An Information-Theoretic Foundation for the Weighted Updating Model

TL;DR

This paper establishes an information-theoretic foundation for the Weighted Updating model, showing how exponential weighting of distributions affects their entropy and interpretation in decision-making.

Contribution

It provides a rigorous basis linking weighted Bayesian updating to changes in information entropy, clarifying the meaning of weights in biased belief formation.

Findings

Weights > 1 reduce entropy, indicating more informative beliefs.

Weights < 1 increase entropy, indicating less informative beliefs.

The model offers a systematic way to interpret biased updating in decision processes.

Abstract

Weighted Updating generalizes Bayesian updating, allowing for biased beliefs by weighting the likelihood function and prior distribution with positive real exponents. I provide a rigorous foundation for the model by showing that transforming a distribution by exponential weighting (and normalizing) systematically affects the information entropy of the resulting distribution. For weights greater than one the resulting distribution has less information entropy than the original distribution, and vice versa. The entropy of a distribution measures how informative a decision maker is treating the underlying observation(s), so this result suggests a useful interpretation of the weights. For example, a weight greater than one on a likelihood function models an individual who is treating the associated observation(s) as being more informative than a perfect Bayesian would.