{"id":"555de4ea-2394-4f70-85d9-646159bddd27","arxiv_id":"2505.22587","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A Dirichlet-process-plus-Gamma construction enables MCMC inference of jump measures for subordinator-driven state-space models.","lead":"This paper introduces a Bayesian non-parametric method to infer the Levy measure of subordinators and normal variance-mean processes inside a linear state-space model, using a Dirichlet process on jump sizes and a Gamma prior on jump rate. The method is tested on synthetic infinite-activity processes and tick-level FX data, where it shows competitive forecasting versus a Gaussian Langevin baseline.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Gibbs update for lambda in Eq. (39) omits the residual waiting time from the last jump to the end of the observation window; the correct posterior is Gamma(alpha_lambda+M, beta_lambda+T), so the claimed sampler does not target the stated posterior.","rationale":"The paper's central claim is that the augmented Gibbs sampler draws posterior samples of the Levy measure Q(x) under the model in Eq. (36). I examined the three Gibbs steps. The Kalman-based marginal likelihood in Eq. (23) and the Student-t marginalization leading to Eq. (54) both look internally consistent, and the MH acceptance in Eq. (45) correctly uses a likelihood ratio when the proposal is the prior. However, the conditional update for lambda in Eq. (39) is wrong: for a Poisson process on a fixed observation window, the empty interval from the last jump to the end of the window is part of the data and contributes exp(-lambda(T-V_M)) to the likelihood. Omitting it yields a Gamma posterior with rate beta_lambda + V_M instead of beta_lambda + T. This is not a matter of posterior consistency or high-frequency asymptotics; it is a finite-sample algebraic error that biases every Q sample. The reader's weakest assumption concerned the finite-activity truncation of infinite-activity processes; I agree that truncation is unvalidated, but the lambda-update error is more load-bearing because it invalidates the sampler even for data generated exactly from the finite-activity model. A corrected sampler would need to replace V_M with the total observation length, and the experiments would need to be rerun; the current manuscript does not support the central claim as written. Hence I recommend REJECT rather than CONDITIONAL: the flaw is concrete and the reported posterior estimates are not samples from the stated target, although a corrected version may be salvageable.","tokens_in":19335,"tokens_out":14017,"duration_ms":178656,"concrete_test":"Simulate a Poisson process with known lambda on [0,1] (e.g., lambda=0.5, T=1), record M and V_M, and compare the posterior in Eq. (39), Gamma(alpha_lambda+M, beta_lambda+V_M), to the correct Gamma(alpha_lambda+M, beta_lambda+1). For many simulated datasets, the Eq. (39) posterior mean is biased upward by roughly (beta_lambda+1)/(beta_lambda+V_M) and its credible intervals under-cover the true lambda. This directly settles whether the sampler's lambda update targets the stated posterior.","verdict_should_be":"REJECT","load_bearing_attack":"The most load-bearing problem is an internal inconsistency in the core sampler. In Section 4.2, Eqs. (38)-(39) define T_i = V_i - V_{i-1} with V_0 = 0 and state p(lambda | {V_i}_{i=1}^M) = Gamma(alpha_lambda + M, beta_lambda + sum_{i=1}^M T_i). For a Poisson process observed over a fixed window [0,T] with M jumps, the joint likelihood of (M, V_1,...,V_M) is lambda^M exp(-lambda T). The sum of the first M interarrival times equals V_M, not T; the final censored interval T - V_M, during which no jump occurs, is informative and must be added. The correct posterior is therefore Gamma(alpha_lambda + M, beta_lambda + T). Using V_M in place of T makes the rate parameter too small and biases lambda upward, with the bias most severe when the last jump occurs well before the end of the observation window. Because every Levy measure sample is formed as Q^(l) = lambda^(l) f^(l)(x), all inferred measures inherit this bias. This is not a modeling approximation or a tuning choice; it is an incorrect conditional distribution, so the augmented Gibbs sampler does not have the generative model in Eq. (36) as its stationary distribution. The finite-activity truncation concern raised by the reader is real, but this technical error is more fundamental: even under the finite-activity model, the sampler is inconsistent.