{"id":"c37c7cd5-a238-4dd7-b0ea-64f7ab0805ad","arxiv_id":"2507.20153","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A hidden-state Markov-switching discrete Hawkes process is identifiable, estimable by EM, and fits bat call data better than Poisson or homogeneous Hawkes alternatives.","lead":"The paper introduces a statistical model for event sequences whose rate shifts between hidden states, coupling a Hawkes self-excitation term with a Markov-switching baseline. It proves the model is identifiable, gives an EM algorithm, and tests it on bat echolocation recordings.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The EM M-step as printed in §3.2 differentiates a Poisson mean μ_q+βU_k, not the model's μ_q+U_k, so the published algorithm is not an EM for Model (2).","rationale":"Good-faith reading: the paper's contribution is a new model plus an EM algorithm, with supporting simulations and a bat application. The hidden-Markov representation in Proposition 2 is sound, and the public code plus simulation study provide some independent support. The most load-bearing soft spot, however, is the M-step specification: the derivatives printed in §3.2 do not correspond to the model defined in §2.1 and Proposition 2. Deriving from (10)–(11) using U_k = αY_{k−1} + βU_{k−1} gives gradients with denominators μ_q + U_k and no extra β factors; the paper's formulas have denominators μ_q + βU_k and factors β, which is the gradient for a different Poisson mean. This is not a stylistic issue: the EM algorithm's defining property is that the M-step increases the complete-data Q function of the stated model. If the printed equations are used, the algorithm is not an EM for Model (2), so the simulation and application results cannot be reproduced from the paper alone. The public code may well implement the correct update; in that case the fix is a correction of the equations. If it implements the printed update, the simulation results themselves need scrutiny. A single audit plus one likelihood-monotonicity check settles this. The identifiability issue raised by the reader (missing distinct-μ assumption) is real but less load-bearing, because it affects only the theorem's statement and is easily repaired by adding the assumption or stating identifiability up to label permutation. The AIC degree-of-freedom miscount (D_Q should be Q^2 + Q + 1, not Q^2 + 2) is also real and affects the application's model ranking, but it does not undermine the inference method itself. Overall, the paper is promising but should remain conditional pending correction of the M-step equations and an audit of the code.","tokens_in":19623,"tokens_out":16249,"duration_ms":173054,"concrete_test":"Audit the public repository (github.com/scj-robin/CodeHawkesDiscreteHMM): compare its M-step update for α, β, and μ against the analytical gradient of (11) with U_k defined by (8); then on one simulated dataset from Section 4, run one EM iteration using (i) the printed formulas and (ii) the code, and print the observed-data likelihood before and after. If the printed M-step does not increase the likelihood, or the code's update differs from the printed derivatives, the published algorithm is not an EM for Model (2) and the Section 4 and 5 numbers must be regenerated from the corrected code.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 3.2's M-step is not the maximizer for Model (2). With the model's own recurrence (8), U_k = αY_{k−1} + βU_{k−1}, and emission (9), Y_k | Z_k, U_k ∼ P(μ_{Z_k} + U_k), the objective in (11) is Σ_{k,q} τ_{kq}^{(h)}[Y_k log(μ_q + U_k) − (μ_q + U_k)] + const. The correct partial derivative with respect to α is therefore ∂_αQ = Σ_k (∂_αU_k)[Σ_q τ_{kq}Y_k/(μ_q + U_k) − 1], with ∂_αU_k = Y_{k−1} + β∂_αU_{k−1}, and no extra factor β. The printed formula instead contains an extra factor β and denominators μ_q + βU_k; it is exactly the gradient for a model with Poisson mean μ_q + βU_k, not Model (2). The same mismatch appears in the ∂_β and ∂_μ formulas, whose denominators are μ_q + βU_k. A reader implementing the printed equations would be maximizing a different objective, so the EM's likelihood-monotonicity property and the simulation results rest on code that is not specified by the paper. This is more directly consequential than the missing distinct-μ assumption in Proposition 1, though both need fixing.