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A novel viewpoint for Bayesian inversion based on the Poisson point process

Zhiliang Deng, Zhiyuan Wang, Xiaomei Yang, Xiaofei Guan

Abstract

We present a novel Bayesian framework for inverse problems in which the pos terior distribution is interpreted as the intensity measure of a Poisson point process (PPP). The posterior density is approximated using kernel density estimation, and the superposition property of PPPs is then exploited to enable efficient sampling from each kernel component. This methodology offers a new means of exploring the posterior distribution and facilitates the generation of independent and identically distributed samples, thereby enhancing the analysis of inverse problem solutions.

A novel viewpoint for Bayesian inversion based on the Poisson point process

Abstract

We present a novel Bayesian framework for inverse problems in which the pos terior distribution is interpreted as the intensity measure of a Poisson point process (PPP). The posterior density is approximated using kernel density estimation, and the superposition property of PPPs is then exploited to enable efficient sampling from each kernel component. This methodology offers a new means of exploring the posterior distribution and facilitates the generation of independent and identically distributed samples, thereby enhancing the analysis of inverse problem solutions.

Paper Structure

This paper contains 10 sections, 7 theorems, 30 equations, 5 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Poisson point process
  3. The perspective of PPP on Bayesian inversion
  4. A decomposition sampling approach
  5. Numerical examples
  6. Two-dimensional unimodal example
  7. Two-dimensional bimodal example.
  8. 2D inverse heat conduction
  9. High dimensional example
  10. Conclusion

Key Result

Proposition 1

Last2017 Let $\mu$ be a probability measure on $\mathbb{X}$ and let $\gamma\geq 0$. Suppose that $\eta$ is a mixed binomial process with mixing distribution $Poisson(\gamma)$ and sampling distribution $\mu$. Then $\eta$ is a PPP with intensity measure $\gamma\mu$.

Figures (5)

  • Figure 1: Numercial illustration for 2D unimodal example.
  • Figure 2: Numerical illustration for 2D bimodal example.
  • Figure 3: Heat conduction setups.
  • Figure 4: Numerical illustration for 2D inverse heat conduction.
  • Figure 5: Numerical illustration for high dimensional example.

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 1
  • Theorem 2.1: Superposition theorem
  • Theorem 2.2: Mapping theorem
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4