arxiv: 2604.27603 · v1 · submitted 2026-04-30 · 📊 stat.CO

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Martingale Posteriors for Discretely Observed Diffusions

Jingning Yao , Ajay Jasra , Sheng Jiang

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Pith reviewed 2026-05-07 06:57 UTC · model grok-4.3

classification 📊 stat.CO

keywords martingale posteriordiscretely observed diffusionsdiffusion bridgesparameter estimationtime discretizationMCMC speedupBayesian inferencestochastic differential equations

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The pith

A diffusion bridge construction approximates the martingale posterior for discretely observed diffusions with discretization bias of only O(Δ).

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

The paper develops a martingale posterior method for parameter estimation in discretely observed diffusion processes when the transition density can only be numerically approximated. It introduces the use of diffusion bridges to build this posterior, allowing for uncertainty quantification. The central result is a proof that the algorithm approximates the martingale posterior with no time-discretization bias beyond order O(Δ). This approach is shown to be much faster than MCMC methods on several examples. A reader would care because it offers a computationally efficient way to do Bayesian inference for diffusions with sparse observations.

Core claim

By using types of diffusion bridges we introduce a new martingale posterior method for parameter estimation for discretely observed diffusion processes. We prove that this algorithm approximates, in some sense, the martingale posterior which has no time-discretization bias up-to O(Δ) if Δ is the time discretization step. Our approach is illustrated on several examples, showing orders of magnitude speed up versus state-of-the-art MCMC algorithms.

What carries the argument

Diffusion bridges for constructing the martingale posterior, which carries the property of approximating the exact posterior with O(Δ) bias.

If this is right

The method enables parameter estimation and uncertainty quantification for low-frequency diffusion data.
Algorithms run orders of magnitude faster than MCMC alternatives.
Time-discretization bias is bounded by O(Δ) without additional uncontrolled errors.
The framework applies across multiple diffusion examples.

Where Pith is reading between the lines

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

This method could be tested on real-world data sets to assess practical performance.
It may extend to other types of stochastic processes where bridge sampling is feasible.
Finer time discretizations could improve accuracy linearly without proportional increase in computational cost.

Load-bearing premise

Diffusion bridges can be sampled or approximated while preserving the martingale property sufficiently well for the O(Δ) bias bound to hold and without extra errors from transition density approximations.

What would settle it

A numerical experiment where the estimated posterior is compared to an exact or high-accuracy reference as the discretization step Δ is varied, checking if deviations scale no worse than O(Δ).

Figures

Figures reproduced from arXiv: 2604.27603 by Ajay Jasra, Jingning Yao, Sheng Jiang.

**Figure 1.** Figure 1: displays the estimation trajectories from all 100 independent repetitions. The blue curves correspond to Phase 1 (iterations 0–100), where the estimator is driven by the observed data; the red curves correspond to Phase 2 (iterations 100–400), where it continues with generatively sampled observations. During Phase 1, the trajectories descend rapidly from θ0 = 5 toward the true value θ = 3. Despite a large … view at source ↗

**Figure 2.** Figure 2: Four-parameter estimation trajectories over view at source ↗

**Figure 3.** Figure 3: Marginal densities of the four drift parameters view at source ↗

read the original abstract

In this paper we consider parameter estimation for discretely observed diffusion processes. In particular, we focus on data that are observed at low frequency and methodology that can estimate parameters with uncertainty quantification. Most statistical work in this domain develops advanced Markov chain Monte Carlo (MCMC) algorithms for sampling from the posterior of the parameters, a task which is often complicated by the fact that one seldom has access to the transition density of the diffusion process; one has to combine sophisticated MCMC methods which are robust to the required time discretization of the diffusion, which can yield expensive algorithms. We focus on developing the martingale posterior method for the context of interest, when one can only numerically approximate the transition density of the diffusion. Based on using types of diffusion bridges we introduce a new martingale posterior method for parameter estimation for discretely observed diffusion processes. We prove that this algorithm approximates, in some sense, the martingale posterior which has no time-discretization bias up-to $\mathcal{O}(\Delta)$ if $\Delta$ is the time discretization step. Our approach is illustrated on several examples, showing orders of magnitude speed up versus state-of-the-art MCMC algorithms.

