Sampling on Discrete Spaces with Temporal Point Processes

Cameron A. Stewart (1); London; Maneesh Sahani (1) ((1) Gatsby Computational Neuroscience Unit; U.K.); University College London

arxiv: 2603.09089 · v2 · pith:3ANW2OZSnew · submitted 2026-03-10 · 📊 stat.CO · math.PR· q-bio.NC

Sampling on Discrete Spaces with Temporal Point Processes

Cameron A. Stewart (1) , Maneesh Sahani (1) ((1) Gatsby Computational Neuroscience Unit , University College London , London , U.K.) This is my paper

Pith reviewed 2026-05-21 12:42 UTC · model grok-4.3

classification 📊 stat.CO math.PRq-bio.NC

keywords temporal point processesdiscrete samplingmultivariate countsinfinite-server queuesstochastic neural networksdownward-closed supportbirth-death processes

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The pith

A system of coupled infinite-server queues with deterministic service times samples any multivariate count distribution whose support is downward-closed.

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

The paper constructs a temporal point process for sampling from discrete multivariate count distributions. For any target distribution whose support is closed under taking lower counts, the authors build a network of possibly coupled infinite-server queues with fixed service times. Event counts inside a sliding time window of fixed length converge in distribution to the target as time advances. This structure gives the process a discrete form of momentum that reduces the random-walk behavior common in other samplers. The same construction yields a recurrent stochastic neural network whose dynamics implement the sampling and display features such as relative refractory periods.

Core claim

For any target multivariate count distribution with downward-closed support, there exists a multivariate temporal point process structured as a system of potentially coupled infinite-server queues with deterministic service times whose event-count vector in a fixed-length sliding window converges in distribution to the target as time tends to infinity. The admissible families permit both reversible and non-reversible dynamics. Introducing auxiliary randomness reduces the sampler to a birth-death process that shares the same limiting distribution.

What carries the argument

System of potentially coupled infinite-server queues with deterministic service times, whose sliding-window event counts converge to the target distribution.

If this is right

The construction yields a recurrent stochastic neural network that performs sampling-based computation while exhibiting relative refractory periods and oscillations.
The sampler always outperforms pure birth-death processes in multivariate effective sample size across 63 tested distributions.
Non-reversible dynamics are admissible, broadening the class of usable processes beyond reversible ones.
The same limiting distribution is recovered when auxiliary randomness collapses the process to a birth-death process.

Where Pith is reading between the lines

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

The momentum property may reduce autocorrelation times compared with standard MCMC on discrete spaces, especially when normalized by computation time.
Coupling the queues could be generalized to enforce additional marginal constraints without changing the joint limit.
The sliding-window construction suggests a natural way to embed discrete sampling inside continuous-time dynamical systems for hybrid inference tasks.

Load-bearing premise

The target count distribution must have downward-closed support.

What would settle it

Run the queue construction on a small multivariate count distribution that lacks downward-closed support and check whether the sliding-window counts still converge in distribution to the target.

read the original abstract

Temporal point processes offer a powerful framework for sampling from discrete distributions, yet they remain underutilized in existing literature. We show how to construct, for any target multivariate count distribution with downward-closed support, a multivariate temporal point process whose event-count vector in a fixed-length sliding window converges in distribution to the target as time tends to infinity. Structured as a system of potentially coupled infinite-server queues with deterministic service times, the sampler exhibits a discrete form of momentum that suppresses random-walk behaviour. The admissible families of processes permit both reversible and non-reversible dynamics. As an application, we derive a recurrent stochastic neural network whose dynamics implement sampling-based computation and exhibit some biologically plausible features, including relative refractory periods and oscillations. The introduction of auxiliary randomness reduces the sampler to a birth-death process, establishing the latter as a degenerate case with the same limiting distribution. In simulations on 63 target distributions, our sampler always outperforms these birth-death processes and frequently outperforms Zanella processes in multivariate effective sample size, with further gains when normalized by CPU time.

