arXiv subjects

Frank van der Meulen

Publications and source records attributed to Frank van der Meulen.

At least 19 recordsLinked to original sources

arXiv2026

Conditioning diffusions on a terminal bearing

We study sampling of a diffusion on $[0,T]$ conditioned on its terminal position lying on a prescribed ray, i.e. on an exactly observed bearing. For a linear drift we derive closed-form expressions for the Doob $h$-function and its gradient. For a broad class of tracking models admitting a Kronecker block structure, the guiding term is obtained from a single row of a matrix exponential, and the conditioned process decomposes into independent radial and angular components. For a nonlinear drift, we construct a guided process using the $h$-function of the linear model and prove absolute continuity of the law of the true conditioned process with respect to the law of the guided process, with an explicit likelihood ratio. This result is used to devise an auxiliary particle filter for bearings observed sequentially in time without noise. Numerical experiments show that the proposed filter approximates the filtering distribution and estimates the log-likelihood considerably more accurately than the shifted Rayleigh filter, at the cost of a higher computing time.

math.PR↗

arXiv2026

Neural Backward Filtering Forward Guiding

Inference in nonlinear continuous stochastic processes on trees is challenging, particularly when observations are sparse and the topology is complex. Exact smoothing via Doob's $h$-transform is intractable for general nonlinear dynamics. We propose Neural Backward Filtering Forward Guiding (NBFFG), a unified framework for both discrete transitions and continuous diffusions. Our method constructs a variational posterior by leveraging a proxy linear-Gaussian process. This proxy process yields a closed-form backward filter that serves as a guide, steering the generative path toward high-likelihood regions. We then learn a neural residual to capture the non-linear discrepancies. This formulation allows for an unbiased pathwise subsampling scheme, reducing the training complexity from tree-size dependent to path-length dependent. Empirical results show that NBFFG outperforms baselines on synthetic benchmarks, and we demonstrate the method on a high-dimensional inference task in phylogenetic analysis with reconstruction of ancestral butterfly wing shapes.

stat.ML↗

arXiv2026

Compositionality in algorithms for smoothing

Backward Filtering Forward Guiding (BFFG) is a bidirectional algorithm proposed in Mider et al. [2021] and studied more in depth in a general setting in Van der Meulen and Schauer [2022]. In category theory, optics have been proposed for modelling systems with bidirectional data flow. We connect BFFG with optics by demonstrating that the forward and backwards map together define a functor from a category of Markov kernels into a category of optics, which is furthermore lax monoidal in the case when the guiding kernels coincide with the generative dynamics

math.CT↗

arXiv2026

Semi-parametric Bernstein-von Mises Theorem in a Parabolic PDE Problem

We consider the heat equation with absorption in a bounded domain of $\mathbb{R}^d$, where both the scalar diffusivity and the absorption function are unknown. We investigate a Bayesian approach for recovering the diffusivity from a noisy observation of the solution to the PDE over the domain. Given a Gaussian process prior on the absorption function, we derive a Bernstein-von Mises theorem for the marginal posterior distribution of the diffusivity under assumptions on the prior and on smoothness properties of the absorption.

math.ST↗

arXiv2025

Guided filtering and smoothing for infinite-dimensional diffusions

We consider the filtering and smoothing problems for an infinite-dimensional diffusion process X, observed through a finite-dimensional representation at discrete points in time. At the heart of our proposed methodology lies the construction of a path measure, termed the guided distribution of X, that is absolutely continuous with respect to the law of X, conditioned on the observations. We show that this distribution can be incorporated as a potent proposal measure for both sequential Monte Carlo as well as Markov Chain Monte Carlo schemes to tackle the filtering and smoothing problems respectively. In the offline setting, we extend our approach to incorporate parameter estimation of unknown model parameters. The proposed methodology is numerically illustrated in a case study for the stochastic Amari equation.

math.PR↗

arXiv2025

Neural Guided Diffusion Bridges

We propose a novel method for simulating conditioned diffusion processes (diffusion bridges) in Euclidean spaces. By training a neural network to approximate bridge dynamics, our approach eliminates the need for computationally intensive Markov Chain Monte Carlo (MCMC) methods or score modeling. Compared to existing methods, it offers greater robustness across various diffusion specifications and conditioning scenarios. This applies in particular to rare events and multimodal distributions, which pose challenges for score-learning- and MCMC-based approaches. We introduce a flexible variational family, partially specified by a neural network, for approximating the diffusion bridge path measure. Once trained, it enables efficient sampling of independent bridges at a cost comparable to sampling the unconditioned (forward) process.