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a Bayesian nonparametric framework for inferring the Lévy measures of a subordinator process and of the normal variance-mean (NVM) process that it drives, within a linear Lévy state-space model with additive Gaussian observation noise. The Lévy measure is modeled as Q(x) = λ f(x), where λ has a Gamma prior and f has a Dirichlet process prior (referred to as the IGSDP in the abstract), so that the inference target is the Lévy measure of a finite-activity compound Poisson approximation. Inference is performed with an augmented Gibbs sampler that alternates between (i) sampling Q together with the DP concentration parameter conditional on an augmented jump series {Z_i, V_i}; (ii) updating the jump series via MH-in-Gibbs using the model prior as the proposal and the Rao-Blackwellized marginal likelihood (a Kalman filter with the NVM parameters μ_w and σ_w marginalized) in the acceptance ratio; and (iii) updating the system parameters. Conditional on the jump series, μ_w and σ_w are marginalized in closed form, and the NVM Lévy measure is represented as an unnormalized mixture of Student-t distributions. The method is tested on simulated tempered-stable data and on tick-level GBP/USD data, with a forecasting comparison against a Gaussian Langevin SSM and a naive forecaster.","tokens_in":19722,"tokens_out":26996,"duration_ms":288286,"significance":"If the two concerns below are resolved, this would be a useful methodological contribution: it appears to be the first MCMC scheme that targets the Lévy measure of both the subordinator and the NVM process in a linear Lévy state-space model, and several components are assembled correctly. In particular, the DP posterior update in Eq. (40), the Student-t marginalization in Eq. (54), and the Kalman-based marginal likelihood of Eqs. (23)/(68) are standard results that appear derived correctly, and the paper is commendably explicit about its conditional-update structure. The empirical validation is also sensible: the synthetic experiments exercise an infinite-activity tempered stable driving process, and the real-data forecasting comparison (Table 1) is a meaningful practical check. However, the significance is conditional: the λ-posterior error in Eq. (39) means the sampler currently does not target the stated posterior even under the finite-activity model, and the unquantified truncation approximation limits the strength of the claims in Sections 1.3 and 6. The paper ships no code, so correctness rests entirely on the derivations and experiments.","major_comments":[{"comment":"The conditional posterior for λ stated in Eq. (39) is incorrect for the observation scheme used throughout the paper. With T_i = V_i − V_{i−1} and V_0 = 0, one has Σ_{i=1}^M T_i = V_M, the time of the last jump, so the claimed posterior Gamma(α_λ + M, β_λ + Σ T_i) discards the censored interval between the last jump and the end of the observation window. For a Poisson process of rate λ observed on a fixed window of length L (here L = t_N − t_1, the span of the N discrete observations), the likelihood of the count M and the ordered event times is λ^M exp(−λL); the residual interval (V_M, L] is informative about λ and must be included. The correct full conditional is Gamma(α_λ + M, β_λ + L). Because every Lévy measure sample is formed as Q^(l) = λ^(l) f^(l)(x) in Eqs. (41)–(42), the upward bias in λ (the denominator β_λ + V_M is smaller than β_λ + L with positive probability) propagates to all reported Lévy measures and to the NVM measure (54)–(55). I verified this concern independently: the augmented Gibbs sampler does not have the generative model (36) as its stationary distribution, even under the finite-activity assumption. The fix is local, but it is load-bearing, and the Section 5.2 experiments (high-activity tempered stable case) do not stress the regime where the bias is largest, namely when few jumps occur and V_M is well before the window end.","section":"4.2, Eqs. (38)–(39)"},{"comment":"The inference target is the finite-activity compound Poisson approximation Q(x) = λ f(x), not the Lévy measure of the infinite-activity process that generates the data. Section 3 states that it is natural to perform inference under the finite-activity assumption, and Appendix D defines an explicit truncation threshold ε for the ground truth, but ε and the DP truncation level K are fixed by the user and are never learned, validated, or subjected to sensitivity analysis, and their numerical values are not reported for the synthetic or real-data experiments. When the driving process is infinite-activity (tempered stable, Section 5.2), the model is misspecified, and the posterior concentrates on an approximating measure whose bias is controlled by ε; Figures 1–2 therefore validate the sampler for the approximating model, not the unqualified claim in Sections 1.3 and 6 that the procedure infers the Lévy measures of subordinators and NVM processes. I request (i) a scoping statement in Sections 1.3 and 6 that the object of inference is the truncated measure, (ii) a sensitivity analysis over ε (and K) with the chosen values reported, and (iii) at least one posterior check of the truncation, such as a comparison of the observed number of jumps with the predictive distribution implied by the inferred λ.","section":"3, 5.2, Appendix D"},{"comment":"The abstract advertises two contributions that are absent from the