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Nonparametric Bayesian inference for reversible multi-dimensional diffusions

Matteo Giordano, Kolyan Ray

TL;DR

We address nonparametric Bayesian inference for reversible multidimensional diffusions with periodic drift by modeling the potential $B$ (with $b=\nabla B$) and deriving posterior contraction rates for the gradient $\nabla B$ as $T\to\infty$, leveraging the invariant density $\mu_B \propto e^{2B}$. A general contraction theorem for gradient vector fields is developed via plug-in estimators of $\mu_B$ and concentration inequalities, and instantiated with Gaussian priors and $p$-exponential priors to obtain minimax-optimal rates $T^{-s/(2s+d)}$ over Sobolev classes in any dimension. The MAP and posterior mean attain the same rates under the stated conditions. The work further discusses generalizations to adaptive priors, non-periodic potentials, and non-constant diffusivity, highlighting the potential for Bayesian inverse problems approaches in stochastic dynamics.

Abstract

We study nonparametric Bayesian models for reversible multi-dimensional diffusions with periodic drift. For continuous observation paths, reversibility is exploited to prove a general posterior contraction rate theorem for the drift gradient vector field under approximation-theoretic conditions on the induced prior for the invariant measure. The general theorem is applied to Gaussian priors and $p$-exponential priors, which are shown to converge to the truth at the minimax optimal rate over Sobolev smoothness classes in any dimension.

Nonparametric Bayesian inference for reversible multi-dimensional diffusions

TL;DR

We address nonparametric Bayesian inference for reversible multidimensional diffusions with periodic drift by modeling the potential (with ) and deriving posterior contraction rates for the gradient as , leveraging the invariant density . A general contraction theorem for gradient vector fields is developed via plug-in estimators of and concentration inequalities, and instantiated with Gaussian priors and -exponential priors to obtain minimax-optimal rates over Sobolev classes in any dimension. The MAP and posterior mean attain the same rates under the stated conditions. The work further discusses generalizations to adaptive priors, non-periodic potentials, and non-constant diffusivity, highlighting the potential for Bayesian inverse problems approaches in stochastic dynamics.

Abstract

We study nonparametric Bayesian models for reversible multi-dimensional diffusions with periodic drift. For continuous observation paths, reversibility is exploited to prove a general posterior contraction rate theorem for the drift gradient vector field under approximation-theoretic conditions on the induced prior for the invariant measure. The general theorem is applied to Gaussian priors and -exponential priors, which are shown to converge to the truth at the minimax optimal rate over Sobolev smoothness classes in any dimension.

Paper Structure

This paper contains 33 sections, 16 theorems, 180 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Basic notation and definitions
  4. Diffusions with periodic drift and Bayesian inference
  5. Gaussian process priors
  6. Contraction rates
  7. Examples of Gaussian priors
  8. Conjugacy of Gaussian priors
  9. $p$-exponential priors
  10. A general contraction theorem for multi-dimensional diffusions with gradient vector field drift
  11. Generalizations and extensions
  12. Adaptation
  13. Non-periodic potentials
  14. Models with non-constant diffusivity
  15. Proof of Theorem \ref{['thm:contraction_b']}
  16. ...and 18 more sections

Key Result

Theorem 2.1

Let $\Pi =\Pi_T$ be the rescaled Gaussian process prior for $B$ in prior with $W\sim\Pi_W$ satisfying Condition GP_condition for some $s>(d-1)\vee(1/2)$, some $\kappa>0$, and RKHS $\mathbb{H}$. Suppose that $B_0\in H^{s+1}(\mathbb{T}^d)$ and that there exists a sequence $B_{0,T} \in \mathbb{H}$ such

Figures (1)

  • Figure 1: An example of a spatially heterogeneous potential $B$ (left) and the corresponding gradient vector field $\nabla B$ (right). Note the axis scales are different for clarity.

Theorems & Definitions (31)

  • Theorem 2.1
  • Corollary 2.2
  • Remark 2.1: Minimax rates
  • Example 2.1: Periodic Matérn process
  • Example 2.2: Truncated Gaussian series
  • Lemma 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Remark 2.2: Loss functions
  • Remark 2.3: Posterior sampling
  • ...and 21 more