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Statistical algorithms for low-frequency diffusion data: A PDE approach

Matteo Giordano, Sven Wang

TL;DR

The paper introduces a PDE-based framework for nonparametric inference of multi-dimensional diffusion coefficients from low-frequency data by treating transition densities as heat-kernel solutions to the Fokker-Planck equation. A central advance is the closed-form gradient of the likelihood, derived via a variation-of-constants approach and expressed in a spectral basis of the elliptic generator, which reduces computation to finite-element-based elliptic eigenvalue problems. This enables efficient likelihood evaluation and gradient calculations, supporting both gradient-based (ULA, MAP) and gradient-free (pCN) posterior inference under Gaussian process priors, with extensive simulations demonstrating accurate conductivity recovery. The work also outlines extensions to more general reversible models with potential energy, discusses gradient-stability-based complexity considerations, and connects to small-time asymptotics and diffusion-generative modeling, highlighting broad methodological and practical implications. Overall, the PDE perspective provides scalable, principled tools for nonparametric diffusion inference in low-frequency regimes where traditional likelihood methods struggle.

Abstract

We consider the problem of making nonparametric inference in a class of multi-dimensional diffusions in divergence form, from low-frequency data. Statistical analysis in this setting is notoriously challenging due to the intractability of the likelihood and its gradient, and computational methods have thus far largely resorted to expensive simulation-based techniques. In this article, we propose a new computational approach which is motivated by PDE theory and is built around the characterisation of the transition densities as solutions of the associated heat (Fokker-Planck) equation. Employing optimal regularity results from the theory of parabolic PDEs, we prove a novel characterisation for the gradient of the likelihood. Using these developments, for the nonlinear inverse problem of recovering the diffusivity, we then show that the numerical evaluation of the likelihood and its gradient can be reduced to standard elliptic eigenvalue problems, solvable by powerful finite element methods. This enables the efficient implementation of a large class of popular statistical algorithms, including (i) preconditioned Crank-Nicolson and Langevin-type methods for posterior sampling, and (ii) gradient-based descent optimisation schemes to compute maximum likelihood and maximum-a-posteriori estimates. We showcase the effectiveness of these methods via extensive simulation studies in a nonparametric Bayesian model with Gaussian process priors, in which both the proposed optimisation and sampling schemes provide good numerical recovery. The reproducible code is available online at https://github.com/MattGiord/LF-Diffusion.

Statistical algorithms for low-frequency diffusion data: A PDE approach

TL;DR

The paper introduces a PDE-based framework for nonparametric inference of multi-dimensional diffusion coefficients from low-frequency data by treating transition densities as heat-kernel solutions to the Fokker-Planck equation. A central advance is the closed-form gradient of the likelihood, derived via a variation-of-constants approach and expressed in a spectral basis of the elliptic generator, which reduces computation to finite-element-based elliptic eigenvalue problems. This enables efficient likelihood evaluation and gradient calculations, supporting both gradient-based (ULA, MAP) and gradient-free (pCN) posterior inference under Gaussian process priors, with extensive simulations demonstrating accurate conductivity recovery. The work also outlines extensions to more general reversible models with potential energy, discusses gradient-stability-based complexity considerations, and connects to small-time asymptotics and diffusion-generative modeling, highlighting broad methodological and practical implications. Overall, the PDE perspective provides scalable, principled tools for nonparametric diffusion inference in low-frequency regimes where traditional likelihood methods struggle.