body. First, the 'Independent Gamma-scaled Dirichlet Process (IGSDP)' is named only in the abstract; the model actually used is Q = λ f with f ~ DP(α, H_η) and λ ~ Gamma (model (36), Eqs. (41)–(42)), and the claim that the well-known Gamma process is a special case is not established anywhere in the text. As stated it is in tension with the discreteness of DP realizations: Q is almost surely a discrete measure in Eq. (42), whereas the Gamma process has an infinite-activity, absolutely continuous Lévy measure; a precise limiting or embedding statement is needed. Second, the abstract promises 'an explicit characterization of the parameter contour for NVM processes' yielding 'an identifiable parameterization of the model,' but no such characterization appears in the body or in Appendices A–D. Identifiability of the NVM parameters is nontrivial — the transformation (μ_w, σ_w, ν(dz)) → (μ_w/c, σ_w/√c, ν(dz/c)) leaves the law of J(t) in (13) invariant — so the advertised characterization should be supplied if it exists, or these claims should be removed from the abstract and Section 1.3.","section":"Abstract vs. body (also 2.5, 4.6)"}],"minor_comments":[{"comment":"The text says the Dirichlet process 'generates discrete densities with probability 1' and therefore 'cannot be used to estimate a continuous density'; a DP generates discrete probability measures rather than densities, and since the paper's estimator is the discrete random measure (42), the repeated use of 'Lévy density' for Q(x) (Section 2.1, Figure 2 captions) should be reconciled with the discrete-measure interpretation.","section":"2.5 (paragraph after Eq. (29))"},{"comment":"The hit-rate metric is never defined, and the naive forecaster's reported hit rate of 0.0000 needs clarification: if a hit requires a strictly nonzero predicted direction, a naive prediction of zero change would always be a miss, but the definition should be stated explicitly; reporting standard errors or intervals for the MSE comparison would also strengthen the claim that the Lévy model outperforms the naive benchmark.","section":"5.3, Table 1"},{"comment":"No numerical values are reported for the truncation threshold ε, the DP truncation level K, or the prior hyperparameters (a, b, α_λ, β_λ, α_w, β_w, k_w, and the initial subordinator parameters), so the synthetic and real-data experiments are not reproducible as described.","section":"5.2, Appendix D"},{"comment":"The symbol α is overloaded: it denotes the DP concentration parameter in (36), the extended state vector α(t) in (59), and appears in the inverse-Gamma shape α_w in (36) and (65); this makes Sections 4.2, 4.5, and Appendix B unnecessarily hard to follow.","section":"4.2, 4.5, B"},{"comment":"The sentence claiming consistent results for various parameter combinations and random initializations is not backed by any reported results; a summary of the repeated runs (e.g., posterior means and their ranges across initializations) should be included in an appendix.","section":"5.2"},{"comment":"The efficiency claim for the prior-proposal MH update rests on the heuristic of weak temporal dependence; reporting the average acceptance rate of the blocked updates and the functional autocorrelation times for the subordinator series itself (as opposed to θ) would make the claim verifiable.","section":"4.3"},{"comment":"Several bibliographic entries are incomplete ([4], [5] lack venues/years; [19] lacks the author's initials), and there are numerous typos, e.g., 'a random variabel N' in Algorithm 1, 'non-negativesubordinator' in Section 2.2, and 'we make use of a fact that...' in Section 3; a careful proofread is needed.","section":"References; Algorithm 1"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about Eq. (39) lands; I verified it independently. The fix is one line, so I recommend major revision rather than rejection, but the authors must re-run the experiments with the corrected conditional and should add the ε/K sensitivity analysis before the paper can be accepted. The abstract substantially oversells the contributions (IGSDP, parameter contour), and the AE may wish to check the novelty claims against [7], [26], [57], and [59]. The manuscript is otherwise clearly written and the conditional-marginalization derivations check out."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea here is sound: use a Dirichlet process on the normalized jump-size distribution, a Gamma prior on the rate, and an MH-in-Gibbs scheme with a Rao-Blackwellized Kalman filter to learn Lévy measures in linear state-space models. The Student-t mixture representation for the NVM density is a neat and correct-looking piece of work, and the paper does a good job explaining why the shot-noise representation makes augmentation tractable. On the build side, the derivations in Section 2.4 and Appendix B for marginalizing out μw and σw are coherent. The paper is also honest about the finite-activity approximation being a modeling choice, and the experiments, while not exhaustive, are at least set up to check the right things: hidden states, Lévy measures, mixing, and out-of-sample forecasting on real FX ticks. Credit where it’s due: this is a competent, sensible application of known Bayesian nonparametric machinery to a useful problem class.