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This study introduces a novel Bayesian inference framework for stochastic systems with limited data. It enables data-driven prior selection, simplifying property estimation without needing handcrafted priors.

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Area of Science:

  • Statistical Inference
  • Machine Learning
  • Information Theory

Background:

  • Estimating properties of large stochastic systems is challenging with insufficient data.
  • Bayesian inference requires prior distributions, which are often difficult to specify.
  • Existing methods use property-specific priors for entropy, mutual information, and correlations.

Purpose of the Study:

  • To develop a general framework for selecting priors in Bayesian inference for arbitrary properties.
  • To simplify the inference process when dealing with limited sample data.
  • To reduce the need for manually defined priors in complex systems.

Main Methods:

  • Expanding the prior distribution as a linear combination of indexed priors derived from maximum entropy principles.
  • Constraining the mean values of the property of interest within the maximum entropy framework.
  • Demonstrating that only specific aspects of the prior influence the inference process.

Main Results:

  • A general framework for selecting priors applicable to arbitrary properties was proposed.
  • The method showed that often only one or a few components of the prior expansion are significant.
  • The data selects the relevant prior component, eliminating the need for handcrafted priors.
  • The proposed approximation performed well compared to existing ad-hoc methods.

Conclusions:

  • The developed framework offers a data-driven approach to prior selection in Bayesian inference.
  • This method simplifies property estimation in stochastic systems with limited data.
  • The approach reveals connections between Bayesian inference and equilibrium statistical mechanics, relating relevant priors to system temperature.