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 builds a temporal point process sampler from coupled infinite-server queues with fixed service times to hit downward-closed count distributions, adding momentum and beating birth-death baselines in simulations.

read the letter

The core idea here is to realize sampling from multivariate count distributions with downward-closed support as the limiting event counts in a sliding window of a temporal point process. They model it as a system of infinite-server queues with deterministic service times, which introduces a discrete momentum effect and allows both reversible and non-reversible dynamics. Adding auxiliary randomness collapses it to a birth-death process that shares the same limit. They also sketch a recurrent neural network implementation with refractory-like behavior and oscillations. In simulations across 63 target distributions, the method improves multivariate effective sample size over the birth-death case every time and often over Zanella processes, with gains persisting after normalizing for CPU time. That empirical record is the strongest part of what is shown. The queue framing with deterministic services is a clean way to control memory and avoid pure random-walk behavior, and the reduction to birth-death processes is a useful sanity check. The neural network angle is a reasonable downstream application for stochastic computation models. The main soft spot is the universality claim inside the downward-closed class. The abstract states the construction works for any such distribution, but the stress-test concern about explicit parameterization of arrivals and couplings is fair: deterministic service times fix a finite memory horizon, and it is not obvious from the given details whether arbitrary higher-order dependencies can always be encoded through the coupling rules. Without a clear map from target pmf to queue parameters or a proof sketch, the expressivity part needs verification. The simulation protocol and choice of the 63 distributions also deserve a closer look for reproducibility. This is aimed at researchers working on discrete sampling methods or stochastic neural models who want alternatives to standard MCMC. It has enough new construction and empirical backing to deserve a serious referee rather than a desk reject.

Referee Report

3 major / 2 minor

Summary. The paper claims to construct, for any multivariate count distribution with downward-closed support, a multivariate temporal point process structured as a system of potentially coupled infinite-server queues with deterministic service times such that the event-count vector in a fixed-length sliding window converges in distribution to the target as time tends to infinity. The admissible families permit reversible and non-reversible dynamics; an auxiliary-randomness reduction yields birth-death processes as a degenerate case. The construction is applied to a recurrent stochastic neural network exhibiting relative refractory periods and oscillations. Simulations on 63 target distributions report that the sampler always outperforms birth-death baselines and frequently outperforms Zanella processes in multivariate effective sample size, with further gains when normalized by CPU time.

Significance. If the explicit parameterization and convergence argument are supplied and verified, the work would offer a novel point-process framework for sampling on discrete spaces that incorporates a discrete form of momentum via deterministic service times. This could improve mixing relative to memoryless alternatives and enable biologically motivated sampling architectures. The empirical gains on 63 distributions and the reduction to birth-death processes are concrete strengths that would support adoption in statistical computing if the universality claim is made rigorous.

major comments (3)

[Abstract / Construction] Abstract and construction section: the claim that a suitable system of coupled infinite-server queues exists for every downward-closed multivariate count distribution is asserted without an explicit map from the target pmf to the arrival intensities, coupling rules, and deterministic service times. Because deterministic service times fix a finite memory horizon, it is not immediate that arbitrary dependence structures within the downward-closed class can be realized; this map and a supporting argument are load-bearing for the central universality statement.
[Theoretical development] Convergence argument: the manuscript states that the event-count vector converges in distribution to the target as t → ∞ but supplies neither a proof sketch, conditions on the coupling, nor an error analysis. The limiting distribution is matched by construction, yet the non-Markovian dynamics induced by deterministic service times require explicit verification that the finite-memory process can attain the full class of admissible joint distributions.
[Numerical experiments] Simulation protocol: the reported outperformance on 63 distributions in multivariate ESS (and CPU-normalized ESS) lacks a precise description of how the queue parameters were chosen for each target, the sliding-window length, the burn-in horizon, and the exact comparison protocol against Zanella processes. Without these details the empirical claims cannot be reproduced or assessed for fairness.

minor comments (2)

[Introduction] The definition of downward-closed support should be stated formally with an explicit mathematical condition (componentwise inequality) at its first appearance.
[Preliminaries] Notation for the multivariate counting process and the sliding-window vector should be introduced with a single consistent symbol and an equation reference.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our manuscript. We address each of the major comments below and have made revisions to incorporate the suggested improvements where appropriate. We believe these changes will clarify the construction, strengthen the theoretical foundations, and enhance the reproducibility of the empirical results.

read point-by-point responses

Referee: [Abstract / Construction] Abstract and construction section: the claim that a suitable system of coupled infinite-server queues exists for every downward-closed multivariate count distribution is asserted without an explicit map from the target pmf to the arrival intensities, coupling rules, and deterministic service times. Because deterministic service times fix a finite memory horizon, it is not immediate that arbitrary dependence structures within the downward-closed class can be realized; this map and a supporting argument are load-bearing for the central universality statement.

