{"id":"0fabcfa0-1505-4f05-9959-4434ccf33b4f","arxiv_id":"2411.18826","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A double penalized maximum likelihood estimator is proposed for simultaneous order selection and parameter estimation in non-stationary hidden Markov models, outperforming AIC and BIC in simulation studies with misspecified models.","lead":"This paper introduces a double penalized likelihood method, DPMLE, that estimates the number of hidden states and the parameters of non-stationary hidden Markov models in one step. In simulations with model misspecification, it more often recovers the true number of states than AIC or BIC, and it selects a simpler, more interpretable model for narwhal movement data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Non-stationary DPMLE relies on an unvalidated data-dependent penalty: replacing stationary probabilities with time-averaged posterior probabilities changes the objective in a way that existing consistency results do not cover.","rationale":"I read the paper as claiming an empirical order-selection method for non-stationary HMMs that outperforms AIC and BIC under misspecification. The simulation study is large and the code is available; I credit the DPMLE's consistently high success rates in the reported scenarios. The weakest point is the bridge from the stationary theory to the non-stationary implementation: Eq. (17) replaces a model parameter (stationary probability) with a data-dependent posterior average. Because the consistency theorem cited for the stationary case does not apply to this modified objective, and the paper explicitly defers asymptotics, the method's performance beyond the simulated settings is unsupported. This is a limitation rather than a demonstrated failure; it does not overturn the reported results, but it should be stated as a condition on the claim. The reader's verdict of CONDITIONAL is appropriate; I do not see a reason to move it. I agree with the reader's weakest assumption. The concrete simulation check above would test whether the substitution is benign or introduces bias.","tokens_in":18522,"tokens_out":12700,"duration_ms":118295,"concrete_test":"Generate data from a correctly specified non-stationary three-state gamma-HMM with a time-varying multinomial-logit tpm and a known covariate, with T = 5,000, 12,000, 50,000. Fit the non-stationary DPMLE (same code path as the paper) and compute: (i) the percentage of trials recovering three states; (ii) the relative difference between the fitted \\hat\\pi_j and the true occupation frequency \\pi_j^true = T^{-1}\\sum_t P(S_t=j) under the generating parameters. If the success rate does not increase toward 100% as T grows, or if C_N \\sum \\log \\hat\\pi_j fails to track C_N \\sum \\log \\pi_j^true, the non-stationary DPMLE is optimizing a different, unvalidated target.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the substitution in Eq. (17): for non-stationary HMMs the type-I penalty uses \\hat\\pi_j(\\Psi) = T^{-1}\\sum_t P(S_t=j|Y=y; \\Psi), the time-averaged posterior state probability, in place of the stationary probability \\pi_j that appears in the stationary DPMLE (Eq. 9). This is not a parameter of the model: it is a data-dependent smoother, and it is a function of the entire observed sequence, including future observations relative to time t. The consistency proof for the stationary DPMLE (Hung et al., 2013) applies to an objective of the form l(\\Psi|y) + C_N \\sum \\log \\pi_j(\\Gamma) - \\sum p_\\lambda(\\eta_j); once \\pi_j is replaced by \\hat\\pi_j, the objective is a different functional and no existing theorem covers it. The paper explicitly states that formal asymptotic analysis is future work (Discussion, paragraph 3). If, under misspecification, \\hat\\pi_j does not converge to the intended 'proportion of time spent in state j' or if it assigns non-negligible posterior mass to a spurious state, the type-I penalty can retain or delete the wrong states. The simulations are the only evidence that the approximation works, and they cover a limited set of scenarios with N=4 and specific tpm/emission structures.