# Asymptotic Behavior of Bayesian Generalization Error in Multinomial Mixtures

arXiv: https://arxiv.org/abs/2203.06884 · HTML: https://mlsanity.com/paper/2203.06884/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 24 references that are in the map.
- First posted on arXiv 14 Mar 2022.

## Builds on (most cited first, 2 of 2)

- [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)

## Built on by (most cited first, 9 of 9)

- [Mathematical Theory of Bayesian Statistics for Unknown Information Source](https://mlsanity.com/paper/2206.05630/) (2022)
- [Recent Advances in Algebraic Geometry and Bayesian Statistics](https://mlsanity.com/paper/2211.10049/) (2022)
- [Bayesian Generalization Error in Linear Neural Networks with Concept Bottleneck Structure and Multitask Formulation](https://mlsanity.com/paper/2303.09154/) (2023)
- [Review and Prospect of Algebraic Research in Equivalent Framework between Statistical Mechanics and Machine Learning Theory](https://mlsanity.com/paper/2406.10234/) (2024)
- [Blow-up Algorithm for Sum-of-Products Polynomials and Real Log Canonical Thresholds](https://mlsanity.com/paper/2303.11619/) (2023)
- [Upper Bound of Bayesian Generalization Error in Partial Concept Bottleneck Model (CBM): Partial CBM outperforms naive CBM](https://mlsanity.com/paper/2403.09206/) (2024)
- [Upper Bounds for Local Learning Coefficients of Three-Layer Neural Networks](https://mlsanity.com/paper/2603.12785/) (2026)
- [Upper Bound of Real Log Canonical Threshold of Tensor Decomposition and its Application to Bayesian Inference](https://mlsanity.com/paper/2303.05731/) (2023)
- [Statistical inference for quantum singular models](https://mlsanity.com/paper/2411.16396/) (2024)

## Abstract (arXiv)

Multinomial mixtures are widely used in the information engineering field, however, their mathematical properties are not yet clarified because they are singular learning models. In fact, the models are non-identifiable and their Fisher information matrices are not positive definite. In recent years, the mathematical foundation of singular statistical models are clarified by using algebraic geometric methods. In this paper, we clarify the real log canonical thresholds and multiplicities of the multinomial mixtures and elucidate their asymptotic behaviors of generalization error and free energy.