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How to implement the Bayes' formula in the age of ML?

Amirhossein Taghvaei, Prashant G. Mehta

Abstract

This chapter contains a self-contained introduction to the significance of Bayes' formula in the context of nonlinear filtering problems. Both discrete-time and continuous-time settings of the problem are considered in a unified manner. In control theory, the focus on optimization-based solution approaches is stressed together with a discussion of historical developments in this area (from 1960s onwards). The heart of this chapter contains a presentation of a novel optimal transportation formulation for the Bayes formula (developed recently by the first author) and its relationship to some of the prior joint work (feedback particle filter) from the authors. The presentation highlights how optimal transportation theory is leveraged to overcome some of the numerical challenges of implementing Bayes' law by enabling the use of machine learning (ML) tools.

How to implement the Bayes' formula in the age of ML?

Abstract

This chapter contains a self-contained introduction to the significance of Bayes' formula in the context of nonlinear filtering problems. Both discrete-time and continuous-time settings of the problem are considered in a unified manner. In control theory, the focus on optimization-based solution approaches is stressed together with a discussion of historical developments in this area (from 1960s onwards). The heart of this chapter contains a presentation of a novel optimal transportation formulation for the Bayes formula (developed recently by the first author) and its relationship to some of the prior joint work (feedback particle filter) from the authors. The presentation highlights how optimal transportation theory is leveraged to overcome some of the numerical challenges of implementing Bayes' law by enabling the use of machine learning (ML) tools.

Paper Structure

This paper contains 28 sections, 8 theorems, 60 equations, 5 figures.

Table of Contents

  1. How to implement the Bayes' formula in the age of ML?
  2. Introduction
  3. Implementing Bayes' formula in 1960s
  4. Aims of this article
  5. Paper outline
  6. Nonlinear filtering and Bayes' formula
  7. Nonlinear filter
  8. Particle filter
  9. Fundamental challenges in implementing the Bayes update
  10. Bayes update using importance sampling and the curse of dimensionality
  11. Bayesian update for the ensemble Kalman filter
  12. Bayes update in feedback particle filter (FPF)
  13. Summary
  14. Bayes update with optimal transport maps (a.k.a. "implementing Bayes update in 2020s")
  15. Numerical approximation of the OT map
  16. ...and 13 more sections

Key Result

Proposition 1

Consider the SIR procedure for approximating the conditional distribution. Suppose $\overline X$ is an independent copy of $X$. Assume $\mathbb E[\frac{\ell(Y|\overline X)^2}{\mathbb E[\ell(Y|\overline X)|Y]^2}]<\infty$. Then, where $V(g):=\mathbb E[\frac{\ell(Y|\overline X)^2}{\mathbb E[\ell(Y|\overline X)|Y]^2}(g(\overline X)-\mathbb E[g(X)|Y])^2]$. In particular, for the special case where $X=

Figures (5)

  • Figure 1: This exposition is concerned with particle filter algorithms that seek to approximate the posterior with the empirical distribution of an ensemble of particles.
  • Figure 2: Neural net architectures for the function classes $\mathcal{F}$ and $\mathcal{T}$ within our proposed algorithm.
  • Figure 3: Numerical results for the static example in Sec. \ref{['sec:Static_Example']}. (a) top-left: Samples $\{X^i\}_{i=1}^N$ from the prior $P_X$; bottom-left: samples $\{(X^i,Y^i)\}_{i=1}^N$ from the joint distribution $P_{X,Y}$ in comparison with the transported samples $\{(T(X^{\sigma_i},Y^i),Y^i)\}_{i=1}^N$; rest of the panels: transported samples for $Y=1$ for different values of $N$ and three different algorithms. (b) Similar results to panel (a) but for a smaller $\lambda_w$.
  • Figure 4: Numerical results for the dynamic example \ref{['eq:model-example']}. The left panel shows the trajectory of the particles $\{X^1_t,\ldots,X^N_t\}$ along with the trajectory of the true state $X_t$ for EnKF, OT, and SIR algorithms, respectively. The second panel shows the MMD distance with respect to the exact conditional distribution. The last two panels show MMD variation with dimension and the number of particles.
  • Figure 5: Numerical results for the Lorenz 63 example. The left panel shows the trajectory of the unobserved component of the true state and the particles. The right panel shows the MSE comparison.

Theorems & Definitions (21)

  • Remark 1
  • Definition 1
  • Proposition 1
  • Theorem : Proof
  • Proposition 2
  • Proof 1
  • Proposition 3
  • Proof 2
  • Example 1: Noiseless observation
  • Proposition 4
  • ...and 11 more