stat.ML↗

arXiv2025

Backward Filtering Forward Guiding

We develop a general methodological framework for probabilistic inference in discrete- and continuous-time stochastic processes evolving on directed acyclic graphs (DAGs). The process is observed only at the leaf nodes, and the challenge is to infer its full latent trajectory: a smoothing problem that arises in fields such as phylogenetics, epidemiology, and signal processing. Our approach combines a backward information filtering step, which constructs likelihood-informed potentials from observations, with a forward guiding step, where a tractable process is simulated under a change of measure constructed from these potentials. This Backward Filtering Forward Guiding (BFFG) scheme yields weighted samples from the posterior distribution over latent paths and is amenable to integration with MCMC and particle filtering methods. We demonstrate that BFFG applies to both discrete- and continuous-time models, enabling probabilistic inference in settings where standard transition densities are intractable or unavailable. Our framework opens avenues for incorporating structured stochastic dynamics into probabilistic programming. We numerically illustrate our approach for a branching diffusion process on a directed tree.

stat.ME↗

arXiv2025

Guided simulation of conditioned chemical reaction networks

Let $X$ be a chemical reaction process, modeled as a multi-dimensional continuous-time jump process. Assume that at given times $0< t_1 < \cdots <t_n$, linear combinations $v_i = L_i X(t_i),\, i=1,\dots ,n$ are observed for given matrices $L_i$. We show how the process that is conditioned on hitting the states $v_1,\dots, v_n$ is obtained by a change of measure on the law of the unconditioned process. This results in an algorithm for obtaining weighted samples from the conditioned process. Our results are illustrated by numerical simulations.

math.PR↗

arXiv2025

Simulation of infinite-dimensional diffusion bridges

Let X be the mild solution to a semilinear stochastic partial differential equation. In this article, we develop methodology to sample from the infinite-dimensional diffusion bridge that arises from conditioning X on a linear transformation LXT of the final state XT at some time T > 0. This solves a problem that has so far not been attended to in the literature. Our main contribution is the derivation of a path measure that is absolutely continuous with respect to the path measure of the infinite-dimensional diffusion bridge. This lifts previously known results for stochastic ordinary differential equations to the setting of infinite-dimensional diffusions and stochastic partial differential equations. We demonstrate our findings through numerical experiments on stochastic reaction-diffusion equations.

math.PR↗

arXiv2025

On a class of exponential changes of measure for stochastic PDEs

Given a mild solution $X$ to a semilinear stochastic partial differential equation (SPDE), we consider an exponential change of measure based on its infinitesimal generator $L$, defined in the topology of bounded pointwise convergence. The changed measure $\mathbb{P}^h$ depends on the choice of a function $h$ in the domain of $L$. In our main result, we derive conditions on $h$ for which the change of measure is of Girsanov-type. The process $X$ under $\mathbb{P}^h$ is then shown to be a mild solution to another SPDE with an extra additive drift-term. We illustrate how different choices of $h$ impact the law of $X$ under $\mathbb{P}^h$ in selected applications. These include the derivation of an infinite-dimensional diffusion bridge as well as the introduction of guided processes for SPDEs, generalizing results known for finite-dimensional diffusion processes to the infinite-dimensional case.

math.PR↗

arXiv2025

Simulating conditioned diffusions on manifolds

To date, most methods for simulating conditioned diffusions are limited to the Euclidean setting. The conditioned process can be constructed using a change of measure known as Doob's $h$-transform. The specific type of conditioning depends on a function $h$ which is typically unknown in closed form. To resolve this, we extend the notion of guided processes to a manifold $M$, where one replaces $h$ by a function based on the heat kernel on $M$. We consider the case of a Brownian motion with drift, constructed using the frame bundle of $M$, conditioned to hit a point $x_T$ at time $T$. We prove equivalence of the laws of the conditioned process and the guided process with a tractable Radon-Nikodym derivative. Subsequently, we show how one can obtain guided processes on any manifold $N$ that is diffeomorphic to $M$ without assuming knowledge of the heat kernel on $N$. We illustrate our results with numerical simulations of guided processes and Bayesian parameter estimation based on discrete-time observations. For this, we consider both the torus and the Poincaré disk.

math.PR↗

arXiv2024

Nonparametric Bayesian volatility learning under microstructure noise

In this work, we study the problem of learning the volatility under market microstructure noise. Specifically, we consider noisy discrete time observations from a stochastic differential equation and develop a novel computational method to learn the diffusion coefficient of the equation. We take a nonparametric Bayesian approach, where we \emph{a priori} model the volatility function as piecewise constant. Its prior is specified via the inverse Gamma Markov chain. Sampling from the posterior is accomplished by incorporating the Forward Filtering Backward Simulation algorithm in the Gibbs sampler. Good performance of the method is demonstrated on two representative synthetic data examples. We also apply the method on a EUR/USD exchange rate dataset. Finally we present a limit result on the prior distribution.

stat.ME↗

arXiv2023

Nonparametric Bayesian inference for stochastic processes with piecewise constant priors

