# The Exact Asymptotic Form of Bayesian Generalization Error in Latent Dirichlet Allocation

arXiv: https://arxiv.org/abs/2008.01304 · HTML: https://mlsanity.com/paper/2008.01304/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Aug 2020.

## Built on by (most cited first, 14 of 14)

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- [Review and Prospect of Algebraic Research in Equivalent Framework between Statistical Mechanics and Machine Learning Theory](https://mlsanity.com/paper/2406.10234/) (2024)
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- [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 Bound of Real Log Canonical Threshold of Tensor Decomposition and its Application to Bayesian Inference](https://mlsanity.com/paper/2303.05731/) (2023)
- [An Asymptotic Equation Linking WAIC and WBIC in Singular Models](https://mlsanity.com/paper/2505.13902/) (2025)
- [A Statistical Difference between Single-Layer Learning and Hierarchical Learning in Wide Neural Networks](https://mlsanity.com/paper/2607.23397/) (2026)
- [Singularities of Non-negative Matrix Factorization and their application to Bayesian inference](https://mlsanity.com/paper/2609.34043/) (2026)

## Abstract (arXiv)

Latent Dirichlet allocation (LDA) obtains essential information from data by using Bayesian inference. It is applied to knowledge discovery via dimension reducing and clustering in many fields. However, its generalization error had not been yet clarified since it is a singular statistical model where there is no one-to-one mapping from parameters to probability distributions. In this paper, we give the exact asymptotic form of its generalization error and marginal likelihood, by theoretical analysis of its learning coefficient using algebraic geometry. The theoretical result shows that the Bayesian generalization error in LDA is expressed in terms of that in matrix factorization and a penalty from the simplex restriction of LDA's parameter region. A numerical experiment is consistent to the theoretical result.