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A Robust Solution to Variational Importance Sampling of Minimum Variance.

Jerónimo Hernández-González1, Jesús Cerquides2

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Entropy (Basel, Switzerland)
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Summary

Importance sampling, a Monte Carlo method, uses proposal distributions to reduce variance. This study proposes an approximate projection method for discrete distributions, offering a practical alternative to costly exact computations.

Keywords:
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Area of Science:

  • Computational Statistics
  • Machine Learning
  • Probabilistic Graphical Models

Background:

  • Importance sampling is a Monte Carlo technique that utilizes an alternative proposal distribution to focus sampling and reduce variance.
  • Optimizing the proposal distribution for minimum variance can be framed as an optimization problem, often addressed using variational inference.
  • Variational inference seeks a distribution within a given family that minimizes divergence from the target distribution.

Purpose of the Study:

  • To propose and evaluate an approximate projection method for importance sampling in discrete distributions that factorize over probabilistic graphical models.
  • To address the computational cost associated with the Rényi projection of order 2, which yields the minimum variance importance sampling estimator.
  • To explore practical alternatives for optimizing proposal distributions in importance sampling.

Main Methods:

  • Developed an approximate projection method for importance sampling tailored to discrete distributions within probabilistic graphical models.
  • Evaluated the proposed method's performance and efficiency compared to existing techniques.
  • Investigated the use of variational approaches to solve the optimization problem of selecting minimum variance proposal distributions.

Main Results:

  • The proposed approximate projection method offers a computationally feasible alternative for achieving minimum variance importance sampling.
  • The study demonstrates the effectiveness of the approximate method for discrete, factorizing distributions.
  • A hybrid proposal distribution combining information projection and approximate Rényi projection of order 2 shows practical promise.

Conclusions:

  • Approximate projection methods provide a viable solution to the computational challenges of minimum variance importance sampling.
  • The proposed method is particularly relevant for discrete distributions in probabilistic graphical models.
  • Future work could explore the practical implementation and benefits of mixed proposal distributions for enhanced importance sampling.