Scaling of Model Approximation Errors and Expected Entropy Distances

TL;DR

This paper derives formulas for the expected Kullback-Leibler divergence between probability distributions and various statistical models under different priors, providing benchmarks for model approximation errors.

Contribution

It introduces closed-form expressions for expected divergence from models under uniform and Dirichlet priors, depending on model and sample space dimensions.

Findings

Expected divergence from models with uniform prior is bounded by 1−γ.

Bounds are approached with large state spaces and controlled model dimensions.

Results serve as reference benchmarks for complex statistical models.

Abstract

We compute the expected value of the Kullback-Leibler divergence to various fundamental statistical models with respect to canonical priors on the probability simplex. We obtain closed formulas for the expected model approximation errors, depending on the dimension of the models and the cardinalities of their sample spaces. For the uniform prior, the expected divergence from any model containing the uniform distribution is bounded by a constant $1-\gamma$, and for the models that we consider, this bound is approached if the state space is very large and the models' dimension does not grow too fast. For Dirichlet priors the expected divergence is bounded in a similar way, if the concentration parameters take reasonable values. These results serve as reference values for more complicated statistical models.