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

arXiv: https://arxiv.org/abs/1702.01522 · HTML: https://mlsanity.com/paper/1702.01522/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Feb 2017.

## Built on by (most cited first, 30 of 36)

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- [The autoregressive neural network architecture of the Boltzmann distribution of pairwise interacting spins systems](https://mlsanity.com/paper/2302.08347/) (2023)
- [Inferring effective couplings with Restricted Boltzmann Machines](https://mlsanity.com/paper/2309.02292/) (2023)
- [Feature Programming for Multivariate Time Series Prediction](https://mlsanity.com/paper/2306.06252/) (2023)
- [Inferring Higher-Order Couplings with Neural Networks](https://mlsanity.com/paper/2501.06108/) (2025)
- [Quantifying Relevance in Learning and Inference](https://mlsanity.com/paper/2202.00339/) (2022)
- [Network reconstruction via the minimum description length principle](https://mlsanity.com/paper/2405.01015/) (2024)
- [A theoretical framework for overfitting in energy-based modeling](https://mlsanity.com/paper/2501.19158/) (2025)
- [Ising Model Selection Using ℓ1-Regularized Linear Regression: A Statistical Mechanics Analysis](https://mlsanity.com/paper/2102.03988/) (2021)
- [Quantum Approximate Optimization Algorithm applied to the binary perceptron](https://mlsanity.com/paper/2112.10219/) (2021)
- [CoRMF: Criticality-Ordered Recurrent Mean Field Ising Solver](https://mlsanity.com/paper/2403.03391/) (2024)
- [Self-attention as an attractor network: transient memories without backpropagation](https://mlsanity.com/paper/2409.16112/) (2024)
- [Scalable network reconstruction in subquadratic time](https://mlsanity.com/paper/2401.01404/) (2024)
- [Entropy, concentration, and learning: a statistical mechanics primer](https://mlsanity.com/paper/2409.18630/) (2024)
- [Discrete distributions are learnable from metastable samples](https://mlsanity.com/paper/2410.13800/) (2024)
- [Pseudo-likelihood produces associative memories able to generalize, even for asymmetric couplings](https://mlsanity.com/paper/2507.05147/) (2025)
- [STAR-Teaming: A Strategy-Response Multiplex Network Approach to Automated LLM Red Teaming](https://mlsanity.com/paper/2604.18976/) (2026)
- [On Model Selection Consistency of Lasso for High-Dimensional Ising Models](https://mlsanity.com/paper/2110.08500/) (2021)
- [Optimal regularizations for data generation with probabilistic graphical models](https://mlsanity.com/paper/2112.01292/) (2021)
- [Categorical Distributions of Maximum Entropy under Marginal Constraints](https://mlsanity.com/paper/2204.03406/) (2022)
- [Biases in Inverse Ising Estimates of Near-Critical Behaviour](https://mlsanity.com/paper/2301.05556/) (2023)
- [Uncertainty quantification and posterior sampling for network reconstruction](https://mlsanity.com/paper/2503.07736/) (2025)
- [Inferring entropy production in many-body systems using nonequilibrium maximum entropy](https://mlsanity.com/paper/2505.10444/) (2025)
- [Correspondence Between Ising Machines and Neural Networks](https://mlsanity.com/paper/2511.00746/) (2025)
- [Inference of stochastic time series with missing data](https://mlsanity.com/paper/2101.11816/) (2021)
- [Nonequilibrium thermodynamics of self-supervised learning](https://mlsanity.com/paper/2106.08981/) (2021)
- [Data-Driven Interaction Analysis of Line Failure Cascading in Power Grid Networks](https://mlsanity.com/paper/2112.01061/) (2021)

## Abstract (arXiv)

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.