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Semiparametric Bernstein-von Mises theorems for reversible diffusions

Matteo Giordano, Kolyan Ray

TL;DR

This work develops a general semiparametric Bernstein–von Mises framework for Bayesian nonparametric priors in continuous-time reversible diffusions with periodic drift, enabling robust uncertainty quantification for low-dimensional functionals of the potential $B$ (or the invariant measure $\mu_B$) from long-time observations. The authors convert the inference problem into a LAN-based analysis linked to an elliptic PDE $A_{\mu_0}$, and prove that the marginal posterior for $\sqrt{T}(\Psi(B)-\hat{\Psi}_T)$ converges to a Gaussian with variance $\|\nabla A_{\mu_0}^{-1}\psi\|_{\mu_0}^2$ under suitable concentration and change-of-measure conditions. They instantiate the theory for Gaussian and Besov-Laplace priors, deriving conjugate posterior forms in the Gaussian case and establishing semiparametric BvM for a broad class of nonlinear functionals, including entropy and invariants of the diffusion. Numerical experiments with Gaussian priors corroborate the asymptotic Gaussianity of plug-in functionals and demonstrate improving coverage and shrinking interval lengths as the horizon $T$ grows. The results justify Bayesian uncertainty quantification for semiparametric inference in reversible diffusion models and provide a methodological bridge between stochastic dynamics, elliptic PDE theory, and nonparametric Bayesian inference.

Abstract

We establish a general semiparametric Bernstein-von Mises theorem for Bayesian nonparametric priors based on continuous observations in a periodic reversible multidimensional diffusion model. We consider a wide range of functionals satisfying an approximate linearization condition, including several nonlinear functionals of the invariant measure. Our result is applied to Gaussian and Besov-Laplace priors, showing these can perform efficient semiparametric inference and thus justifying the corresponding Bayesian approach to uncertainty quantification. Our theoretical results are illustrated via numerical simulations.

Semiparametric Bernstein-von Mises theorems for reversible diffusions

TL;DR

This work develops a general semiparametric Bernstein–von Mises framework for Bayesian nonparametric priors in continuous-time reversible diffusions with periodic drift, enabling robust uncertainty quantification for low-dimensional functionals of the potential (or the invariant measure ) from long-time observations. The authors convert the inference problem into a LAN-based analysis linked to an elliptic PDE , and prove that the marginal posterior for converges to a Gaussian with variance under suitable concentration and change-of-measure conditions. They instantiate the theory for Gaussian and Besov-Laplace priors, deriving conjugate posterior forms in the Gaussian case and establishing semiparametric BvM for a broad class of nonlinear functionals, including entropy and invariants of the diffusion. Numerical experiments with Gaussian priors corroborate the asymptotic Gaussianity of plug-in functionals and demonstrate improving coverage and shrinking interval lengths as the horizon grows. The results justify Bayesian uncertainty quantification for semiparametric inference in reversible diffusion models and provide a methodological bridge between stochastic dynamics, elliptic PDE theory, and nonparametric Bayesian inference.

Abstract

We establish a general semiparametric Bernstein-von Mises theorem for Bayesian nonparametric priors based on continuous observations in a periodic reversible multidimensional diffusion model. We consider a wide range of functionals satisfying an approximate linearization condition, including several nonlinear functionals of the invariant measure. Our result is applied to Gaussian and Besov-Laplace priors, showing these can perform efficient semiparametric inference and thus justifying the corresponding Bayesian approach to uncertainty quantification. Our theoretical results are illustrated via numerical simulations.

Paper Structure

This paper contains 19 sections, 13 theorems, 141 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation.
  3. Main results
  4. Reversible diffusions with periodic drift and Bayesian inference
  5. A general semiparametric Bernstein--von Mises theorem
  6. Gaussian priors
  7. Besov-Laplace priors
  8. Numerical illustrations with Gaussian priors
  9. Data generation.
  10. Prior specification and posterior inference.
  11. Results.
  12. Proofs
  13. Proof of Theorem \ref{['thm:BvM']}: general semiparametric BvM
  14. Auxiliary results
  15. Proof of Theorem \ref{['Theo:GaussBvM']}: Gaussian priors
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\Pi=\Pi_T$ be a prior for $B$ that is supported on $\dot{C}^2(\mathbb{T}^d)$. Suppose $B_0 \in \dot{C}^{(d/2+1+\kappa)\vee 2}(\mathbb{T}^d)$ for some $\kappa>0$, and let $\Psi:\dot{C}^2(\mathbb{T}^d) \to \mathbb{R}$ be a functional satisfying the expansion Eq:psi_exp with representor $\psi \in for some $M>0$ and $\varepsilon_T,\zeta_T,\xi_T \to 0$ with $\sqrt{T}\varepsilon_T\to\infty$. Let $

Figures (3)

  • Figure 1: Left: a periodic potential energy field $B$. Right: a continuous trajectory $(X_t, \ 0\le t\le T)$ started at $X_0=(1,1)$ and run until time $T=1$.
  • Figure 2: Top row: the three potential energy fields $B^{(i)}$, $i=1,2,3$, from \ref{['Eq:Truths']}. Bottom row: the corresponding posterior means $\bar{B}^{(i)}_{T}$, $i=1,2,3$, at time $T=100$. The relative $L^2$-estimation errors $\|B^{(i)} - \bar{B}^{(i)}_T\|_2/\|B^{(i)}\|_2$ are equal to 0.21, 0.03, 0.12, respectively.
  • Figure 3: Top row: plug-in posterior distributions of $\Psi_i(B)|X^T$, $i=1,2,3$, at $T=50$. Bottom row: plug-in posteriors at $T=100$. The vertical red lines indicate the 'ground truths' $\Psi_i(B_1)$, for $B_1$ as in Figure \ref{['Fig:Truths']} (top-left). The vertical blue lines identify the $95\%$ credible intervals. The green line corresponds to a normal PDF centred at the posterior mean and with variance equal to the posterior variance.

Theorems & Definitions (41)

  • Definition 1
  • Theorem 1: Semiparametric Bernstein--von Mises
  • Example 1: Linear functionals
  • Example 2: Square functional
  • Example 3: Power functionals
  • Remark 1: Posterior contraction rates and remainder
  • Example 4: Linear functionals of the invariant measure
  • Example 5: Entropy of the invariant measure
  • Example 6: Square-root of the invariant measure
  • Example 7: Power functionals of the invariant measure
  • ...and 31 more