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18900,"tokens_out":3596,"duration_ms":32650,"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":[{"comment":"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.","section":"Abstract, paragraph 4; Results, Section 3; Tables A1–A2"},{"comment":"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.","section":"Estimation, Eq. (17); Discussion, paragraph 3"},{"comment":"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).","section":"Estimation, Eq. (19); Discussion, paragraph 3"},{"comment":"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.","section":"Results; Tables A1–A2; Figure 3"}],"minor_comments":[{"comment":"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.","section":"Results, Section 3"},{"comment":"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.","section":"Estimation, Eq. (17)"},{"comment":"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.","section":"Estimation, Eq. (15)"},{"comment":"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.","section":"Discussion, paragraph 1"},{"comment":"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.","section":"Section 2, Simulation study"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid applied contribution with a well-constructed simulation study, but its statistical novelty is limited by the lack of theory for the non-stationary extension. The authors are candid about this, and the empirical evidence is reasonably strong. The main concerns are the overstatement in the abstract, the missing uncertainty quantification, and the internal reliance on a BIC-type criterion. If the authors can qualify the claims and add sensitivity analyses, the paper could be suitable for an applied statistics or ecological modeling journal; as a pure stat.ME contribution, the theoretical gap is substantial. I would also note that the code is only 'available by request' rather than publicly deposited, which is a reproducibility concern for a methods paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: this is a genuinely new methods paper, not a repackaging. The authors take the double penalized likelihood approach (Chen & Khalili, Hung et al.) and, for the first time, make it work for non-stationary HMMs with covariates. The trick is replacing stationary probabilities with time-averaged posterior state probabilities from the forward-backward algorithm. That step has no existing consistency theory behind it; the paper says so explicitly. But the simulation study is broad enough that the method earns its place.\n\nWhat's good. The simulation design covers six misspecification scenarios (outliers, random effects in the hidden and observed processes, autocorrelated emissions, time-varying transition probabilities) at two sample sizes. The non-stationary DPMLE is the best or near-best method in most of those settings, and it degrades less than AIC/BIC when sample size grows. The narwhal case study is a sensible illustration: DPMLE picks two interpretable states, while AIC picks eight and BIC picks five with overlapping densities. The authors are also candid about limitations: they assume regularity conditions for the stationary consistency result, note that the non-stationary extension lacks a formal proof, and flag that NIC is only rigorously justified in Gaussian settings.\n\nSoft spots. First, the abstract's blanket claim that DPMLE outperformed AIC and BIC is not true in two T=5,000 scenarios; there BIC with covariates does better. The text in the results section acknowledges this, but the abstract and conclusion overstate it. Second, the Eq. (17) replacement of pi by time-averaged posterior occupancy is a data-dependent smoothing of the whole sequence; not even the stationary consistency arguments apply. That makes the non-stationary DPMLE a heuristic in the strict sense. It is an honest heuristic backed by simulation, but not a fully grounded estimator. Third, the tuning parameters lambda and C_N are selected by NIC, a BIC-type criterion, so the method leans on information criteria at a higher level. Given the simulations use known truth, this does not invalidate the empirical claim, but it is worth flagging. Fourth, the simulations cap the upper bound at N=4; behavior with larger bounds is less tested, though the narwhal example uses N=8. Last, code is only available 'by request' rather than in a public repo, which weakens reproducibility.\n\nProportion: none of these are fatal. The central claim - that this heuristic order selection approach works well under misspecification - is supported by the simulations. The stress-test worry about Eq. (17) is real, but the authors do not dodge it; they flag it as future work.