\n\nThe problem is that the sampler does not target the stated posterior. I checked Eq. (39) directly, and the stress-test note is correct. For a Poisson process observed on a fixed window [0,T], the complete-data likelihood is λ^M exp(−λT). The paper’s posterior Gamma(αλ + M, βλ + Σ_i T_i) uses only the interarrival times up to the last jump, i.e. V_M, and drops the censored interval T − V_M. That is not a modeling approximation; it is an incorrect conditional distribution. Every Lévy measure sample inherits the resulting upward bias in λ, and the bias gets worse when the last jump lands well before the end of the observation window. This is more fundamental than the finite-activity truncation concern, which is real but secondary: even under the finite-activity model the chain would be inconsistent.\n\nTwo smaller issues. First, the abstract announces an \"IGSDP,\" a Gamma-process special case with conjugacy, and an explicit parameter contour; none of these appear in the body. The reader is right that the abstract overclaims relative to the text. Second, there is no code, no data release, and no precise tuning details, which makes the empirical claims hard to reproduce. The finite-activity truncation level and the DP truncation level K are set by the user and never validated or varied; the paper should at least show sensitivity to both.\n\nThe fix to Eq. (39) is simple: replace Σ T_i with T. With that correction, plus an aligned abstract and a reproducible supplement, this could be a useful contribution to the Lévy state-space literature. As it stands, the central algorithm is not trustworthy, so I would not cite it yet, but I would send it to review rather than desk-reject: the idea is worth refereeing, and the main flaw is both load-bearing and fixable.","headline":"There is a load-bearing bug in the core sampler — the conditional posterior for the jump rate λ omits the censored interval — and the abstract promises things the body never delivers; the underlying modeling idea is still sensible and worth refereeing if the fix is straightforward.","tokens_in":20224,"tokens_out":1814,"would_cite":false,"duration_ms":27254,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G51","62F15","62M05","62G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that Lévy jump measures inside linear state-space models can be learned from noisy data by an augmented Gibbs sampler, factorizing $Q(x)=\\lambda f(x)$ into a Gamma rate and a Dirichlet-process jump-size law.","keywords":["Lévy measure","Dirichlet process","Bayesian nonparametrics","state-space model","subordinator","normal variance-mean process","augmented Gibbs sampler","shot-noise representation"],"falsifier":"Generate data from an infinite-activity Lévy process with a known Lévy density, run the algorithm at several user-chosen truncation thresholds, and compare the posterior-mean subordinator tail functions at small jump sizes: if the inferred measure shifts materially with the threshold, the threshold rather than the data is determining the recovered jumps. A complementary check is a posterior predictive test that simulates new subordinator series from the inferred posterior measure and compares their small-jump activity with that of the true series.","tokens_in":19098,"feed_emoji":"📈","tokens_out":15311,"duration_ms":130030,"temperature":0.7,"pith_summary":"Lévy processes are the standard model for dynamics that mix smooth motion with sudden jumps, but the measure that drives them — the rate and size law of the jumps — is almost never known, and estimation is hard because the likelihood is intractable except in special cases. This paper claims the first Bayesian non-parametric inference for that Lévy measure inside a linear state-space model, covering both the subordinator that supplies the random time change and the normal variance-mean (NVM) process built on it. The method factorizes the measure as a total rate times a probability law, $Q(x)=\\lambda f(x)$, puts a Gamma prior on $\\lambda$ and a Dirichlet process on $f$, and treats the jump series as augmented variables; given the jumps the model is Gaussian, so a Rao-Blackwellized Kalman filter supplies the likelihood in closed form. The posterior yields the subordinator Lévy density as a weighted discrete measure and the NVM density as an unnormalized mixture of Student-$t$ distributions. Tests on synthetic tempered-stable data and on tick-level GBP/USD prices show that the inferred measures track the ground truth away from the smallest jumps, and the resulting one-step forecasts beat a Gaussian Langevin baseline.","feed_headline":"Bayesian sampler recovers hidden jump laws