We present a survey of some of our recent results on Bayesian nonparametric inference for a multitude of stochastic processes. The common feature is that the prior distribution in the cases considered is on suitable sets of piecewise constant or piecewise linear functions, that differ for the specific situations at hand. Posterior consistency and in most cases contraction rates for the estimators are presented. Numerical studies on simulated and real data accompany the theoretical results.

math.ST↗

arXiv2022

Automatic Backward Filtering Forward Guiding for Markov processes and graphical models

We incorporate discrete and continuous time Markov processes as building blocks into probabilistic graphical models with latent and observed variables. We introduce the automatic Backward Filtering Forward Guiding (BFFG) paradigm (Mider et al., 2021) for programmable inference on latent states and model parameters. Our starting point is a generative model, a forward description of the probabilistic process dynamics. We backpropagate the information provided by observations through the model to transform the generative (forward) model into a pre-conditional model guided by the data. It approximates the actual conditional model with known likelihood-ratio between the two. The backward filter and the forward change of measure are suitable to be incorporated into a probabilistic programming context because they can be formulated as a set of transformation rules. The guided generative model can be incorporated in different approaches to efficiently sample latent states and parameters conditional on observations. We show applicability in a variety of settings, including Markov chains with discrete state space, interacting particle systems, state space models, branching diffusions and Gamma processes.

stat.CO↗

arXiv2022

Introduction to Automatic Backward Filtering Forward Guiding

In this document I aim to give an informal treatment of automatic Backward Filtering Forward Guiding, a general algorithm for conditional sampling from a Markov process on a directed acyclic graph. I'll show that the underlying ideas can be understood with a basic background in probability and statistics. The more technical treatment is the paper Automatic backward filtering forward guiding for Markov processes and graphical models (Van der Meulen and Schauer, 2021). I specifically assume some background knowledge on likelihood based inference and Bayesian statistics. The final sections are more demanding and assume familiarity with continuous-time stochastic processes constructed from their infinitesimal generator. Clearly, all work discussed here is the result of research carried out over the past decade together with various collaborators, most importantly Moritz Schauer (Chalmers University of Technology and University of Gothenburg, Sweden). Section 8 is based on joint work with Marcin Mider (Trium Analysis Online GmbH, Germany) and Frank Schäfer (University of Basel, Switzerland) as well.

stat.CO↗

arXiv2022

Sticky PDMP samplers for sparse and local inference problems

We construct a new class of efficient Monte Carlo methods based on continuous-time piecewise deterministic Markov processes (PDMPs) suitable for inference in high dimensional sparse models, i.e. models for which there is prior knowledge that many coordinates are likely to be exactly $0$. This is achieved with the fairly simple idea of endowing existing PDMP samplers with 'sticky' coordinate axes, coordinate planes etc. Upon hitting those subspaces, an event is triggered during which the process sticks to the subspace, this way spending some time in a sub-model. This results in non-reversible jumps between different (sub-)models. While we show that PDMP samplers in general can be made sticky, we mainly focus on the Zig-Zag sampler. Compared to the Gibbs sampler for variable selection, we heuristically derive favourable dependence of the Sticky Zig-Zag sampler on dimension and data size. The computational efficiency of the Sticky Zig-Zag sampler is further established through numerical experiments where both the sample size and the dimension of the parameter space are large.

stat.CO↗

arXiv2022

Conditioning continuous-time Markov processes by guiding

A continuous-time Markov process $X$ can be conditioned to be in a given state at a fixed time $T > 0$ using Doob's $h$-transform. This transform requires the typically intractable transition density of $X$. The effect of the $h$-transform can be described as introducing a guiding force on the process. Replacing this force with an approximation defines the wider class of guided processes. For certain approximations the law of a guided process approximates - and is equivalent to - the actual conditional distribution, with tractable likelihood-ratio. The main contribution of this paper is to prove that the principle of a guided process, introduced in Schauer et al. (2017) for stochastic differential equations, can be extended to a more general class of Markov processes. In particular we apply the guiding technique to jump processes in discrete state spaces. The Markov process perspective enables us to improve upon existing results for hypo-elliptic diffusions.

math.PR↗

arXiv2021

Diffusion bridges for stochastic Hamiltonian systems and shape evolutions

Stochastically evolving geometric systems are studied in shape analysis and computational anatomy for modelling random evolutions of human organ shapes. The notion of geodesic paths between shapes is central to shape analysis and has a natural generalisation as diffusion bridges in a stochastic setting. Simulation of such bridges is key to solve inference and registration problems in shape analysis. We demonstrate how to apply state-of-the-art diffusion bridge simulation methods to recently introduced stochastic shape deformation models thereby substantially expanding the applicability of such models. We exemplify these methods by estimating template shapes from observed shape configurations while simultaneously learning model parameters.

math.NA↗