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Efficient Incremental Belief Updates Using Weighted Virtual Observations

David Tolpin

TL;DR

The paper tackles the challenge of incremental belief updates in Monte Carlo Bayesian inference for probabilistic programs by conditioning the model on a small set of weighted virtual observations to reproduce the original posterior $p(x|y^*)$. It develops a theoretically grounded framework that starts from exact results for conjugate models and extends to efficient approximate solutions for single-level and multi-level models, accompanied by a practical optimization procedure. A reference implementation demonstrates the method across didactic examples and real case studies, achieving substantial data compression and notable speedups while preserving posterior fidelity. The work also situates itself relative to empirical Bayes, Bayesian coresets, inference compilation, and federated learning, offering a flexible, model-agnostic approach to efficient incremental inference.

Abstract

We present an algorithmic solution to the problem of incremental belief updating in the context of Monte Carlo inference in Bayesian statistical models represented by probabilistic programs. Given a model and a sample-approximated posterior, our solution constructs a set of weighted observations to condition the model such that inference would result in the same posterior. This problem arises e.g. in multi-level modelling, incremental inference, inference in presence of privacy constraints. First, a set of virtual observations is selected, then, observation weights are found through a computationally efficient optimization procedure such that the reconstructed posterior coincides with or closely approximates the original posterior. We implement and apply the solution to a number of didactic examples and case studies, showing efficiency and robustness of our approach. The provided reference implementation is agnostic to the probabilistic programming language or the inference algorithm, and can be applied to most mainstream probabilistic programming environments.

Efficient Incremental Belief Updates Using Weighted Virtual Observations

TL;DR

The paper tackles the challenge of incremental belief updates in Monte Carlo Bayesian inference for probabilistic programs by conditioning the model on a small set of weighted virtual observations to reproduce the original posterior . It develops a theoretically grounded framework that starts from exact results for conjugate models and extends to efficient approximate solutions for single-level and multi-level models, accompanied by a practical optimization procedure. A reference implementation demonstrates the method across didactic examples and real case studies, achieving substantial data compression and notable speedups while preserving posterior fidelity. The work also situates itself relative to empirical Bayes, Bayesian coresets, inference compilation, and federated learning, offering a flexible, model-agnostic approach to efficient incremental inference.

Abstract

We present an algorithmic solution to the problem of incremental belief updating in the context of Monte Carlo inference in Bayesian statistical models represented by probabilistic programs. Given a model and a sample-approximated posterior, our solution constructs a set of weighted observations to condition the model such that inference would result in the same posterior. This problem arises e.g. in multi-level modelling, incremental inference, inference in presence of privacy constraints. First, a set of virtual observations is selected, then, observation weights are found through a computationally efficient optimization procedure such that the reconstructed posterior coincides with or closely approximates the original posterior. We implement and apply the solution to a number of didactic examples and case studies, showing efficiency and robustness of our approach. The provided reference implementation is agnostic to the probabilistic programming language or the inference algorithm, and can be applied to most mainstream probabilistic programming environments.
Paper Structure (18 sections, 5 theorems, 29 equations, 8 figures, 2 algorithms)

This paper contains 18 sections, 5 theorems, 29 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem
  3. General Problem
  4. Hardness of General Case
  5. Solution for Conjugate Models
  6. Problem Admitting Efficient Approximate Solution
  7. Solution
  8. Single-Level Models
  9. Example: Beta-Bernoulli Model
  10. Example: Normal Model with Non-Informative Prior
  11. Multi-Level Models
  12. Example: Normal Model With Hyperprior on The Mean
  13. Case Studies
  14. Parallel Experiments in Eight Schools
  15. Tumor Incidence in Rats
  16. ...and 3 more sections

Key Result

Theorem 1

The problem of observation reconstruction is NP-hard.

Figures (8)

  • Figure 1: Beta-Bernoulli model: posteriors and weights.
  • Figure 2: Normal model: posteriors and weights.
  • Figure 3: Normal model with hyperprior: posteriors.
  • Figure 4: Eight schools: posteriors and weights.
  • Figure 5: Eight schools: cross-validation.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 1
  • Theorem 3
  • proof
  • Definition 2
  • Theorem 4
  • proof
  • ...and 3 more