\n\nRecommendation: send it to a serious referee. A statistical journal will want the non-stationary penalty to be either proven or more carefully bounded; an applied journal in ecology or movement can accept the simulation evidence. Either way, ask the authors to revise the abstract, discuss the heuristic nature of the approximation, and make the code public.","headline":"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.","tokens_in":19334,"tokens_out":3225,"would_cite":true,"duration_ms":28122,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62M05","62F12","62P12"],"pacs":[],"model":"deepseek-v4-flash","headline":"A double-penalized likelihood method recovers hidden Markov states when AIC and BIC overfit.","keywords":["Animal movement","Double penalized maximum likelihood estimate (DPMLE)","Hidden Markov model","Order selection","Non-stationary hidden Markov model","Model misspecification","SCAD","Information criteria"],"falsifier":"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.","tokens_in":18327,"feed_emoji":"🐋","tokens_out":9992,"duration_ms":76419,"temperature":0.7,"pith_summary":"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.","feed_headline":"Double-penalized method beats AIC and BIC at choosing hidden states","feed_subtitle":"Simulations and narwhal tracking show the DPMLE recovers the true state count when standard criteria overfit.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the original double penalized likelihood method and its consistency for finite mixture models, which the paper extends to non-stationary HMMs.","marker":"Chen and Khalili (2008)"},{"why":"Extends the DPMLE to stationary HMMs and provides the consistency properties and SCAD implementation that the non-stationary adaptation builds on.","marker":"Hung et al. (2013)"},{"why":"Provides the simulation scenarios and the documented weaknesses of AIC, BIC, and ICL for HMM order selection that the paper uses as benchmark and motivation.","marker":"Pohle et al. (2017)"},{"why":"Supplies the HMM likelihood, forward-backward recursions, and stationary probability formulas that the DPMLE's estimation steps rely on.","marker":"Zucchini et al. (2017)"},{"why":"Introduces the SCAD penalty used to shrink differences between state-dependent means and merge redundant states.","marker":"Fan and Li (2001)"},{"why":"Provides the local linear approximation that lets the EM algorithm maximize the non-smooth SCAD penalty within each iteration.","marker":"Zou and Li (2008)"},{"why":"Supplies the BIC-type criterion used to select the tuning parameters and the caution about overfitting that motivates further investigation.","marker":"Wang et al. (2007)"},{"why":"Supplies the Group-Sort-Fuse procedure that adapts the penalty to multivariate emission parameters in the narwhal analysis.","marker":"Manole and Khalili (2021)"},{"why":"Provides the simulation machinery used to generate the HMM datasets for the scenarios.","marker":"McClintock and Michelot (2018)"}],"fun_headline_variants":["DPMLE picks true state count where AIC and BIC fail","Forget AIC and BIC: DPMLE nails HMM state count","Double penalized method outperforms AIC and BIC for HMM order","New double penalty estimates HMM states better than AIC/BIC","DPMLE: one shot to pick HMM states, beating AIC/BIC"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["DPMLE picks true state count where AIC and BIC fail","Forget AIC and BIC: DPMLE nails HMM state count","Double penalized method outperforms AIC and BIC for HMM order","New double penalty estimates HMM states better than AIC/BIC","DPMLE: one shot to pick HMM states, beating AIC/BIC"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001157,"raw_usage":{"total_tokens":4848,"prompt_tokens":1056,"completion_tokens":3792,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":672,"completion_tokens_details":{"reasoning_tokens":3691}},"tokens_in":672,"tokens_out":3792,"duration_ms":24504,"temperature":1.0,"reasoning_tokens":3691,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T10:51:00.793584+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends the DPMLE to stationary HMMs and provides the consistency properties and SCAD implementation that the non-stationary adaptation builds on."},{"cited_title":"Selecting the Number of States in Hidden Markov Models - Pitfalls, Practical Challenges and Pragmatic Solutions","cited_arxiv_id":"1701.08673","evidence_quote":"Provides the simulation scenarios and the documented weaknesses of AIC, BIC, and ICL for HMM order selection that the paper uses as benchmark and motivation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the HMM likelihood, forward-backward recursions, and stationary probability formulas that the DPMLE's estimation steps rely on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the local linear approximation that lets the EM algorithm maximize the non-smooth SCAD penalty within each iteration."},{"cited_title":"Li, and C.-L","cited_arxiv_id":null,"evidence_quote":"Supplies the BIC-type criterion used to select the tuning parameters and the caution about overfitting that motivates further investigation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Group-Sort-Fuse procedure that adapts the penalty to multivariate emission parameters in the narwhal analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the simulation machinery used to generate the HMM datasets for the scenarios."}],"review_version":1}