On Maximum Entropy and Inference

TL;DR

This paper investigates how maximum entropy principles can be used both to infer relevant variables directly from data and to estimate models, especially in complex spin systems, emphasizing the role of data distribution over parameter count.

Contribution

It introduces a method to infer relevant interactions in spin models using maximum entropy, highlighting the importance of data distribution in the inference process.

Findings

Successfully recovers correct models in prototype cases

Demonstrates applicability on real dataset

Highlights the role of data distribution in inference

Abstract

Maximum Entropy is a powerful concept that entails a sharp separation between relevant and irrelevant variables. It is typically invoked in inference, once an assumption is made on what the relevant variables are, in order to estimate a model from data, that affords predictions on all other (dependent) variables. Conversely, maximum entropy can be invoked to retrieve the relevant variables (sufficient statistics) directly from the data, once a model is identified by Bayesian model selection. We explore this approach in the case of spin models with interactions of arbitrary order, and we discuss how relevant interactions can be inferred. In this perspective, the dimensionality of the inference problem is not set by the number of parameters in the model, but by the frequency distribution of the data. We illustrate the method showing its ability to recover the correct model in a few prototype cases and discuss its application on a real dataset.