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Guided smoothing and control for diffusion processes

Oskar Eklund, Annika Lang, Moritz Schauer

TL;DR

This work tackles smoothing for diffusion processes observed continuously by deriving data-dependent drift terms through enlargement of filtrations and introducing guided processes to approximate the conditional dynamics. It couples a principled change of measure (Girsanov) with tractable linear-guided proposals, enabling MC/SMC sampling of $P(X|Y)$ via weights and MH in Wiener space. In the linear Gaussian case, explicit backward filters yield Gaussian likelihoods and explicit smoothed SDEs; in general, the guiding term is obtained from linearizations and backward filtering to produce practical, scalable inference in high dimensions. The authors demonstrate the approach on a 100-dimensional reaction–diffusion system, achieving accurate posterior means and feasible computation, and show how variational and MH-based methods can complement guided proposals.

Abstract

The smoothing distribution is the conditional distribution of the diffusion process in the space of trajectories given noisy observations made continuously in time. It is generally difficult to sample from this distribution. We use the theory of enlargement of filtrations to show that the conditional process has an additional drift term derived from the backward filtering distribution that is moving or guiding the process towards the observations. This term is intractable, but its effect can be equally introduced by replacing it with a heuristic, where importance weights correct for the discrepancy. From this Markov Chain Monte Carlo and sequential Monte Carlo algorithms are derived to sample from the smoothing distribution. The choice of the guiding heuristic is discussed from an optimal control perspective and evaluated. The results are tested numerically on a stochastic differential equation for reaction-diffusion.

Guided smoothing and control for diffusion processes

TL;DR

This work tackles smoothing for diffusion processes observed continuously by deriving data-dependent drift terms through enlargement of filtrations and introducing guided processes to approximate the conditional dynamics. It couples a principled change of measure (Girsanov) with tractable linear-guided proposals, enabling MC/SMC sampling of via weights and MH in Wiener space. In the linear Gaussian case, explicit backward filters yield Gaussian likelihoods and explicit smoothed SDEs; in general, the guiding term is obtained from linearizations and backward filtering to produce practical, scalable inference in high dimensions. The authors demonstrate the approach on a 100-dimensional reaction–diffusion system, achieving accurate posterior means and feasible computation, and show how variational and MH-based methods can complement guided proposals.

Abstract

The smoothing distribution is the conditional distribution of the diffusion process in the space of trajectories given noisy observations made continuously in time. It is generally difficult to sample from this distribution. We use the theory of enlargement of filtrations to show that the conditional process has an additional drift term derived from the backward filtering distribution that is moving or guiding the process towards the observations. This term is intractable, but its effect can be equally introduced by replacing it with a heuristic, where importance weights correct for the discrepancy. From this Markov Chain Monte Carlo and sequential Monte Carlo algorithms are derived to sample from the smoothing distribution. The choice of the guiding heuristic is discussed from an optimal control perspective and evaluated. The results are tested numerically on a stochastic differential equation for reaction-diffusion.

Paper Structure

This paper contains 13 sections, 8 theorems, 93 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Smoothing for continuously observed diffusions
  3. Decoupling change of measure for the joint process
  4. The dynamics of the conditional process
  5. Connection to stochastic PDEs
  6. Backward filter for the likelihood process in the linear case
  7. Guided processes
  8. Guiding based on a linear approximation
  9. Monte Carlo method
  10. Bellman principle and optimal control for the smoothing distribution
  11. Optimizing the guide
  12. Application: Reaction-diffusion
  13. Proofs

Key Result

Proposition 2.3

For bounded measurable real test functions $f$ on the path space $C([0,T], \mathbb R^d)$, it holds

Figures (5)

  • Figure 1: Forward simulation of system \ref{['reaction1']}. Left: Heatmap of $(X^{(j)}_{t})$. (Horizontal axis: time. Vertical: Coordinates. Color: Value.) Right: Coordinate process $X^{(50)}$ and, in bold, the corresponding observation process $Y^{(50)}$.
  • Figure 2: Left: Estimate $\mathbb{E} [X_t \mid Y, \zeta]$. Right: A sample of $X \mid Y, \zeta$.
  • Figure 3: Left: Forward simulation of the system \ref{['reaction1']}, $d=20$. Centre: Posterior mean computed with Metropolis--Hastings for a noisy observation of the left sample. Right: Mean of variational approximation of the same posterior.
  • Figure 4: Left: Loss curve (negative reward versus iteration). Center: $B(\theta^\star)$ versus $-5\Lambda$. Right: $m(\theta^\star)$.
  • Figure 5: Left: Error $|\mu^{\star,(i)}_t-\mu^{\circ,(i)}_t|$ corresponding to the ad hoc choice. Right: Error $|\mu^{\star,(i)}_t-\hat{\mu}^{(i)}_t|$ corresponding to the variational optimum.

Theorems & Definitions (17)

  • Proposition 2.3
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • Proposition 2.6
  • Remark 2.7
  • Remark 2.8
  • Proposition 2.9
  • Theorem 3.1
  • proof
  • ...and 7 more