# Recent Advances in Algebraic Geometry and Bayesian Statistics

arXiv: https://arxiv.org/abs/2211.10049 · HTML: https://mlsanity.com/paper/2211.10049/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 70 references that are in the map.
- First posted on arXiv 18 Nov 2022.

## Builds on (most cited first, 9 of 9)

- [Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC](https://mlsanity.com/paper/1507.04544/) (2015)
- [Understanding predictive information criteria for Bayesian models](https://mlsanity.com/paper/1307.5928/) (2013)
- [Deep Learning is Singular, and That's Good](https://mlsanity.com/paper/2010.11560/) (2020)
- [A Bayesian information criterion for singular models](https://mlsanity.com/paper/1309.0911/) (2013)
- [The Exact Asymptotic Form of Bayesian Generalization Error in Latent Dirichlet Allocation](https://mlsanity.com/paper/2008.01304/) (2020)
- [Upper Bound of Bayesian Generalization Error in Non-negative Matrix Factorization](https://mlsanity.com/paper/1612.04112/) (2016)
- [Asymptotic Behavior of Bayesian Generalization Error in Multinomial Mixtures](https://mlsanity.com/paper/2203.06884/) (2022)
- [Mathematical Theory of Bayesian Statistics for Unknown Information Source](https://mlsanity.com/paper/2206.05630/) (2022)
- [Testing Homogeneity for Normal Mixture Models: Variational Bayes Approach](https://mlsanity.com/paper/1912.07041/) (2019)

## Built on by (most cited first, 4 of 4)

- [Information-Theoretic Framework for Understanding Modern Machine-Learning](https://mlsanity.com/paper/2506.07661/) (2025)
- [Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective](https://mlsanity.com/paper/2601.06597/) (2026)
- [Breaking the Illusion: When Positive Meets Negative in Multimodal Decoding](https://mlsanity.com/paper/2605.06679/) (2026)
- [Empirical Evaluation of No Free Lunch Violations in Permutation-Based Optimization](https://mlsanity.com/paper/2603.03613/) (2026)

## Abstract (arXiv)

This article is a review of theoretical advances in the research field of algebraic geometry and Bayesian statistics in the last two decades. Many statistical models and learning machines which contain hierarchical structures or latent variables are called nonidentifiable, because the map from a parameter to a statistical model is not one-to-one. In nonidentifiable models, both the likelihood function and the posterior distribution have singularities in general, hence it was difficult to analyze their statistical properties. However, from the end of the 20th century, new theory and methodology based on algebraic geometry have been established which enables us to investigate such models and machines in the real world. In this article, the following results in recent advances are reported. First, we explain the framework of Bayesian statistics and introduce a new perspective from the birational geometry. Second, two mathematical solutions are derived based on algebraic geometry. An appropriate parameter space can be found by a resolution map, which makes the posterior distribution be normal crossing and the log likelihood ratio function be well-defined. Third, three applications to statistics are introduced. The posterior distribution is represented by the renormalized form, the asymptotic free energy is derived, and the universal formula among the generalization loss, the cross validation, and the information criterion is established. Two mathematical solutions and three applications to statistics based on algebraic geometry reported in this article are now being used in many practical fields in data science and artificial intelligence.