# Estimating Real Log Canonical Thresholds

arXiv: https://arxiv.org/abs/1906.01341 · HTML: https://mlsanity.com/paper/1906.01341/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2019.

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

- [The Local Learning Coefficient: A Singularity-Aware Complexity Measure](https://mlsanity.com/paper/2308.12108/) (2023)
- [Estimating the Local Learning Coefficient at Scale](https://mlsanity.com/paper/2402.03698/) (2024)
- [Bayesian Generalization Error in Linear Neural Networks with Concept Bottleneck Structure and Multitask Formulation](https://mlsanity.com/paper/2303.09154/) (2023)
- [Learning under Singularity: An Information Criterion improving WBIC and sBIC](https://mlsanity.com/paper/2402.12762/) (2024)
- [Upper Bound of Bayesian Generalization Error in Partial Concept Bottleneck Model (CBM): Partial CBM outperforms naive CBM](https://mlsanity.com/paper/2403.09206/) (2024)
- [Statistical Learning Analysis of Physics-Informed Neural Networks](https://mlsanity.com/paper/2602.11097/) (2026)
- [Upper Bounds for Local Learning Coefficients of Three-Layer Neural Networks](https://mlsanity.com/paper/2603.12785/) (2026)
- [An Asymptotic Equation Linking WAIC and WBIC in Singular Models](https://mlsanity.com/paper/2505.13902/) (2025)
- [Noise-Debiased Thermodynamic Variance for Local Learning Coefficient Probes](https://mlsanity.com/paper/2606.22389/) (2026)

## Abstract (arXiv)

Evaluation of the marginal likelihood plays an important role in model selection problems. The widely applicable Bayesian information criterion (WBIC) and singular Bayesian information criterion (sBIC) give approximations to the log marginal likelihood, which can be applied to both regular and singular models. When the real log canonical thresholds are known, the performance of sBIC is considered to be better than that of WBIC, but only few real log canonical thresholds are known. In this paper, we propose a new estimator of the real log canonical thresholds based on the variance of thermodynamic integration with an inverse temperature. In addition, we propose an application to make sBIC widely applicable. Finally, we investigate the performance of the estimator and model selection by simulation studies and application to real data.