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Sampling from mixture distributions based on regime-switching diffusions

M. V. Tretyakov

TL;DR

The paper develops a regime-switching diffusion framework (SDEwS) to sample finite mixture distributions by coupling a diffusion with a state-dependent Markov switching mechanism. It adopts a backward Kolmogorov PDE approach to establish optimal weak convergence results for an explicit fixed-step Euler discretization, proving first-order accuracy in both finite time and the ergodic limit. It analyzes ensemble- and time-averaging estimators for ergodic limits, derives bias and variance bounds, and demonstrates the method on multiple mixture examples, including nonglobally Lipschitz potentials where trajectory rejection is used. The work contributes a principled, efficient SDE-based sampler for mixtures that does not require the normalization constant and provides rigorous weak-error guarantees and practical guidance for parallel computation in Monte Carlo contexts.

Abstract

It is proposed to use stochastic differential equations with state-dependent switching rates (SDEwS) for sampling from finite mixture distributions. An Euler scheme with constant time step for SDEwS is considered. It is shown that the scheme converges with order one in weak sense and also in the ergodic limit. Numerical experiments illustrate the use of SDEwS for sampling from mixture distributions and confirm the theoretical results.

Sampling from mixture distributions based on regime-switching diffusions

TL;DR

The paper develops a regime-switching diffusion framework (SDEwS) to sample finite mixture distributions by coupling a diffusion with a state-dependent Markov switching mechanism. It adopts a backward Kolmogorov PDE approach to establish optimal weak convergence results for an explicit fixed-step Euler discretization, proving first-order accuracy in both finite time and the ergodic limit. It analyzes ensemble- and time-averaging estimators for ergodic limits, derives bias and variance bounds, and demonstrates the method on multiple mixture examples, including nonglobally Lipschitz potentials where trajectory rejection is used. The work contributes a principled, efficient SDE-based sampler for mixtures that does not require the normalization constant and provides rigorous weak-error guarantees and practical guidance for parallel computation in Monte Carlo contexts.

Abstract

It is proposed to use stochastic differential equations with state-dependent switching rates (SDEwS) for sampling from finite mixture distributions. An Euler scheme with constant time step for SDEwS is considered. It is shown that the scheme converges with order one in weak sense and also in the ergodic limit. Numerical experiments illustrate the use of SDEwS for sampling from mixture distributions and confirm the theoretical results.
Paper Structure (10 sections, 8 theorems, 72 equations, 7 figures, 6 tables)

This paper contains 10 sections, 8 theorems, 72 equations, 7 figures, 6 tables.

Table of Contents

  1. Introduction
  2. SDEs with switching \newlabelsec:prel0
  3. Ergodic case
  4. Numerical method, estimators and their properties \newlabelsec:meth0
  5. Finite-time case \newlabelsec:mft0
  6. Ergodic case \newlabelsec:merg0
  7. Numerical experiments \newlabelsec:exp0
  8. Proofs \newlabelsec:proof0
  9. Finite-time convergence \newlabelsec:ftconv0
  10. Ergodic limits convergence \newlabelsec:ergproof0

Key Result

Proposition 2.2

\newlabelprop:rates0 Let Assumption ass:erg hold for the functions $U(x,m)$ and for the matrix $Q(x)$ satisfying (eq:q_choice). Then the solution $(X_{0,x,m}(t);{ \if@compatibility \mathchar"0116 {} \mathchar"0116 } _{0,x,m}(t))$ of the SDEwS (eq:erg), (eq:erg3) with $Q(x)$ satisfying (eq:q_c

Figures (7)

  • Figure 1: Example 1: mixture of two univariate Gaussian distributions (\ref{['eq:uGm2']}). The global error and the total variation distance for the Euler method. Error bars indicate the Monte Carlo error. \newlabelfig:2Gauerr0
  • Figure 2: Example 1. Plot of the exact density ${ \if@compatibility \mathchar"011A {} \mathchar"011A } (x)$ (\ref{['eq:uGm2']}) (orange) and the normalized histogram corresponding to the Euler approximation (blue) with $h=0.025$, $T=100$ and $M=4 \cdot10^8$. \newlabelfig:2Gaudist0
  • Figure 3: Example 2: mixture distribution (\ref{['eq:exa2']}). Global error and the total variation distance for the Euler method. Error bars indicate the Monte Carlo error. \newlabelfig:Example2Conv0
  • Figure 4: Example 2. Plot of the exact density ${ \if@compatibility \mathchar"011A {} \mathchar"011A } (x)$ (\ref{['eq:exa2']}) (orange) and the normalized histogram corresponding to the Euler approximation (blue) with $h=0.025$, $T=200$ and $M=10^8$. \newlabelfig:Example2dist0
  • Figure 5: Example 3: mixture distribution ${ \if@compatibility \mathchar"011A {} \mathchar"011A }$ from (\ref{['eq:exa3']}) with parameters as in Table \ref{['Tparam3']}. \newlabelfig:Example3rho0
  • ...and 2 more figures

Theorems & Definitions (13)

  • Proposition 2.2
  • Definition 1
  • Definition 2
  • Theorem 3.2
  • Remark 1
  • Theorem 3.4
  • Theorem 3.5
  • Remark 2
  • Lemma 5.1
  • Lemma 5.2
  • ...and 3 more