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a discrete-time Hawkes process with a latent Markov chain driving the baseline immigration rate (Model (2)), motivated by bat echolocation sequences. The authors prove identifiability of the parameter vector θ from the joint distribution of (Y1,Y2,Y3) (Proposition 1), establish a Markovian representation via an auxiliary process U (Proposition 2), and derive an EM algorithm with a recursive M-step. A simulation study evaluates parameter recovery, state recovery, and AIC-based selection of the number of hidden states, and an application to 1,555 bat cry sequences compares the proposed 'Hawkes-HMM' with homogeneous Poisson, Poisson-HMM, and homogeneous Hawkes models. The paper claims that for 1,144 of 1,555 sequences the proposed model attains the maximum AIC.","tokens_in":19980,"tokens_out":3907,"duration_ms":39947,"significance":"The modeling idea is timely and the HMM representation in Proposition 2 is a clean and valuable contribution: by introducing the deterministic auxiliary process U, the infinite-memory model becomes a finite-state HMM with state-dependent Poisson emissions, enabling standard forward-backward recursions. The simulation design is honest, since data are generated from a continuous-time process and then discretized rather than drawn directly from the fitted model. The application to bat behavior is novel and the authors provide public code. However, several load-bearing technical parts of the manuscript are not correct as printed: the M-step equations maximize a different objective than Model (2), the forward recursion uses the wrong emission mean, the identifiability proposition is broader than its proof, and the AIC degrees-of-freedom count is wrong. These issues do not invalidate the conceptual framework, but they must be fixed before the claims can be accepted.","major_comments":[{"comment":"The printed partial derivatives for α, β, and μ are not the derivatives of Q(θ|θ^(h)) for Model (2). Under the model's own recurrence (8) and emission (9), Y_k has mean μ_{Z_k} + U_k, with U_k = αY_{k−1} + βU_{k−1}. The published formulas contain denominators μ_q + βU_k and an extra factor β in ∂_αQ and ∂_βQ; these are exactly the derivatives for a model with Poisson mean μ_q + βU_k. A reader implementing the printed equations would be maximizing a different objective, so the EM monotonicity property for Model (2) is not guaranteed by the text and the simulation results are not reproducible from the paper alone. Please replace the three displayed derivatives with the correct ones, e.g. ∂_αQ = Σ_k (∂_αU_k)(Σ_q τ_{kq}Y_k/(μ_q+U_k) − 1) and similarly for ∂_βQ and ∂_μ_qQ.","section":"§3.2, M-step (displayed derivatives of Q)"},{"comment":"The M-step update for the transition matrix is printed as π_{qℓ}^{(h+1)} = η_{kqℓ}^{(h)} / Σ_ℓ η_{kqℓ}^{(h)}, which depends on an arbitrary time k and omits the sum over k. The correct maximizer of the complete-data expected log-likelihood is π_{qℓ}^{(h+1)} = [Σ_{k=2}^n η_{kqℓ}^{(h)}] / [Σ_{k=2}^n τ_{k−1,q}^{(h)}]. Separately, the forward recursion in Eq. (14) writes the emission factor as P(Y_k; μ_q + U_k), but the current state in the recursion is ℓ, so the factor should be P(Y_k; μ_ℓ + U_k). Both errors affect the algorithm that is actually implemented and must be corrected.","section":"§3.2, π update; Appendix A.2, Eq. (14)"},{"comment":"The proposition states identifiability of θ for Model (2) without any restriction on μ, but the proof in Appendix A.1 relies on the ordering 'without loss of generality, suppose that μ_1 < μ_2 < ... < μ_Q'. This distinctness assumption is not stated in the proposition. If two hidden states share the same baseline rate, the Poisson mixtures in (13) collapse and the parameters of those states are not identifiable from (Y1,Y2,Y3); the linear-independence argument for the functions f_{qℓ} also fails. The proposition should either include the distinct-μ assumption explicitly or the proof must be extended to handle coincident baselines.","section":"Proposition 1 and Appendix A.1"},{"comment":"The number of free parameters is reported as D_Q = Q^2 + 2, which is incorrect for Model (2). A direct count gives (Q−1) free parameters for ν, Q(Q−1) for the transition matrix π, Q for μ, and 1 each for α and β, totaling Q^2 + Q + 1. Using D_Q = Q^2 + 2 changes the AIC penalty and may affect the simulation conclusions about selecting Q as well as the application claim that the proposed model wins for 1,144 of 1,555 sequences. Please recompute the AIC comparisons with D_Q = Q^2 + Q + 1 and report whether the conclusions change.","section":"§4.1, AIC criterion (D_Q)"}],"minor_comments":[{"comment":"The complete log-likelihood uses the symbol K in the first sum ('KX q=1') where the number of states is elsewhere denoted Q; K is undefined and should be Q.","section":"§3.2, Eq. (10)"},{"comment":"The sentence 'as the set Poisson distributions is additively closed).include Based on