arxiv: 2605.03331 · v1 · submitted 2026-05-05 · 📊 stat.ME · stat.CO

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Bayesian Modelling of Nonstationary Extreme Values Using a Nonparametric Hawkes Process

Gordon J. Ross , Dean Markwick

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Pith reviewed 2026-05-07 14:15 UTC · model grok-4.3

classification 📊 stat.ME stat.CO

keywords nonstationary extremesHawkes processBayesian nonparametricDirichlet processgeneralized Pareto distributionpoint processextreme value theorypredictive performance

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The pith

A Bayesian nonparametric Hawkes process with hierarchical GPD marks achieves the best held-out predictive performance for nonstationary extreme events.

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

The paper builds a point process model for extremes whose rate and size both change over time. It models the occurrence rate with a self-exciting Hawkes process whose excitation kernel is learned nonparametrically through a Dirichlet process mixture, allowing data-driven clustering without fixed functional forms. Event magnitudes are modeled by a generalized Pareto distribution whose parameters are partially pooled across clusters in a hierarchical structure. An MCMC sampler is derived for the full posterior, and both simulations and four real datasets show that the combined flexible components yield better forecasts than simpler stationary or parametric alternatives.

Core claim

We develop a Bayesian model for nonstationary extremes that uses a Hawkes process with a Dirichlet process mixture prior on the excitation kernel to capture clustering and a hierarchical GPD mark model to allow magnitude distributions to vary across clusters while sharing strength through partial pooling. The resulting hierarchical specification is sampled via MCMC. Simulation experiments confirm that each flexible element improves predictive accuracy when the corresponding structure is present in the data-generating process, and on four real datasets the full nonparametric Hawkes model with hierarchical GPD marks attains the highest held-out predictive performance among the variants tested.

What carries the argument

A Hawkes process whose temporal excitation pattern is learned via a Dirichlet process mixture, coupled with a hierarchical generalized Pareto distribution on event magnitudes that induces partial pooling across clusters.

If this is right

The nonparametric excitation kernel can represent arbitrary clustering patterns in the timing of extremes.
Hierarchical GPD marks improve magnitude estimation for small clusters by borrowing strength across clusters.
When the data-generating process contains self-excitation or cluster-specific tails, the added flexibility demonstrably raises predictive accuracy.
The MCMC algorithm produces full posterior samples that quantify uncertainty in both rates and magnitudes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same structure could be applied to financial crash data or climate records where clustering of extremes is suspected but the precise form is unknown.
Replacing the Dirichlet process with other nonparametric priors might further relax assumptions on cluster shapes.
Predictive gains suggest direct use in operational risk or disaster-warning systems that require accurate tail forecasts.
Extending the model to include spatial marks or multivariate extremes would address joint occurrences of different types of events.

Load-bearing premise

The observed extremes are produced by a self-exciting point process whose clustering pattern is well approximated by a Dirichlet process mixture and whose sizes are adequately described by a hierarchical GPD.

What would settle it

A collection of extreme events generated from a non-self-exciting process or from magnitude distributions that depart markedly from the GPD, on which the proposed model shows no predictive gain over a stationary Poisson process with a single GPD.

read the original abstract

Modelling and forecasting the occurrence of extreme events is especially difficult when the event process is nonstationary, with changes in both the rate at which extremes occur and the magnitude of the extremes when they occur. We approach this task by developing a Bayesian point process model for extreme events, which uses a self-exciting Hawkes process to model the rate at which extremes occur. The Hawkes process has a structure which allows events to occur in clusters, making it realistic for many types of data. We use a flexible Bayesian nonparametric approach based on the Dirichlet process to learn the temporal excitation pattern from the data. Further, we build on Extreme Value Theory by using a Generalised Pareto Distribution (GPD) to model the magnitudes of the extremes, with a hierarchical mark model allowing these magnitudes to vary across Hawkes-induced clusters. A hierarchical specification of the model results in partial pooling, allowing for more accurate GPD estimation even in clusters with only a small number of observations. We develop an MCMC algorithm to sample from the resulting hierarchical model. A simulation study confirms that the two flexible components improve prediction when the corresponding features are present in the data-generating mechanism, and across four real data sets the nonparametric Hawkes model with hierarchical GPD marks gives the best held-out predictive performance among the model variants considered.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper combines a Dirichlet-process Hawkes process for self-exciting extremes with a hierarchical GPD mark model that pools across clusters, and the version with both pieces shows the best held-out predictive performance on the four real datasets they tested.

read the letter

The central contribution is the joint nonparametric treatment of the excitation pattern and the cluster-specific tail parameters. The Dirichlet process lets the model learn how past events raise the intensity without committing to a fixed kernel shape, while the hierarchical GPD allows the shape and scale to differ by cluster yet borrow strength when a cluster has few points. That combination is not standard in the extremes literature they cite, and the simulation recovers the expected improvement when both self-excitation and cluster-wise tail variation are present in the data-generating process. On the real data the full model wins on held-out log predictive density or similar scores against the simpler variants they compare to. That is useful evidence for a practitioner who needs to forecast clustered extremes whose sizes are not stationary. The MCMC is described at a high level and the abstract claims it works, but the usual diagnostics for mixing and prior sensitivity on the concentration parameter are not visible here, so a referee would want those checks. The comparative claim is narrow and therefore easier to defend, but it still rests on the assumption that the held-out periods are representative and that the baseline models were implemented without hidden advantages. Overall the work is for people who already use Hawkes or GPD models for point processes and want a Bayesian nonparametric upgrade that stays computationally feasible. It is worth sending to peer review because the model is clearly specified, the simulation is on point, and the real-data ranking is reported; the gaps are fixable with standard additions rather than fundamental.

