# Statistical inference for quantum singular models

arXiv: https://arxiv.org/abs/2411.16396 · HTML: https://mlsanity.com/paper/2411.16396/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 121 references that are in the map.
- First posted on arXiv 25 Nov 2024.

## Builds on (most cited first, 10 of 10)

- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Foundational Challenges in Assuring Alignment and Safety of Large Language Models](https://mlsanity.com/paper/2404.09932/) (2024)
- [The Local Learning Coefficient: A Singularity-Aware Complexity Measure](https://mlsanity.com/paper/2308.12108/) (2023)
- [Quantum Hamiltonian-Based Models and the Variational Quantum Thermalizer Algorithm](https://mlsanity.com/paper/1910.02071/) (2019)
- [WAIC, but Why? Generative Ensembles for Robust Anomaly Detection](https://mlsanity.com/paper/1810.01392/) (2018)
- [Quantum neural networks form Gaussian processes](https://mlsanity.com/paper/2305.09957/) (2023)
- [Asymptotic Behavior of Bayesian Generalization Error in Multinomial Mixtures](https://mlsanity.com/paper/2203.06884/) (2022)
- [Bayesian Free Energy of Deep ReLU Neural Network in Overparametrized Cases](https://mlsanity.com/paper/2303.15739/) (2023)
- [Free energy of Bayesian Convolutional Neural Network with Skip Connection](https://mlsanity.com/paper/2307.01417/) (2023)
- [Posterior covariance information criterion for general loss functions](https://mlsanity.com/paper/2206.05887/) (2022)

## Abstract (arXiv)

Deep learning has seen substantial achievements, with numerical and theoretical evidence suggesting that singularities of statistical models are considered a contributing factor to its performance. From this remarkable success of classical statistical models, it is naturally expected that quantum singular models will play a vital role in many quantum statistical tasks. However, while the theory of quantum statistical models in regular cases has been established, theoretical understanding of quantum singular models is still limited. To investigate the statistical properties of quantum singular models, we focus on two prominent tasks in quantum statistical inference: quantum state estimation and model selection. In particular, we base our study on classical singular learning theory and seek to extend it within the framework of Bayesian quantum state estimation. To this end, we define quantum generalization and training loss functions and give their asymptotic expansions through algebraic geometrical methods. The key idea of the proof is the introduction of a quantum analog of the likelihood function using classical shadows. Consequently, we construct an asymptotically unbiased estimator of the quantum generalization loss, the quantum widely applicable information criterion (QWAIC), as a computable model selection metric from given measurement outcomes.