this' contains a stray '.include' and a missing period; please clean up the proof text.","section":"Appendix A.1"},{"comment":"The caption of Table 1 says the results are obtained with C = 2, and Figure 8 shows sensitivity of the selected Q to C. It would be helpful to state explicitly whether the 1,144/1,555 comparison is robust to the choice of discretization coefficient, given that the alluvial plot indicates non-negligible variation with C.","section":"§5, Table 1 and Figure 8"}],"recommendation":"major_revision","confidential_remarks":"The core idea and the Markov representation are sound, and the paper is likely salvageable. The main barrier is that the EM algorithm as printed is not an EM for the stated model, and the AIC penalty count and the identifiability statement are also wrong as stated. I did not inspect the GitHub code, so I cannot verify whether the simulations used the corrected equations; the paper must be self-contained and the printed equations must match the implementation. The authors should also double-check the forward recursion emission index and the π update before resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the core modeling idea is solid and worth engaging with: a discrete-time Hawkes process whose baseline rate switches according to a hidden Markov chain, with an auxiliary variable U_k that turns the infinite-memory process into an HMM. Second, the printed inference algorithm is not an EM for the model as written. The M-step derivatives in §3.2 contain an extra factor β and Poisson denominators μ_q + βU_k; those are the derivatives for a different model where the emission mean is μ_q + βU_k, not μ_q + U_k. That directly contradicts Eq. (9). The simulation results probably come from code that does something else, but the paper does not say.\n\nWhat is new: the paper's own literature review identifies Zhou et al. (2022) as the only prior Hawkes-with-Markov-switching work, and there the state is observed. Here the state is hidden, which changes the inference problem. The HMM representation (Prop. 2) is correct and useful. The identifiability proof via finite Poisson mixtures is a legitimate classical route, and the simulation design—simulate continuous-time, then discretize—is honest. The bat call application is motivating and the comparison to simpler models makes sense.\n\nWhere it is soft, in increasing order of concern:\n\n- The AIC degrees-of-freedom count is wrong: D_Q = Q^2 + 2 should be Q^2 + Q + 1 once ν, π, μ, α, β are counted. The model selection section depends on this.\n- The π update in the M-step is missing the sum over k; as printed it uses a single time point.\n- Proposition 1 silently assumes distinct μ_q. The proof's \"without loss of generality\" ordering is not without loss of generality; if two states share a baseline rate, the Poisson mixture collapses and the parameters are not identifiable. The assumption needs to be in the statement.\n- The M-step mismatch is the load-bearing issue. A reader implementing the printed equations maximizes a different objective, so likelihood monotonicity does not hold for the printed algorithm. The authors must reconcile the derivatives with the model, and ideally confirm the public code matches the corrected formulas.\n\nMy take: this paper deserves a serious referee. The model is new, identifiable under a stated assumption, and the application is genuinely useful. The inference section needs a major revision before the results can be trusted. I would cite the model and the HMM representation once the M-step is corrected; right now I would be cautious about citing the algorithm.","headline":"Valuable new model with a printed M-step that does not match the likelihood; fixable, but needs referee attention before the results are trusted.","tokens_in":20466,"tokens_out":5938,"would_cite":true,"duration_ms":57309,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62M05","60G55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a Markov-switching discrete-time Hawkes process is identifiable from the joint law of its first three counts, and shows the parameters can be learned with an EM algorithm that reads the hidden states as behavioral…","keywords":["Markov-switching Hawkes process","discrete-time Hawkes process","hidden Markov model","identifiability","EM algorithm","bat echolocation","AIC model selection","point process"],"falsifier":"Set $Q=2$ with $\\mu_1=\\mu_2$ but two different transition matrices $\\pi$, and compute the joint distribution of $(Y_1,Y_2,Y_3)$ for both parameter sets: if the two three-count laws coincide, Proposition 1 as stated is false. A complementary check is to simulate long