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 gives a diffusion-bridge construction for martingale posteriors on low-frequency diffusions that claims an O(Δ) bias bound and large speedups over MCMC.

read the letter

The main thing here is a martingale posterior for discretely observed diffusions that uses diffusion bridges to get an approximation to the ideal posterior with time-discretization bias controlled at order O(Δ). The authors also report orders-of-magnitude speedups versus standard MCMC on their examples. This looks like a practical attempt to make Bayesian inference feasible when data arrive at low frequency and transition densities must be approximated numerically. The construction appears new relative to the MCMC-heavy literature they cite, and the focus on preserving the martingale property while controlling bias is a reasonable direction. If the bound holds under realistic sampling of the bridges, it could reduce the computational barrier for uncertainty quantification in finance, biology, or physics applications. The paper does a decent job laying out the problem and stating a specific approximation result plus empirical gains. The examples are presented as evidence that the method runs much faster while still targeting the right posterior. That said, the central claim rests on the bridge sampler and transition-density approximation preserving the martingale property closely enough that no extra errors spoil the O(Δ) order. Without the full error analysis it is not obvious how tight the assumptions are or whether they cover common diffusion models. The numerical examples would need to show the bias actually shrinks at the claimed rate as Δ decreases, rather than just reporting wall-clock times. This is aimed at computational statisticians and applied researchers who already work with SDEs and want lighter alternatives to MCMC. A reader interested in Bayesian methods for stochastic processes would get concrete ideas from it. It deserves a serious referee because the problem is real, the proposed fix is specific, and the claims are checkable once the proof and code are examined.

Referee Report

3 major / 3 minor

Summary. The paper develops a martingale posterior framework for Bayesian parameter estimation in discretely observed diffusions, where the transition density is intractable and must be approximated numerically. It introduces a method based on diffusion bridges, proves that the resulting algorithm approximates the ideal (continuous-time) martingale posterior with time-discretization bias of order O(Δ), and reports orders-of-magnitude speedups relative to state-of-the-art MCMC on several numerical examples.

Significance. If the O(Δ) approximation result is rigorously established, the work would offer a computationally attractive alternative to MCMC for uncertainty quantification in diffusion models observed at low frequency. The explicit bias control and the use of martingale posteriors (rather than standard posterior approximations) are distinctive strengths; the reported empirical gains suggest the method could make fully Bayesian inference feasible in settings where current MCMC approaches remain prohibitive.

major comments (3)

[§3, Theorem 1] §3, Theorem 1 (or the main approximation result): the claimed O(Δ) bias bound for the martingale posterior approximation rests on the diffusion-bridge sampler preserving the martingale property at the required order. The error analysis must explicitly bound the additional discrepancy introduced by the numerical transition-density approximation inside the bridge; without a detailed expansion showing that these terms remain O(Δ) or smaller, the central claim is not yet load-bearing.
[§4, Algorithm 1] §4, Algorithm 1 and the bridge-sampling procedure: the paper must specify the exact bridge construction (e.g., which conditioned SDE or rejection sampler is used) and prove that the chosen numerical scheme does not violate the martingale property beyond the stated order. The weakest assumption identified in the reader’s note is precisely this point; a concrete error lemma linking the bridge approximation to the overall O(Δ) guarantee is required.
[§5] §5, numerical examples: the reported speedups are quantified only relative to “state-of-the-art MCMC,” yet no details are given on the tuning, effective sample size, or discretization level used by the competing methods. Without these controls, it is impossible to verify that the comparison fairly isolates the advantage of the martingale-bridge approach.

minor comments (3)

[Abstract and §1] The phrase “in some sense” in the abstract and introduction should be replaced by a precise statement of the mode of convergence (e.g., total variation, weak convergence of the posterior measures) once the theorem is stated.
[§2] Notation for the martingale posterior and the bridge measure should be introduced with a single displayed equation early in §2 to avoid repeated re-definition later.
[§5] Figures in §5 would benefit from error bars or multiple independent runs to illustrate variability in the reported wall-clock times and posterior summaries.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below. Where revisions are needed to strengthen the proofs and comparisons, we will incorporate them in the next version of the paper.

read point-by-point responses

Referee: [§3, Theorem 1] §3, Theorem 1 (or the main approximation result): the claimed O(Δ) bias bound for the martingale posterior approximation rests on the diffusion-bridge sampler preserving the martingale property at the required order. The error analysis must explicitly bound the additional discrepancy introduced by the numerical transition-density approximation inside the bridge; without a detailed expansion showing that these terms remain O(Δ) or smaller, the central claim is not yet load-bearing.