Authors: We agree with the referee that providing an explicit map is essential to substantiate the universality claim. In the revised manuscript, we have expanded the construction section to include a precise mapping from the target probability mass function to the arrival intensities, coupling rules, and deterministic service times. We also include a supporting argument that demonstrates how the finite memory horizon induced by deterministic service times can still realize arbitrary dependence structures within the downward-closed class through suitable choices of couplings. revision: yes
Referee: [Theoretical development] Convergence argument: the manuscript states that the event-count vector converges in distribution to the target as t → ∞ but supplies neither a proof sketch, conditions on the coupling, nor an error analysis. The limiting distribution is matched by construction, yet the non-Markovian dynamics induced by deterministic service times require explicit verification that the finite-memory process can attain the full class of admissible joint distributions.

Authors: We appreciate this observation regarding the convergence argument. The revised manuscript now includes a detailed proof sketch that verifies the convergence in distribution under the specified conditions on the coupling. We also provide an error analysis and explicitly address how the non-Markovian dynamics with finite memory can attain the full class of admissible joint distributions. revision: yes
Referee: [Numerical experiments] Simulation protocol: the reported outperformance on 63 distributions in multivariate ESS (and CPU-normalized ESS) lacks a precise description of how the queue parameters were chosen for each target, the sliding-window length, the burn-in horizon, and the exact comparison protocol against Zanella processes. Without these details the empirical claims cannot be reproduced or assessed for fairness.

Authors: We thank the referee for highlighting the need for more detailed simulation protocols. In the revised version, we have added a comprehensive description of the experimental setup, including the method for selecting queue parameters for each of the 63 target distributions, the chosen sliding-window length, the burn-in horizon, and the precise protocol used for comparisons with Zanella processes. These additions should facilitate reproducibility and allow for a fair evaluation of the results. revision: yes

Circularity Check

0 steps flagged

No significant circularity: construction sets limit by design but remains self-contained

full rationale

The paper constructs a specific family of temporal point processes (coupled infinite-server queues with deterministic service times) whose sliding-window count vector is engineered to converge to any chosen target distribution with downward-closed support. This matching is the explicit goal of the construction rather than a hidden reduction; the derivation supplies an explicit mechanism (arrival intensities and coupling rules) that realizes the limit, and the claim is then validated externally by simulation on 63 independent target distributions where the new sampler is compared to birth-death and Zanella baselines on effective sample size. No self-citation is load-bearing for the central existence result, no parameter is fitted to the evaluation data and then re-labeled as a prediction, and the downward-closed support condition is stated as an assumption rather than derived from the construction itself. The overall argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of a point-process construction for every downward-closed support and on the convergence of the sliding-window count vector; these are stated as given without further derivation in the abstract.

axioms (1)

domain assumption Every multivariate count distribution with downward-closed support admits a temporal point process realization whose sliding-window counts converge to it.
Stated directly in the abstract as the scope of the construction.

invented entities (1)

discrete momentum via deterministic service times no independent evidence
purpose: Suppresses random-walk behavior in the sampler
Introduced in the abstract as a property of the queue structure.

pith-pipeline@v0.9.0 · 5728 in / 1347 out tokens · 49861 ms · 2026-05-21T12:42:05.265707+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1. … λ*_i(t) = f(S−[t]+e_i)/f(S−[t]) η*_i(t) … π is the limiting distribution of S(t) as t↑∞. Structured as a system of potentially coupled infinite-server queues with deterministic service times.
IndisputableMonolith/Foundation/Breath1024.lean period8 / flipAt512 unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

m-periodic … sliding window of length m … 8-tick micro-structure absent; only generic periodicity m.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.