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A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments

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High dimensional Bernstein-von Mises: simple examples.

Iain M Johnstone¹

¹Department of Statistics, Stanford University, CA 94305.

Institute of Mathematical Statistics Collections

|November 30, 2010

Summary

This summary is machine-generated.

This study explores how posterior and frequentist probabilities align in Gaussian sequence models as data dimensions increase. We demonstrate three scenarios, showing conditions for the Bernstein-von Mises theorem

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Area of Science:

Statistics
Bayesian Inference
Frequentist Statistics

Background:

Understanding the relationship between Bayesian posterior probabilities and frequentist probabilities is crucial in statistical modeling.
The Bernstein-von Mises theorem provides conditions under which posterior distributions approximate frequentist properties.
Investigating this relationship becomes complex as the dimensionality (p) of the model increases relative to the sample size (n).

Purpose of the Study:

To illustrate three distinct perspectives on the matching of posterior and frequentist probabilities in Gaussian sequence models.
To examine how this matching behaves as the dimension (p) increases with the sample size (n).
To explore the implications for the validity of the Bernstein-von Mises theorem under varying conditions.

Main Methods:

Development of simple, illustrative examples within Gaussian sequence models featuring Gaussian priors.
Analysis of three specific settings: convergence of joint posterior distributions, behavior of squared error loss (a non-linear functional), and estimation of linear functionals.
Progressive relaxation of conditions required for the Bernstein-von Mises theorem's applicability.

Main Results:

Demonstration of how posterior and frequentist probabilities converge under specific conditions as p increases with n.
Characterization of the behavior of a non-linear functional (squared error loss) in this high-dimensional context.
Evaluation of the accuracy of linear functional estimation from posterior distributions.

Conclusions:

The study provides concrete examples clarifying the nuanced agreement between Bayesian and frequentist approaches in increasing dimensions.
The findings highlight that the conditions for the Bernstein-von Mises theorem's validity can be relaxed in certain settings.
These results offer practical insights into the behavior of statistical inference as model complexity grows.