Abstract

We consider the problem of making nonparametric inference in a class of multi-dimensional diffusions in divergence form, from low-frequency data. Statistical analysis in this setting is notoriously challenging due to the intractability of the likelihood and its gradient, and computational methods have thus far largely resorted to expensive simulation-based techniques. In this article, we propose a new computational approach which is motivated by PDE theory and is built around the characterisation of the transition densities as solutions of the associated heat (Fokker-Planck) equation. Employing optimal regularity results from the theory of parabolic PDEs, we prove a novel characterisation for the gradient of the likelihood. Using these developments, for the nonlinear inverse problem of recovering the diffusivity, we then show that the numerical evaluation of the likelihood and its gradient can be reduced to standard elliptic eigenvalue problems, solvable by powerful finite element methods. This enables the efficient implementation of a large class of popular statistical algorithms, including (i) preconditioned Crank-Nicolson and Langevin-type methods for posterior sampling, and (ii) gradient-based descent optimisation schemes to compute maximum likelihood and maximum-a-posteriori estimates. We showcase the effectiveness of these methods via extensive simulation studies in a nonparametric Bayesian model with Gaussian process priors, in which both the proposed optimisation and sampling schemes provide good numerical recovery. The reproducible code is available online at https://github.com/MattGiord/LF-Diffusion.
Paper Structure (57 sections, 17 theorems, 191 equations, 9 figures, 5 tables)

This paper contains 57 sections, 17 theorems, 191 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Challenges
  3. Main contributions
  4. Related literature and discussion
  5. Likelihood and gradient computation via PDEs
  6. Parabolic PDE characterisations
  7. Reduction to elliptic eigenvalue problems
  8. Numerical PDE methods
  9. Applications to statistical algorithms
  10. A nonparametric Bayesian approach with Gaussian priors
  11. Parametrisation
  12. Prior and posterior
  13. Gradients of log-posterior densities
  14. Posterior inference via the pCN algorithm
  15. Posterior inference via the unadjusted Langevin algorithm
  16. ...and 42 more sections

Key Result

Theorem 2.1

Suppose that $d\le 3$, that $D>0$ and fix any $x,y\in \mathcal{O}$. Then, the Fréchet derivate of $\Phi$ at $f\in\mathcal{F}$ is given by the following linear operator More specifically, for any $R>0$ and $\kappa>0$, there exist $\zeta>0$ and $C>0$ such that for any $h\in C^2(\bar{\mathcal{O}})$ with $f+h \in \mathcal{F}$ and $\max \{\|f\|_{C^{1+\kappa}},\|f+h\|_{C^{1+\kappa}}\} \le R$, Here, $C

Figures (9)

  • Figure 1: Left to right: the (reparametrised) true conductivity function $F_0$, and the posterior mean estimates $\bar{F}_n$ for $n=500,~2500,~50000$, obtained via the pCN algorithm. Computation times ranged between 55 and 59 minutes.
  • Figure 2: Left: the acceptance ratio along the 25000 iterations of the pCN algorithm, for the case $n=50000$. Centre and right, respectively: the log-likelihood $\log(L_n(f_{\vartheta_m}))$ for the first 2500 chain steps, and for the steps from the $2500^{\textnormal{th}}$ to the $25000^{\textnormal{th}}$, again for $n=50000$.
  • Figure 3: Left: the posterior mean estimate $\bar{F}_n$, for $n=50000$, obtained via the ULA, to be compared to the ground truth $F_0$ shown in Figure \ref{['Fig:PostMean']} (left). The overall computational time was 90 minutes. Right: the log-posterior density $\log\pi(\vartheta_m|X^{(n)})$ for the first 1000 chain steps.
  • Figure 4: Left: the MAP estimate, with $n=50000$, computed by the gradient descent, to be compared to the ground truth $F_0$ shown in Figure \ref{['Fig:PostMean']} (left). The overall computational time was 1 minute. Centre: the distances $|\vartheta_{m+1} - \vartheta_m|$ between consecutive gradient descent iterates. Right: the log-posterior density $\log\pi(\vartheta_m|X^{(n)})$ along the gradient descent steps.
  • Figure 5: Left column: trace-plots for some individual components of 25000 approximate samples from the posterior distribution of $\theta|X^{(n)}$ obtained via the pCN algorithm with different starting points. Right column: resulting approximations to the marginal posterior probability density functions.
  • ...and 4 more figures

Theorems & Definitions (32)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Remark 2.4: Numerical approximation errors
  • Remark 2.5: Computational cost
  • Remark 3.1: $\alpha$-regular Gaussian priors
  • Proposition 3.2
  • Remark 3.3: Multimodality of the posterior
  • Theorem A.1
  • Lemma A.2
  • ...and 22 more