Chen and Khalili 2008; Hung et al. 2013), with no present-author self-citation carrying the statistical derivation. The authors' own prior ecological papers are used only for context, not for the methodological argument. The use of a BIC-type criterion (NIC, Eq. 19) to select tuning parameters is the closest thing to an internal loop, but it selects hyperparameters rather than the order directly, and the paper explicitly discloses that NIC is BIC-type and that its theoretical derivation is limited. The deferred asymptotic analysis of the non-stationary DPMLE (Discussion) is a limitation, not a circular step. No specific equation or fitted parameter is shown to reduce to the input by construction, so no circularity is flagged.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central method relies on two tuning hyperparameters selected by a BIC-type criterion, and on several assumptions that are acknowledged as heuristic or unproven in the paper. No new physical or conceptual entities are introduced.

free parameters (2)
  • lambda_M (SCAD penalty tuning parameter) = selected via NIC from a random search over log(M*lambda_M) in [1,5] (simulations) or 100 random pairs (case study)
    Controls the strength of the penalty that shrinks differences between state-dependent means; its value is data-dependent and chosen by the Narwhal Information Criterion.
  • C_N (stationary probability penalty constant) = selected jointly with lambda_M via NIC; search interval not given on log scale
    Controls the penalty on the stationary probabilities (or their forward-backward proxy) that discourages nearly empty states; chosen by the same BIC-type criterion.
assumptions (3)
  • ad hoc to paper Regularity conditions for the consistency of the stationary DPMLE are met, and the same conditions extend to the non-stationary penalized likelihood.
    Stated in Section 'Estimation': 'we assume the regularity conditions necessary for the consistency of the stationary DPMLE are met although they are often violated in practice.' No proof is offered for the non-stationary case, and the discussion defers formal asymptotics to future work.
  • ad hoc to paper The time-averaged posterior probability (from forward-backward recursions) is a valid substitute for the stationary probability in the type I penalty for non-stationary HMMs.
    Introduced in Eq (17) without theoretical justification; it is motivated heuristically as the proportion of time spent in each state.
  • ad hoc to paper The BIC-type Narwhal Information Criterion (NIC) consistently selects the tuning hyperparameters lambda_M and C_N in this HMM setting.
    The paper notes NIC has only been rigorously derived in Gaussian models (Discussion, paragraph 4), and that overfitting issues highlighted by Wang et al. (2007) warrant further investigation.

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Cite this review

Pith. "Pith review of Improved order selection method for hidden Markov models: a case study with movement data." pith.science (2026). https://pith.science/paper/7YHVSHLU

@misc{pith2026241118826,
  author       = {Pith},
  title        = {Pith review of: Improved order selection method for hidden Markov models: a case study with movement data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7YHVSHLU}},
  note         = {Machine review of arXiv:2411.18826}
}
read the original abstract

Hidden Markov models (HMMs) are a versatile statistical framework commonly used in ecology to characterize behavioural patterns from animal movement data. In HMMs, the observed data depend on a finite number of underlying hidden states, generally interpreted as the animal's unobserved behaviour. The number of states is a crucial parameter, controlling the trade-off between ecological interpretability of behaviours (fewer states) and the goodness of fit of the model (more states). Selecting the number of states, commonly referred to as order selection, is notoriously challenging. Common model selection metrics, such as AIC and BIC, often perform poorly in determining the number of states, particularly when models are misspecified. Building on existing methods for HMMs and mixture models, we propose a double penalized likelihood maximum estimate (DPMLE) for the simultaneous estimation of the number of states and parameters of non-stationary HMMs. The DPMLE differs from traditional information criteria by using two penalty functions on the stationary probabilities and state-dependent parameters. For non-stationary HMMs, forward and backward probabilities are used to approximate stationary probabilities. Using a simulation study that includes scenarios with additional complexity in the data, we compare the performance of our method with that of AIC and BIC. We also illustrate how the DPMLE differs from AIC and BIC using narwhal (Monodon monoceros) movement data. The proposed method outperformed AIC and BIC in identifying the correct number of states under model misspecification. Furthermore, its capacity to handle non-stationary dynamics allowed for more realistic modeling of complex movement data, offering deeper insights into narwhal behaviour. Our method is a powerful tool for order selection in non-stationary HMMs, with potential applications extending beyond the field of ecology.

Figures

Figures reproduced from arXiv: 2411.18826 by the authors.

Figure 1
Figure 1. Simulation scenarios: state 1 (orange), state 2 (blue), and state 3 (green). [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Map showing the location data for eight narwhal tracked from August 2017 to October [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Success rate in estimating the correct number of states (i.e., three) for each scenario, [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Estimates from the HMM of narwhal movement data obtained with the non-stationary [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]
Figure 5
Figure 5. Figure 5: Narwhal locations coloured by state from HMM model selected by BIC (a) and non [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.