Referee Report

2 major / 2 minor

Summary. The paper develops a Bayesian point process model for nonstationary extreme events that combines a self-exciting Hawkes process with a Dirichlet process prior on the excitation kernel to capture clustering and a hierarchical generalized Pareto distribution (GPD) for event magnitudes that allows cluster-specific tail variation with partial pooling. An MCMC algorithm is derived for posterior inference; a simulation study shows that the nonparametric and hierarchical components recover expected gains when present in the data-generating process; and on four real datasets the full nonparametric Hawkes + hierarchical GPD specification yields the best held-out predictive performance among the variants examined.

Significance. If the comparative predictive results hold, the model supplies a principled, data-driven way to handle both nonstationary intensity and heterogeneous tail behavior in extremes, with the hierarchical GPD component offering practical gains for sparse clusters. The simulation study provides direct evidence that the two flexible modeling choices improve forecasts when the assumed features are present, and the held-out evaluation on real data supplies a falsifiable ranking of the model variants.

major comments (2)

[Methods (MCMC algorithm section)] The manuscript provides only a high-level description of the MCMC algorithm. Without explicit statements of the proposal mechanisms, acceptance rates, or convergence diagnostics (e.g., effective sample sizes or Gelman-Rubin statistics for the Dirichlet process concentration and GPD hyperparameters), it is difficult to verify that the reported posterior samples are reliable enough to support the predictive comparisons.
[Simulation study and real-data results sections] The simulation study and real-data results are summarized qualitatively in the abstract. The full results section should report the exact predictive scores (log predictive density, CRPS, or exceedance probabilities) together with standard errors or bootstrap intervals so that the magnitude and statistical significance of the claimed superiority can be assessed directly.

minor comments (2)

[Model specification] Notation for the Dirichlet process concentration parameter and the hierarchical GPD hyperparameters should be introduced once and used consistently; the current description leaves their prior specifications implicit.
[Introduction] Standard references to the original Hawkes process and to classical extreme-value theory (e.g., Pickands or Coles) are missing from the introduction; adding them would clarify the incremental contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments and positive assessment of the significance of our work. We address each major comment below and outline the revisions we will make to the manuscript.

read point-by-point responses

Referee: [Methods (MCMC algorithm section)] The manuscript provides only a high-level description of the MCMC algorithm. Without explicit statements of the proposal mechanisms, acceptance rates, or convergence diagnostics (e.g., effective sample sizes or Gelman-Rubin statistics for the Dirichlet process concentration and GPD hyperparameters), it is difficult to verify that the reported posterior samples are reliable enough to support the predictive comparisons.

Authors: We agree that a more detailed exposition of the MCMC algorithm is warranted to support verification of the posterior samples. In the revised manuscript we will expand the relevant section to specify the proposal distributions and update mechanisms for the Dirichlet process concentration parameter, the GPD hyperparameters, and the remaining model components. We will also report acceptance rates together with convergence diagnostics, including effective sample sizes and Gelman-Rubin statistics, for these parameters. revision: yes
Referee: [Simulation study and real-data results sections] The simulation study and real-data results are summarized qualitatively in the abstract. The full results section should report the exact predictive scores (log predictive density, CRPS, or exceedance probabilities) together with standard errors or bootstrap intervals so that the magnitude and statistical significance of the claimed superiority can be assessed directly.

Authors: We acknowledge the value of presenting quantitative results with measures of uncertainty. While the results sections already contain tables of predictive scores for the model variants, we will revise the manuscript to augment these tables with standard errors or bootstrap intervals for the log predictive density and related metrics in both the simulation study and the real-data analyses. This addition will allow readers to evaluate the magnitude and statistical significance of the reported improvements. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim is a comparative empirical result: after MCMC fitting of the nonparametric Hawkes process with hierarchical GPD marks, the model variant achieves the best held-out predictive performance on four real datasets. This ranking is obtained by direct evaluation on withheld data and is not equivalent by construction to any fitted parameter or self-referential definition. The simulation study separately confirms recovery of known features when they are present in the data-generating process, but does not alter the real-data comparison. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the derivation chain are present; the model specification, inference, and validation remain independent of the target performance metric.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from point process and extreme value theory plus a small number of modeling choices whose values are learned from data.

free parameters (2)

Dirichlet process concentration parameter
Controls the number of distinct excitation patterns learned; given a prior and inferred from data.
Hierarchical hyperparameters for GPD shape and scale
Control partial pooling across clusters; estimated via the posterior.

axioms (2)

domain assumption Extreme events form a self-exciting point process whose intensity depends on past events.
Core modeling choice for capturing clustering, standard in Hawkes literature.
domain assumption Exceedances over a high threshold follow a generalized Pareto distribution.
Standard result from extreme value theory invoked for magnitude modeling.

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Review history (2 revisions) →

discussion (0)

Reference graph

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