of Lévy processes","feed_subtitle":"A new Gibbs scheme learns both the subordinator jump measure and the NVM density as a Student-t mixture.","key_machinery":"The load-bearing object is the factorization of the Lévy density into a total Poisson rate and a jump-size probability law, $Q(x)=\\lambda f(x)$, with $\\lambda\\sim\\mathrm{Gamma}(\\alpha_\\lambda,\\beta_\\lambda)$ and $f\\sim\\mathrm{DP}(\\alpha,H_\\eta)$; the paper calls this random measure the Independent Gamma-scaled Dirichlet Process, of which the Gamma process is a special case. Under the finite-activity assumption the subordinator is a compound Poisson process, so conditioning on the augmented jump series $\\{Z_i,V_i\\}$ decouples the posterior: a conjugate Gamma update for $\\lambda$ from the inter-arrival times, and a conjugate Dirichlet-process update for $f$ from the jump-size samples, drawn with an explicitly truncated stick-breaking construction. The observations enter through the shot-noise representation of the linear SDE, which is Gaussian conditional on the jumps; a Rao-Blackwellized Kalman filter with $\\mu_w$ and $\\sigma_w^2$ marginalized gives the likelihood, and integrating the Gaussian NVM kernels against the inverse-Gamma and normal posteriors produces the unnormalized Student-$t$ mixture (54) that is the final NVM-measure sample.","core_discovery":"The central claim is that the posterior of the Lévy measure $Q(x)$ is tractable once the problem is posed in finite-activity form: writing $Q(x)=\\lambda f(x)$ with $f\\sim\\mathrm{DP}(\\alpha,H_\\eta)$ and $\\lambda\\sim\\mathrm{Gamma}(\\alpha_\\lambda,\\beta_\\lambda)$, the subordinator becomes a compound Poisson process whose jump times and sizes, conditioned on, are independent of the DP hyper-parameters. The inter-arrival times give a conjugate Gamma posterior for $\\lambda$, the jump sizes give a conjugate Dirichlet-process posterior for $f$, and the likelihood linking the jumps to the observations is the closed-form Kalman marginal (23) of the conditionally Gaussian shot-noise representation, with $\\mu_w$ and $\\sigma_w^2$ marginalized. An augmented Gibbs sampler alternates these updates with an MH-in-Gibbs block update of the jump series and an MH step for system parameters. Combining samples as $Q^{(l)}(x)=\\lambda^{(l)}\\sum_j w_j^{(l)}\\delta_{x_j}$ yields the subordinator measure estimate, and integrating the NVM kernels over the posteriors of $\\mu_w$ and $\\sigma_w^2$ yields the NVM measure as an unnormalized Student-$t$ mixture, equation (55). The paper demonstrates this on data generated from a tempered-stable subordinator, an infinite-activity process handled through the finite-activity approximation, and on high-frequency foreign-exchange data, where the inferred momentum signal and forecasts support the inferred measures.","pith_inferences":["The user-fixed truncation threshold could itself be given a prior and sampled, and a posterior-predictive check on small-jump activity would test whether the finite-activity approximation is adequate for a given dataset; the paper fixes the threshold without learning it.","The weak temporal dependence of the augmented jump series is what makes the MH-in-Gibbs update cheap here, so the same augmentation should carry over to nonlinear observation models with a particle-filter estimate replacing the Kalman marginal, at higher computational cost.","The Student-$t$ mixture output is a ready-made plug-in for derivative pricing or risk measurement in time-changed models, since the NVM Lévy density is exactly the object that governs jump risk premia; the paper demonstrates forecasting but not this downstream use."],"forward_implications":["Given discrete, irregularly spaced observations of a linear SDE, one can now obtain full posterior samples for the Lévy measure of the driving subordinator and of the induced NVM process, not just for the system states and parameters.","The NVM Lévy density comes out as an unnormalized Student-$t$ mixture (55), so tail functions and other measure summaries are available in closed form from the MCMC output.","The DP concentration $\\alpha$ is itself sampled through an auxiliary-variable Gibbs step, removing one of the main prior-tuning knobs from user choice.","On tick-level GBP/USD data the inferred Lévy model improves one-step-forecast MSE and directional hit rate over both a Gaussian Langevin state-space model and a naive random walk.","Because the scheme inherits the shot-noise conditional-Gaussian structure, it carries over to other linear SDEs driven by subordinated Brownian motion."],"supporting_citations":[{"why":"Supplies the shot-noise representation of the Lévy-driven linear SDE, the conditional-Gaussian structure on which the Rao-Blackwellized likelihood is built.","marker":"[57]"},{"why":"Defines the Lévy state-space model whose augmented-sampler and Kalman-marginal machinery this work extends from parametric to non-parametric measure inference.","marker":"[26]"},{"why":"Gives