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Kinetic Theories for Metropolis Monte Carlo Methods

Michael Herty, Christian Ringhofer

TL;DR

This work develops a kinetic theory for Metropolis Monte Carlo methods applied to Bayesian-type inverse problems where the target is a distribution $P(x)$ over parameters. It derives two continuum limits of the MMC density evolution: a Boltzmann-type integro-differential equation for small acceptance rates and a Brownian-motion-type Fokker-Planck equation for small proposal increments, enabling macroscopic descriptions of MMC dynamics. A micro–macro decomposition is proposed to accelerate convergence by coupling a microscopic MMC updater with a macroscopic moment-based evolution, reducing computational cost while preserving accuracy. The theory is demonstrated on a Lorenz63 inverse problem, showing that the kinetic approach can yield richer posterior information and improved efficiency, with practical schemes for fixed and running terminal times and guidance for implementing MMC in complex, data-driven settings.

Abstract

We consider generalizations of the classical inverse problem to Bayesien type estimators, where the result is not one optimal parameter but an optimal probability distribution in parameter space. The practical computational tool to compute these distributions is the Metropolis Monte Carlo algorithm. We derive kinetic theories for the Metropolis Monte Carlo method in different scaling regimes. The derived equations yield a different point of view on the classical algorithm. It further inspired modifications to exploit the difference scalings shown on an simulation example of the Lorenz system.

Kinetic Theories for Metropolis Monte Carlo Methods

TL;DR

This work develops a kinetic theory for Metropolis Monte Carlo methods applied to Bayesian-type inverse problems where the target is a distribution over parameters. It derives two continuum limits of the MMC density evolution: a Boltzmann-type integro-differential equation for small acceptance rates and a Brownian-motion-type Fokker-Planck equation for small proposal increments, enabling macroscopic descriptions of MMC dynamics. A micro–macro decomposition is proposed to accelerate convergence by coupling a microscopic MMC updater with a macroscopic moment-based evolution, reducing computational cost while preserving accuracy. The theory is demonstrated on a Lorenz63 inverse problem, showing that the kinetic approach can yield richer posterior information and improved efficiency, with practical schemes for fixed and running terminal times and guidance for implementing MMC in complex, data-driven settings.

Abstract

We consider generalizations of the classical inverse problem to Bayesien type estimators, where the result is not one optimal parameter but an optimal probability distribution in parameter space. The practical computational tool to compute these distributions is the Metropolis Monte Carlo algorithm. We derive kinetic theories for the Metropolis Monte Carlo method in different scaling regimes. The derived equations yield a different point of view on the classical algorithm. It further inspired modifications to exploit the difference scalings shown on an simulation example of the Lorenz system.
Paper Structure (16 sections, 3 theorems, 143 equations, 4 figures)

This paper contains 16 sections, 3 theorems, 143 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Metropolis Monte Carlo algorithm (MMC)
  3. General definition of the method
  4. Evolution of the probability density
  5. Continuum limits for a large number of iterations
  6. The Boltzmann regime
  7. The Brownian motion regime
  8. A micro--macro decomposition
  9. Application to an Inverse Problem for the Lorentz System
  10. Fixed terminal time
  11. Running terminal time
  12. Summary
  13. Appendix
  14. Technical proofs
  15. Implementation details
  16. ...and 1 more sections

Key Result

Proposition 3.1

If the acceptance rate $\alpha$ is uniformly of order $O(h)=O(N^{c-1})$, then the solution $f(x,\kappa ,s)$ will, for $h\rightarrow 0,\ N\rightarrow \infty$ converge against the solution of the kinetic integro - differential equation equation with the integral kernel $K$ given by The proof of Proposition prpcntindx is deferred to the Appendix. We now compute the limiting solution $f(x,\kappa ,\

Figures (4)

  • Figure 1: Fixed terminal time simulated by Metropolis Monte--Carlo using Gaussian proposals. The initial distribution $P_0$ is depicted in blue as a histogram. The histogram of the terminal distribution $P_N$ is shown in red. In the top part we show the histogram of the parameters, in the bottom part the corresponding histogram of the model evaluations, i.e. $\mu$. With the parameters and model evaluations similar, we show $x_2$ and $x_3$ and $v_2$ and $v_3$ in the lower part of the diagram, respectively. The value 'true mean' represents the solution $v(T,x^*)$ for the optimal parameter $x^*.$
  • Figure 2: Fixed terminal time simulated using gradient--based proposal updates. The initial distribution $P_0$ is depicted in blue as a histogram. The histogram of the terminal distribution $P_N$ is shown in red. In the top part we show the histogram of the parameters, in the bottom part the corresponding histogram of the model evaluations, i.e. $\mu$. With the parameters and model evaluations similar, we show $x_2$ and $x_3$ and $v_2$ and $v_3$ in the lower part of the diagram, respectively. The value 'true mean' represents the solution $v(T,x^*)$ for the optimal parameter $x^*.$
  • Figure 3: Running terminal time simulated using Metropolis Monte Carlo method. The initial distribution $P_0$ is depicted in blue as a histogram. The histogram of the terminal distribution $P_N$ is shown in red. In the top part we show the histogram of the parameters, in the bottom part the corresponding histogram of the model evaluations, i.e. $\mu$. With the parameters and model evaluations similar, we show $x_2$ and $x_3$ and $v_2$ and $v_3$ in the lower part of the diagram, respectively. The value 'true mean' represents the solution $v(T,x^*)$ for the optimal parameter $x^*.$
  • Figure 4: Running terminal time simulated using micro--macro decomposition. The initial distribution $P_0$ is depicted in blue as a histogram. The histogram of the terminal distribution $P_N$ is shown in red. In the top part we show the histogram of the parameters, in the bottom part the corresponding histogram of the model evaluations, i.e. $\mu$. With the parameters and model evaluations similar, we show $x_2$ and $x_3$ and $v_2$ and $v_3$ in the lower part of the diagram, respectively. The value 'true mean' represents the solution $v(T,x^*)$ for the optimal parameter $x^*.$

Theorems & Definitions (9)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Remark 3.2
  • Remark 5.1