# Inverse statistical problems: from the inverse Ising problem to data science

## TL;DR

This paper reviews inverse statistical physics problems, especially the inverse Ising problem, highlighting methods for inferring model parameters from data across various scientific fields.

## Contribution

It provides a comprehensive overview of methods and applications for the inverse Ising problem, including equilibrium and non-equilibrium cases, with emphasis on recent advances and practical algorithms.

## Key findings

- Pseudolikelihood is a powerful method for inverse Ising inference.
- Applications include neural network reconstruction and protein structure prediction.
- Various approximate solutions are effective for different data regimes.

## Abstract

Inverse problems in statistical physics are motivated by the challenges of `big data' in different fields, in particular high-throughput experiments in biology. In inverse problems, the usual procedure of statistical physics needs to be reversed: Instead of calculating observables on the basis of model parameters, we seek to infer parameters of a model based on observations. In this review, we focus on the inverse Ising problem and closely related problems, namely how to infer the coupling strengths between spins given observed spin correlations, magnetisations, or other data. We review applications of the inverse Ising problem, including the reconstruction of neural connections, protein structure determination, and the inference of gene regulatory networks. For the inverse Ising problem in equilibrium, a number of controlled and uncontrolled approximate solutions have been developed in the statistical mechanics community. A particularly strong method, pseudolikelihood, stems from statistics. We also review the inverse Ising problem in the non-equilibrium case, where the model parameters must be reconstructed based on non-equilibrium statistics.