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Sampling with time-changed Markov processes

Andrea Bertazzi, Giorgos Vasdekis

TL;DR

This work addresses efficient sampling from challenging targets by introducing time-changed Markov processes, where a base process $Y$ is sped up or slowed down by a state-dependent function $s$ so that $X_t=Y_{r(t)}$ with $r(t)=\int_0^t s(X_u)\,du$. The authors develop a general framework showing that the tilted measure $\tilde{\mu}(dy)\propto s(y)\mu(dy)$ governs the base process, and establish LLN, ergodicity (geometric or uniform), and a functional central limit theorem for time-changed PDMPs, diffusions, and jump processes. They connect time-changes to space transformations via $s$ acting as a Jacobian-like term, and provide three practical strategies for estimating expectations with respect to $\mu$ (direct, path-reweighting, and biased-jump methods). Through detailed analysis of the Zig-Zag process, diffusion variants, and discrete-state jumps, along with numerical experiments on multimodal and heavy-tailed targets, the paper demonstrates improved convergence and tail exploration, offering a flexible toolkit for accelerating MCMC in challenging settings.

Abstract

We study time-changed Markov processes to speed up the convergence of Markov chain Monte Carlo (MCMC) algorithms. The time-changed process is defined by adjusting the speed of time of a base process via a user-chosen, state-dependent function. We explore the properties of such transformations and apply this idea to several Markov processes from the MCMC literature, such as Langevin diffusions and piecewise deterministic Markov processes, obtaining novel modifications of classical algorithms and also re-discovering known MCMC algorithms. We prove theoretical properties of the time-changed process under suitable conditions on the base process, focusing on connecting the stationary distributions and qualitative convergence properties such as geometric and uniform ergodicity, as well as a functional central limit theorem. We also provide a comparison with the framework of space transformations, clarifying the similarities between the approaches. Throughout the paper we give various visualisations and numerical simulations on simple tasks to gain intuition on the method and its performance. Finally, we provide numerical simulations to gain intuition on the method and its performance on benchmark problems. Our results indicate a performance improvement in the context of multimodal distributions and rare event simulation.

Sampling with time-changed Markov processes

TL;DR

This work addresses efficient sampling from challenging targets by introducing time-changed Markov processes, where a base process is sped up or slowed down by a state-dependent function so that with . The authors develop a general framework showing that the tilted measure governs the base process, and establish LLN, ergodicity (geometric or uniform), and a functional central limit theorem for time-changed PDMPs, diffusions, and jump processes. They connect time-changes to space transformations via acting as a Jacobian-like term, and provide three practical strategies for estimating expectations with respect to (direct, path-reweighting, and biased-jump methods). Through detailed analysis of the Zig-Zag process, diffusion variants, and discrete-state jumps, along with numerical experiments on multimodal and heavy-tailed targets, the paper demonstrates improved convergence and tail exploration, offering a flexible toolkit for accelerating MCMC in challenging settings.

Abstract

We study time-changed Markov processes to speed up the convergence of Markov chain Monte Carlo (MCMC) algorithms. The time-changed process is defined by adjusting the speed of time of a base process via a user-chosen, state-dependent function. We explore the properties of such transformations and apply this idea to several Markov processes from the MCMC literature, such as Langevin diffusions and piecewise deterministic Markov processes, obtaining novel modifications of classical algorithms and also re-discovering known MCMC algorithms. We prove theoretical properties of the time-changed process under suitable conditions on the base process, focusing on connecting the stationary distributions and qualitative convergence properties such as geometric and uniform ergodicity, as well as a functional central limit theorem. We also provide a comparison with the framework of space transformations, clarifying the similarities between the approaches. Throughout the paper we give various visualisations and numerical simulations on simple tasks to gain intuition on the method and its performance. Finally, we provide numerical simulations to gain intuition on the method and its performance on benchmark problems. Our results indicate a performance improvement in the context of multimodal distributions and rare event simulation.
Paper Structure (37 sections, 15 theorems, 67 equations, 6 figures)

This paper contains 37 sections, 15 theorems, 67 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Framework
  3. Time-changes of Markov processes
  4. Estimating expectations with time-changed Markov processes
  5. Examples
  6. Time-changes of piecewise deterministic Markov processes (PDMPs)
  7. Running example: the Zig-Zag process
  8. Other piecewise deterministic Markov processes
  9. Diffusion processes
  10. Jump processes on discrete state spaces
  11. Convergence properties of time changed Markov processes
  12. Ergodicity of time changed processes
  13. Functional central limit theorem
  14. Application to the running example
  15. Convergence of time-changed Markov jump processes
  16. ...and 22 more sections

Key Result

Theorem 2.3

Suppose ass:s.integrability and ass:lln_base hold. Then, the process $X$ has $\mu$ as unique stationary distribution. Furthermore, for any $f \in L^1(\mu)$, and all initial conditions $x \in \mathrm{E}$, we have

Figures (6)

  • Figure 1: Simulations for a mixture of two, one-dimensional standard normal random variables with means at $10$ and $-10$, and equal weight. The base process is the standard ZZP. The plots on the left show short simulations, highlighting the different dynamics, while the plots on the right show longer runs of the processes.
  • Figure 2: Simulations for a two-dimensional, isotropic student distribution with mean at the origin and $5$ degrees of freedom. The plot on the right shows the logarithm of the velocity in the colourmap.
  • Figure 3: Comparison between time-change and space transformation, using the standard ZZP as base process and a two-dimensional, standard normal target distribution. The choices of the speed function and diffeomorphism are as in \ref{['eq:speed_figure']} and \ref{['eq:diffeomorphism_figure']}. The colours and the scale on the right of each plot represent the logarithm of the speed of each process, calculated as distance travelled in a small time unit divided by the length of the time unit.
  • Figure 4: Numerical simulations in the context of Section \ref{['ex:gaussianmixture']}. The plots in the second row are obtained as follows. First, we simulate a jump process with initial condition at the origin, jump rate $s$, and jump kernel given by the Metropolis adjusted ZZP bertazzi_splitting with random step size $\mathrm{Exp}(1/\delta)$ for $\delta = 0.1$ and invariant distribution $\tilde{\mu} \propto s\mu$. Then, we discretise the obtained paths with step size $10^{-2}$. In either case, we performed $5\times 10^4$ iterations of the Metropolis adjusted ZZP.
  • Figure 5: Numerical simulations in the context of \ref{['sec:heavytailed']}. The plots show estimates of the probability of the event $\{x\in\mathbb{R}^2: \lvert x\rvert > 150\}$ obtained simulating $5000$ independent runs of the jump process based on the Metropolis-adjusted ZZP for $10^6$ iterations.
  • ...and 1 more figures

Theorems & Definitions (42)

  • Theorem 2.3: Invariance and LLN for the time-changed process
  • proof
  • Remark 2.4
  • Remark 2.5
  • Theorem 3.1: Time change of a PDMP
  • proof
  • Remark 3.2
  • Proposition 3.4: Time-changed ZZP
  • proof
  • Example 3.5: Bouncy Particle Sampler (BPS) BPS
  • ...and 32 more