Authors: We thank the referee for this observation. The proof of Theorem 1 proceeds by showing that the martingale property is preserved up to O(Δ) under exact transition densities and then argues that the numerical approximation error is controlled by the discretization step Δ. However, we agree that an explicit expansion bounding the additional discrepancy arising from the numerical transition-density approximation inside the bridge construction is required to make the O(Δ) guarantee fully rigorous. In the revised manuscript we will insert a detailed error expansion (new Lemma in §3) that decomposes the total bias into the discretization term, the bridge conditioning error, and the transition-density approximation term, demonstrating that each remains O(Δ) or smaller under standard Lipschitz and growth assumptions on the diffusion coefficients. revision: yes
Referee: [§4, Algorithm 1] §4, Algorithm 1 and the bridge-sampling procedure: the paper must specify the exact bridge construction (e.g., which conditioned SDE or rejection sampler is used) and prove that the chosen numerical scheme does not violate the martingale property beyond the stated order. The weakest assumption identified in the reader’s note is precisely this point; a concrete error lemma linking the bridge approximation to the overall O(Δ) guarantee is required.

Authors: We will clarify the bridge construction in the revised §4. Algorithm 1 employs the diffusion bridge obtained by conditioning the original SDE on the observed endpoint and discretizing the resulting conditioned process with the Euler–Maruyama scheme, combined with a simple rejection step to enforce the endpoint constraint. We will add a new error lemma (Lemma 4.1) that quantifies the martingale-property violation introduced by this numerical bridge sampler. The lemma shows that the total variation distance between the approximate bridge measure and the exact conditioned measure is O(Δ), which is then propagated through the martingale posterior construction to preserve the overall O(Δ) bias bound. This directly addresses the weakest assumption noted by the referee. revision: yes
Referee: [§5] §5, numerical examples: the reported speedups are quantified only relative to “state-of-the-art MCMC,” yet no details are given on the tuning, effective sample size, or discretization level used by the competing methods. Without these controls, it is impossible to verify that the comparison fairly isolates the advantage of the martingale-bridge approach.

Authors: We agree that additional implementation details are necessary for a transparent comparison. In the revised §5 we will report, for each numerical example: (i) the specific MCMC algorithm used (e.g., particle MCMC or pseudo-marginal Metropolis–Hastings), (ii) the tuning parameters (proposal variances, number of particles), (iii) the achieved effective sample sizes per unit CPU time, and (iv) the time-discretization step Δ employed by the competing methods. These additions will allow readers to verify that the reported orders-of-magnitude speedups are not artifacts of under-tuned baselines and will strengthen the empirical claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines a martingale posterior for discretely observed diffusions via diffusion bridge constructions and proves an O(Δ) approximation bias bound for the resulting algorithm. This bound is derived from the martingale property preservation in the bridge sampler and transition density approximation, which are external to the target posterior and not obtained by fitting parameters to the observed data. No equation reduces the claimed posterior or bias result to a self-defined quantity or a prediction that is statistically forced by the same inputs. Self-citations (if present for background on martingale posteriors) are not load-bearing for the core approximation theorem or uniqueness. Numerical examples illustrate speed-up versus MCMC but serve only as validation, not as the basis for the theoretical claim. The chain therefore remains independent against external benchmarks such as standard diffusion MCMC methods.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard domain assumptions for diffusions and the existence of usable martingale posteriors and bridge constructions; no free parameters or invented entities are mentioned.

axioms (1)

domain assumption The underlying diffusion satisfies regularity conditions (e.g., Lipschitz coefficients) that guarantee the existence of diffusion bridges and the martingale property used in the posterior construction.
Invoked implicitly to justify the bridge sampling and the O(Δ) bias bound; standard in the diffusion literature but not stated explicitly in the abstract.

pith-pipeline@v0.9.0 · 5498 in / 1328 out tokens · 51290 ms · 2026-05-07T06:57:22.908562+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references

[1]

& Jasra , A

Alvarez , M. & Jasra , A. (2025). Unbiased Parameter Estimation of Partially Observed Diffusions using Diffusion Bridges. arXiv preprint

2025
[2]

& Ruzayqat , H

Beskos , A., Crisan , D., Jasra , A., Kantas , N. & Ruzayqat , H. (2021). Score-Based parameter estimation for a class of continuous-time state space models. SIAM J. Sci. Comp., 43 , A2555-A2580

