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Exact Gibbs sampling for stochastic differential equations with gradient drift and constant diffusion

Xinyi Pei, Minhyeok Kim, Vinayak Rao

TL;DR

We address exact posterior simulation for diffusion paths in gradient-drift SDEs with unit diffusion by developing GibbsEA3, an augmented MCMC that uses Poisson thinning and path-layer concepts to avoid discretization error. The method extends prior exact rejection sampling (EA1) to EA2/EA3 diffusions and integrates Hamiltonian Monte Carlo for efficient path updates, enabling joint inference of latent paths and parameters. Through synthetic experiments on EA2/EA3 models and a real ice-core dataset, GibbsEA3 and its variants show favorable efficiency compared with particle MCMC, especially under informative observations. The framework is modular, supports parameter learning, and leverages Gaussian-process likelihoods, making it widely applicable to continuous-time diffusion modeling.

Abstract

Stochastic differential equations (SDEs) are an important class of time-series models, used to describe stochastic systems evolving in continuous time. Simulating paths from these processes, particularly after conditioning on noisy observations of the latent path, remains a challenge. Existing methods often introduce bias through time-discretization, require involved rejection sampling or debiasing schemes or are restricted to a narrow family of diffusions. In this work, we propose an exact Markov chain Monte Carlo (MCMC) sampling algorithm that is applicable to a broad subset of all SDEs with unit diffusion coefficient; after suitable transformation, this includes an even larger class of multivariate SDEs and most 1-d SDEs. We develop a Gibbs sampling framework that allows exact MCMC for such diffusions, without any discretization error. We demonstrate how our MCMC methodology requires only fairly straightforward simulation steps. Our framework can be extended to include parameter simulation, and allows tools from the Gaussian process literature to be easily applied. We evaluate our method on synthetic and real datasets, demonstrating superior performance to particle MCMC approaches.

Exact Gibbs sampling for stochastic differential equations with gradient drift and constant diffusion

TL;DR

We address exact posterior simulation for diffusion paths in gradient-drift SDEs with unit diffusion by developing GibbsEA3, an augmented MCMC that uses Poisson thinning and path-layer concepts to avoid discretization error. The method extends prior exact rejection sampling (EA1) to EA2/EA3 diffusions and integrates Hamiltonian Monte Carlo for efficient path updates, enabling joint inference of latent paths and parameters. Through synthetic experiments on EA2/EA3 models and a real ice-core dataset, GibbsEA3 and its variants show favorable efficiency compared with particle MCMC, especially under informative observations. The framework is modular, supports parameter learning, and leverages Gaussian-process likelihoods, making it widely applicable to continuous-time diffusion modeling.

Abstract

Stochastic differential equations (SDEs) are an important class of time-series models, used to describe stochastic systems evolving in continuous time. Simulating paths from these processes, particularly after conditioning on noisy observations of the latent path, remains a challenge. Existing methods often introduce bias through time-discretization, require involved rejection sampling or debiasing schemes or are restricted to a narrow family of diffusions. In this work, we propose an exact Markov chain Monte Carlo (MCMC) sampling algorithm that is applicable to a broad subset of all SDEs with unit diffusion coefficient; after suitable transformation, this includes an even larger class of multivariate SDEs and most 1-d SDEs. We develop a Gibbs sampling framework that allows exact MCMC for such diffusions, without any discretization error. We demonstrate how our MCMC methodology requires only fairly straightforward simulation steps. Our framework can be extended to include parameter simulation, and allows tools from the Gaussian process literature to be easily applied. We evaluate our method on synthetic and real datasets, demonstrating superior performance to particle MCMC approaches.
Paper Structure (34 sections, 8 theorems, 39 equations, 11 figures, 1 table, 2 algorithms)

This paper contains 34 sections, 8 theorems, 39 equations, 11 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Background
  4. Notation
  5. Exact Rejection Sampling Algorithms without discretization error
  6. Challenges with the EA3 rejection sampler
  7. Theoretical framework for our MCMC approach
  8. The GibbsEA1 sampler[wang2020exact]
  9. Our proposed GibbsEA3 sampler for EA3 diffusions
  10. Conditionally updating (psi,xi) from P(d psi,d xi | X)
  11. Conditionally updating X,xi from P(dX,dxi | psi)
  12. Comments on the algorithm
  13. Parameter inference
  14. Experiments
  15. EA2 example
  16. ...and 19 more sections

Key Result

Theorem 1

Define a probability $\mathcal{P}$ on the path-point process product space as Then, the probability measure $\mathcal{Q}^+$ (i.e. the SDE posterior) is the marginal of $\mathcal{P}$: $\mathcal{Q}^+(dX) \mathrel{\,\propto\,} \int_{\psi} \mathcal{Z}^+_{h}(dX) \times \exp(-m(X)T) \times \mathcal{M}_{\Delta(X)}(d\psi) \times \prod_{g \in \psi} \left( \frac{M(X) - \phi(X_g)}{\Delt

Figures (11)

  • Figure 1: The effective samples per second vs time lengths of the EA2 SDE under Gaussian noise $\sigma_N = 0.1$ (top row) and $\sigma_N=1$ (bottom row)
  • Figure 2: Effective samples per second vs time lengths of EA3 double-well diffusion under Gaussian noise $\sigma_N = 0.1$ (top), $\sigma_N=1$ (middle) and Poisson noise (bottom)
  • Figure 3: (left) raw, and (center) preprocessed $\delta^{18}\text{O}$ data, along with (right) posterior median and $95\%$ credible interval over paths from our methodology
  • Figure 4: Posterior inference results for the ice-core data. Top row: trace plots of $(p,q,\sigma_N)$. Bottom row: marginal posterior distributions.
  • Figure 5: ESS/s for different HMC settings in EA2 example with Gaussian noise $\sigma_N = 0.1$
  • ...and 6 more figures

Theorems & Definitions (12)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 3
  • proof
  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • ...and 2 more