sequences from this equal-$\\mu$ model and run the EM-AIC pipeline: a sharp drop in state recovery would confirm that the missing distinctness assumption is doing real work.","tokens_in":19442,"feed_emoji":"🦇","tokens_out":8876,"duration_ms":73857,"temperature":0.7,"pith_summary":"The paper introduces a discrete-time Hawkes process whose baseline rate switches between hidden states according to a Markov chain, while the self-exciting memory of past events stays fixed. It proves that the full parameter set is identifiable from the first three observations, and that the infinite-memory process can be rewritten as a hidden Markov model with Poisson emissions by adding the accumulator $U_k = \\alpha Y_{k-1} + \\beta U_{k-1}$. This representation makes maximum-likelihood inference possible through an EM algorithm with a recursive M-step. Simulations show the estimator recovers parameters, selects the correct number of states by AIC, and classifies time bins accurately. On 1555 overnight bat recordings, the proposed model achieves the best AIC on 1144 sequences, beating homogeneous Poisson, Poisson-HMM, and homogeneous Hawkes alternatives.","feed_headline":"Hidden bat behavior states are identifiable from three time bins","feed_subtitle":"A switching model of bat calls fits 1,144 of 1,555 real recordings better than Poisson and Hawkes alternatives.","key_machinery":"The load-bearing object is the accumulator $U_k = \\alpha Y_{k-1} + \\beta U_{k-1}$, which compresses the infinite past of counts into a single number. It makes $(Z_k,U_k,Y_k)$ a Markov chain and turns the emission into a time-dependent Poisson distribution $Y_k \\mid Z_k,U_k \\sim \\mathcal{P}(\\mu_{Z_k}+U_k)$. The identifiability argument relies on finite mixtures of Poisson distributions and on the linear independence of the functions $\\mu_q^x(\\mu_\\ell+\\alpha x)^y$, under the ordering assumption on $\\mu$ used in the proof.","core_discovery":"The central claim is that the parameter set $\\theta=(\\nu,\\pi,\\mu,\\alpha,\\beta)$ of the Markov-switching discrete Hawkes process is identifiable from the joint distribution of $(Y_1,Y_2,Y_3)$: if two parameter sets give the same three-count law, they are the same. The proof extracts the parameters one by one from Poisson mixtures: $Y_1$ identifies $(\\nu,\\mu)$; conditioning on $Y_1=1$ identifies $\\alpha$; conditioning on $Y_1=1,Y_2=0$ identifies $\\beta$; and a linear-independence argument on the functions $\\mu_q^x(\\mu_\\ell+\\alpha x)^y$ identifies $\\pi$. The paper also claims the process is a hidden Markov model once the accumulator $U_k=\\alpha Y_{k-1}+\\beta U_{k-1}$ is adjoined, which justifies the forward-backward EM algorithm. The application claim is that the switching Hawkes model fits most real bat sequences better than its three nested special cases.","pith_inferences":["The proof's ordering assumption $\\mu_1<\\cdots<\\mu_Q$ is stated as 'without loss of generality' in Appendix A.1 but is absent from Proposition 1; a natural reading is that the theorem silently excludes states with equal baseline rates, and the identifiability result may need that condition added explicitly.","Allowing the memory parameters $\\alpha$ and $\\beta$ to depend on the hidden state would preserve the Markovian structure, as the paper notes, so an EM algorithm should remain possible; the practical question is whether the E-step recurrences stay tractable.","The discretization heuristic $N=2\\times$ (number of events) has empirical support from the simulation grids, but a formal consistency result connecting the discrete estimator to the continuous Hawkes parameters would tell practitioners how fine a binning is actually needed.","The informal map comparison between inferred states and species richness suggests a testable multivariate extension: a joint model of several species' call sequences could determine whether species diversity drives the number of states or merely correlates with recording-site characteristics."],"forward_implications":["Because the model nests the homogeneous Poisson, Poisson-HMM, and homogeneous Hawkes models, an AIC comparison among the four directly tests whether both regime switching and self-excitation are needed for a given sequence.","The same fitted model produces per-time-bin state labels through MAP or Viterbi classification, so parameter estimation and behavioral-epoch detection come from one pipeline.","Identifiability from three observations means the inferred states are not merely a convenient fit: the parameters have a unique interpretation, which is what allows the states to be read as real behavioral modes.","The empirical dominance of two states, and