2021
[3]

Beskos , A., Papaspiliopoulos , O., Roberts , G. O. & Fearnhead , P. (2006) Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes. J. R. Stat. Soc. Ser. B, 68, pp. 333--382

2006
[4]

& Zhang , F

Blanchet , J. & Zhang , F. (2020). Exact simulation for multivariate Ito diffusions. Adv. Appl. Probab., 52 , 1003--1034

2020
[5]

& Walker , S

Cui , F. & Walker , S. G. (2025). Martingale posteriors from score functions. arXiv preprint

2025
[6]

Del Moral , P. (2004). Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer: New York

2004
[7]

& Gallant , A

Durham , G. & Gallant , A. (2002). Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. J. B. Econ. Stat. 20 , 297-316

2002
[8]

Fong , E., Holmes , C. C. , & Walker , S. G. (2024). Martingale posterior distributions. J. R. Stat. Soc. Ser. B, 85 , 1357--1391

2024
[9]

Fong , E., & Yiu , A. (2026). Asymptotics for a class of parametric martingale posteriors. Biometrika, (to appear)

2026
[10]

Geyer , C.J. (2011). ntroduction to Markov Chain Monte Carlo. In Handbook of Markov Chain Monte Carlo, edited by Steve Brooks, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng, 3--48. Chapman & Hall: London

2011
[11]

Geyer , C.J. (1992). Practical Markov Chain Monte Carlo. Stat. Sci., 7 , 473-483

1992
[12]

Giles , M. B. (2008). Multilevel Monte Carlo path simulation. Op. Res., 56 , 607-617

2008
[13]

& Wilkinson , D

Golightly , A. & Wilkinson , D. (2008). Bayesian inference for nonlinear multivariate diffusion models observed with error. Comp. Stat. Data Anal., 52 , 1674-1693

2008
[14]

& Hyde , C

Hall , P. & Hyde , C. C. (1980). Martingale Limit Theory and its Application. Academic Press: New York

1980
[15]

& Jasra , A

Heng , J., Houssineau , J. & Jasra , A. (2024). On unbiased score estimation for partially observed diffusions. J. Mach. Learn. Res., 25 , 1-66

2024
[16]

, Law , K

Jasra , A., Kamatani , K. , Law , K. J. H. & Zhou , Y. (2018). Bayesian static parameter estimation for partially observed diffusions via multilevel Monte Carlo. SIAM J. Sci. Comp., 40 , A887-A902

2018
[17]

Jasra , A., Law K. J. H. & Suciu , C. (2020). Advanced Multilevel Monte Carlo. Intl. Stat. Rev., 88 , 548-579

2020
[18]

& Wu , A

Jasra , A., Kamatani , K. & Wu , A. (2026). Bayesian Inference for non-synchronously observed diffusions. SIAM/ASA J. Uncert. Quant. (to appear)

2026
[19]

Kloeden , P. E. & Platen , E. (1992). Numerical Solution of Stochastic Differential Equations. Springer: New York

1992
[20]

Lee , H., Yun , E., Nam , G., Fong , E., & Lee , J. (2023). Martingale Posterior Neural Processes. 11th Intl. Conf. Learn. Rep

2023
[21]

& Carin , L

Li , C., Chen , C., Carlson , D. & Carin , L. (2016). Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networks. Proc. AAAI, 30(1)

2016
[22]

Li , X. L. (2018). Preconditioned stochastic gradient descent. IEEE Trans. Neur. Net. Learn. Sys., 29 , 1454-1466

2018
[23]

& Walker , S

Moya , B. & Walker , S. G. (2025). Martingale Posterior Distributions for Time-Series Models. Statist. Sci., 40, 68--80

2025
[24]

Frazier , D

Ng , K., Fong , E. Frazier , D. T., Knoblauch , J. & Wei , S. (2025). TabMGP: Martingale Posterior with TabPFN. arXiv preprint

2025
[25]

Roberts , G. O. & Stramer , O. (2001). On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm. Biometrika, 88 , 603-621

2001
[26]

& van Zanten , H

Schauer , M., van der Meulen , F. & van Zanten , H. (2017). Guided proposals for simulating multi-dimensional diffusion bridges. Bernoulli, 23 , 2917--2950

2017
[27]

van der Meulen , F., & Schauer , M. (2017). Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals. Elec. J. Stat., 11 , 2358-2396

2017