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Sparse Bayesian Learning With Weakly Informative Hyperprior and Extended Predictive Information Criterion.

IEEE transactions on neural networks and learning systemsยท2021
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Updated: Jun 10, 2025

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
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Kazuaki Murayama1

  • 1Department of Computer and Network Engineering, Graduate School of Informatics and Engineering, <a href="https://ror.org/02x73b849">The University of Electro-Communications</a>, 1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585, Japan.

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Summary
This summary is machine-generated.

This study links statistical physics and Bayes inference using Bayesian linear regression. It finds that successful regression coefficient estimation balances decreasing energy and increasing entropy, akin to thermodynamic equilibrium states.

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Area of Science:

  • Statistical Physics
  • Bayesian Inference
  • Machine Learning

Background:

  • Analogies exist between statistical mechanics and Bayes inference, notably the partition function and marginal likelihood.
  • Previous work proposed associating discrete sample size with inverse temperature.
  • Thermodynamic functions like energy and entropy were suggested as analogs for Bayes estimation.

Purpose of the Study:

  • To incorporate a macroscopic perspective, similar to the thermodynamic limit, into the analogy between statistical physics and Bayes inference.
  • To identify a continuous analog for inverse temperature.
  • To analyze macroscopic thermodynamic functions for Bayesian linear regression and gain physical insights into Bayes estimation.

Main Methods:

  • Applied a macroscopic perspective (thermodynamic limit) to the statistical physics and Bayes inference analogy.
  • Analyzed analogs of macroscopic thermodynamic functions within a Bayesian linear regression model.
  • Investigated the behavior of these functions to understand Bayes estimation from a physics viewpoint.

Main Results:

  • Identified a candidate for a continuous analog of inverse temperature.
  • Regression coefficient estimation is described by a balance between decreasing energy and increasing entropy.
  • Successful estimation occurs when energy decrease dominates (low temperature); failure occurs when entropy increase dominates (high temperature).

Conclusions:

  • Bayes estimation, particularly in Bayesian linear regression, can be understood through physical principles of energy and entropy balance.
  • The macroscopic perspective provides physical insights into the success and failure mechanisms of estimation.
  • This framework offers a novel way to interpret machine learning models from a statistical physics standpoint.