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A weighted Discrepancy Bound of quasi-Monte Carlo Importance Sampling

Josef Dick, Daniel Rudolf, Houying Zhu

TL;DR

The paper addresses the problem of computing expectations under an unnormalized distribution by a deterministic quasi-Monte Carlo importance sampling scheme. It develops a weighted star-discrepancy framework, proves a Koksma–Hlawka-type bound, and relates the weighted discrepancy to the classical star-discrepancy to leverage standard QMC convergence rates. The main result is an explicit bound $|S(f,u) - Q_n(f,u)| \le 4 (\|f\|_{H_1} \|u\|_D / \int u) D_{\lambda_d}(P_n)$, valid when $\|u\|_D<\infty$, showing that the convergence rate is governed by the star-discrepancy of the point set. The paper also provides a Dirichlet-density example to justify the regularity conditions and includes numerical experiments with Halton and Sobol sequences that support the theoretical predictions. This offers a deterministic, provable alternative to standard Monte Carlo importance sampling with explicit error control in terms of star-discrepancy.

Abstract

Importance sampling Monte-Carlo methods are widely used for the approximation of expectations with respect to partially known probability measures. In this paper we study a deterministic version of such an estimator based on quasi-Monte Carlo. We obtain an explicit error bound in terms of the star-discrepancy for this method.

A weighted Discrepancy Bound of quasi-Monte Carlo Importance Sampling

TL;DR

The paper addresses the problem of computing expectations under an unnormalized distribution by a deterministic quasi-Monte Carlo importance sampling scheme. It develops a weighted star-discrepancy framework, proves a Koksma–Hlawka-type bound, and relates the weighted discrepancy to the classical star-discrepancy to leverage standard QMC convergence rates. The main result is an explicit bound , valid when , showing that the convergence rate is governed by the star-discrepancy of the point set. The paper also provides a Dirichlet-density example to justify the regularity conditions and includes numerical experiments with Halton and Sobol sequences that support the theoretical predictions. This offers a deterministic, provable alternative to standard Monte Carlo importance sampling with explicit error control in terms of star-discrepancy.

Abstract

Importance sampling Monte-Carlo methods are widely used for the approximation of expectations with respect to partially known probability measures. In this paper we study a deterministic version of such an estimator based on quasi-Monte Carlo. We obtain an explicit error bound in terms of the star-discrepancy for this method.

Paper Structure

This paper contains 6 sections, 5 theorems, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Weighted Star-discrepancy and error bound
  3. Integration Error and weighted Star-discrepancy
  4. Weighted and classical Star-discrepancy
  5. Explicit error bound
  6. Illustrating Example

Key Result

Theorem 1

Let $\pi$ be a probability measure of the form eq: pi with unnormalized density $u\colon [0,1]^d \to [0,\infty)$. Then, for $P_n=\{{\boldsymbol{x}}_1, \ldots, {\boldsymbol{x}}_n \} \subset [0,1]^d$, arbitrary weight vector ${\boldsymbol{w}}=(w_1,\dots,w_n)\in \mathbb{R}^n$ with $\sum_{i=1}^d w_i=1$,

Figures (2)

  • Figure 1: Plot of the normalized error \ref{['eq: norm_err']} of $Q_n(f_{{\boldsymbol{\gamma}}},u(\cdot,{\boldsymbol{\alpha}}))$ based on the Halton sequence $H_n$ for $d=2,4,6$.
  • Figure 2: Plot of the normalized error \ref{['eq: norm_err']} of $Q_n(f_{{\boldsymbol{\gamma}}},u(\cdot,{\boldsymbol{\alpha}}))$ based on the Sobol sequence $S_n$ for $d=2,4,6$.

Theorems & Definitions (10)

  • Definition 1: Weighted Star-discrepancy
  • Remark 1
  • Theorem 1: Koskma-Hlawka inequality
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Lemma 1
  • proof
  • Corollary 1