the finite-activity decomposition $Q(x)=\\lambda f(x)$, the Lévy-Khintchine background, and the compound-Poisson simulation algorithm used for proposals.","marker":"[15]"},{"why":"Provides the explicitly truncated stick-breaking (finite-DP) Gibbs sampler used to draw the posterior Dirichlet process for the jump-size law.","marker":"[31]"},{"why":"The main existing Bayesian non-parametric treatment, restricted to Gamma-type subordinators, which this framework generalizes beyond the Gamma case.","marker":"[7]"},{"why":"The alternative Gibbs-posterior approach to Lévy-density estimation that this work contrasts with because it keeps the high-frequency sampling assumption.","marker":"[59]"},{"why":"Supplies the point-process simulation algorithm used to initialize the subordinator series before the MCMC burn-in.","marker":"[40]"},{"why":"Foundational definition of the Dirichlet process used as the prior for the jump-size distribution.","marker":"[21]"}],"fun_headline_variants":["Bayesian nonparametric inference for Lévy measures","New sampler learns hidden jump laws of Lévy processes","Tractable posterior for Lévy measures via Gibbs sampler","IGSDP prior yields Bayesian inference for Lévy measures","Inferring Lévy measures from high-frequency financial data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise, stated in Section 3 and used throughout the experiments, is that inference may be conducted under a finite-activity compound-Poisson approximation of the true Lévy process: the small-jump truncation threshold and the Dirichlet-process component count $K$ are fixed by the user and never learned or validated, so what is recovered is the Lévy measure of the approximating process, and jump mass below the threshold is missed.","fun_headline_variants_meta":{"raw":{"variants":["Bayesian nonparametric inference for Lévy measures","New sampler learns hidden jump laws of Lévy processes","Tractable posterior for Lévy measures via Gibbs sampler","IGSDP prior yields Bayesian inference for Lévy measures","Inferring Lévy measures from high-frequency financial data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000319,"raw_usage":{"total_tokens":1859,"prompt_tokens":1064,"completion_tokens":795,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":680,"completion_tokens_details":{"reasoning_tokens":718}},"tokens_in":680,"tokens_out":795,"duration_ms":8074,"temperature":1.0,"reasoning_tokens":718,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:04:04.649394+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate data from an infinite-activity Lévy process with a known Lévy density, run the algorithm at several user-chosen truncation thresholds, and compare the posterior-mean subordinator tail functions at small jump sizes: if the inferred measure shifts materially with the threshold, the threshold rather than the data is determining the recovered jumps. A complementary check is a posterior predictive test that simulates new subordinator series from the inferred posterior measure and compares their small-jump activity with that of the true series.","supporting_citations":[{"cited_title":"Generalised shot noise representations of stochastic systems driven by non-Gaussian L\\'evy processes","cited_arxiv_id":"2305.05931","evidence_quote":"Supplies the shot-noise representation of the Lévy-driven linear SDE, the conditional-Gaussian structure on which the Rao-Blackwellized likelihood is built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Lévy state-space model whose augmented-sampler and Kalman-marginal machinery this work extends from parametric to non-parametric measure inference."},{"cited_title":"Cont and P","cited_arxiv_id":null,"evidence_quote":"Gives the finite-activity decomposition $Q(x)=\\lambda f(x)$, the Lévy-Khintchine background, and the compound-Poisson simulation algorithm used for proposals."},{"cited_title":"Ishwaran and L","cited_arxiv_id":null,"evidence_quote":"Provides the explicitly truncated stick-breaking (finite-DP) Gibbs sampler used to draw the posterior Dirichlet process for the jump-size law."},{"cited_title":"Belomestny, S","cited_arxiv_id":null,"evidence_quote":"The main existing Bayesian non-parametric treatment, restricted to Gamma-type subordinators, which this framework generalizes beyond the Gamma case."},{"cited_title":"Gibbs posterior inference on a Levy density under discrete sampling","cited_arxiv_id":"2109.06567","evidence_quote":"The alternative Gibbs-posterior approach to Lévy-density estimation that this work contrasts with because it keeps the high-frequency sampling assumption."},{"cited_title":"Kındap and S","cited_arxiv_id":null,"evidence_quote":"Supplies the point-process simulation algorithm used to initialize the subordinator series before the MCMC burn-in."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Foundational definition of the Dirichlet process used as the prior for the jump-size distribution."}],"review_version":1}