the positive association between the number of states and species richness, suggest the latent states plausibly separate transit from foraging and that species diversity adds further structure."],"supporting_citations":[{"why":"Defines the Hawkes process that the discrete model extends, with self-exciting intensity from past events.","marker":"Hawkes (1971)"},{"why":"Provides the Poisson-cluster representation used to derive the discrete approximation and the mapping between continuous and discrete parameters.","marker":"Hawkes and Oakes (1974)"},{"why":"Establishes the INAR($\\infty$) representation and convergence of binned counts to continuous Hawkes processes, motivating the discrete-time model and its parameterization.","marker":"Kirchner (2016)"},{"why":"Supplies the identifiability of finite Poisson mixtures on which the extraction of $\\nu$, $\\mu$, $\\alpha$, and $\\beta$ rests.","marker":"Teicher (1961)"},{"why":"Provides the EM framework that the paper instantiates for maximum-likelihood inference.","marker":"Dempster et al. (1977)"},{"why":"Supplies the forward-backward recursions used in the E step for the hidden Markov representation.","marker":"Cappé et al. (2005)"},{"why":"Defines the AIC criterion used to compare models and to select the number of hidden states.","marker":"Akaike (1974)"},{"why":"Introduces Hawkes modeling of bat echolocation calls and the classification task that this paper extends to within-sequence regime switching.","marker":"Denis et al. (2024)"}],"fun_headline_variants":["Bat call switching model: hidden states from 3 time bins","Three-bin identifiability reveals hidden bat behavior states","Switching Hawkes model decodes bat echolocation from cries","Markov-switching Hawkes fits bat calls better than alternatives","Hidden bat states recoverable from minimal three-bin data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The identifiability proof assumes the hidden states have strictly different baseline rates $\\mu_1<\\cdots<\\mu_Q$, a condition not stated in Proposition 1; if two states share the same spontaneous rate, the Poisson mixtures used in the proof can collapse and the theorem as stated may fail.","fun_headline_variants_meta":{"raw":{"variants":["Bat call switching model: hidden states from 3 time bins","Three-bin identifiability reveals hidden bat behavior states","Switching Hawkes model decodes bat echolocation from cries","Markov-switching Hawkes fits bat calls better than alternatives","Hidden bat states recoverable from minimal three-bin data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000538,"raw_usage":{"total_tokens":2594,"prompt_tokens":967,"completion_tokens":1627,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":583,"completion_tokens_details":{"reasoning_tokens":1544}},"tokens_in":583,"tokens_out":1627,"duration_ms":11827,"temperature":1.0,"reasoning_tokens":1544,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:49:20.162987+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Set $Q=2$ with $\\mu_1=\\mu_2$ but two different transition matrices $\\pi$, and compute the joint distribution of $(Y_1,Y_2,Y_3)$ for both parameter sets: if the two three-count laws coincide, Proposition 1 as stated is false. A complementary check is to simulate long sequences from this equal-$\\mu$ model and run the EM-AIC pipeline: a sharp drop in state recovery would confirm that the missing distinctness assumption is doing real work.","supporting_citations":[{"cited_title":", year 1971","cited_arxiv_id":null,"evidence_quote":"Defines the Hawkes process that the discrete model extends, with self-exciting intensity from past events."},{"cited_title":", year 2016","cited_arxiv_id":null,"evidence_quote":"Establishes the INAR($\\infty$) representation and convergence of binned counts to continuous Hawkes processes, motivating the discrete-time model and its parameterization."},{"cited_title":", year 1961","cited_arxiv_id":null,"evidence_quote":"Supplies the identifiability of finite Poisson mixtures on which the extraction of $\\nu$, $\\mu$, $\\alpha$, and $\\beta$ rests."},{"cited_title":", author Laird, N.M","cited_arxiv_id":null,"evidence_quote":"Provides the EM framework that the paper instantiates for maximum-likelihood inference."},{"cited_title":", year 1974","cited_arxiv_id":null,"evidence_quote":"Defines the AIC criterion used to compare models and to select the number of hidden states."},{"cited_title":", author Dion-Blanc, C","cited_arxiv_id":null,"evidence_quote":"Introduces Hawkes modeling of bat echolocation calls and the classification task that this paper extends to within-sequence regime switching."}],"review_version":1}