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Update of Prior Probabilities by Minimal Divergence.

¹Departement Fysica, Universiteit Antwerpen, 2610 Antwerpen, Belgium.

Entropy (Basel, Switzerland)

|December 24, 2021

Summary

This summary is machine-generated.

This study updates probability distributions using new data, balancing new observations with prior knowledge. Optimal methods involve minimizing Hellinger distance or quadratic Bregman divergence, yielding distinct results.

Keywords:

Bregman divergence Hellinger distance Jeffrey conditioning minimal divergence statistical update procedure

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Area of Science:

Statistics
Probability Theory
Data Analysis

Background:

Empirical probability distributions are fundamental in statistical modeling.
Updating these distributions with new data is crucial for adapting models.
Existing methods may not optimally balance new evidence with prior information.

Purpose of the Study:

To investigate methods for updating empirical probability distributions.
To compare updates that incorporate new observations and prior information.
To explore the impact of different divergence measures on the update process.

Main Methods:

Updating empirical probability distributions using new observational data.
Employing minimization of Hellinger distance for optimal update.
Employing minimization of quadratic Bregman divergence for optimal update.
Considering updates incorporating conditional probability information.

Main Results:

The update process successfully reproduces new observations while interpolating with prior information.
Minimizing Hellinger distance and quadratic Bregman divergence yield different optimal updates.
The choice of divergence measure influences the resulting probability distribution.

Conclusions:

Both Hellinger distance and quadratic Bregman divergence provide valid frameworks for updating probability distributions.
The divergence between these methods highlights the importance of selecting an appropriate measure.
Further research can explore the implications of conditional probability updates.