# Foundation of Calculating Normalized Maximum Likelihood for Continuous Probability Models

arXiv: https://arxiv.org/abs/2409.08387 · HTML: https://mlsanity.com/paper/2409.08387/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 25 references that are in the map.
- First posted on arXiv 12 Sep 2024.

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

- [Minimum Description Length Revisited](https://mlsanity.com/paper/1908.08484/) (2019)
- [Detecting Signs of Model Change with Continuous Model Selection Based on Descriptive Dimensionality](https://mlsanity.com/paper/2302.12127/) (2023)

## Built on by (most cited first, 3 of 3)

- [Normalized Maximum Likelihood Code-Length on Riemannian Data Spaces](https://mlsanity.com/paper/2508.21466/) (2025)
- [The Normalized Maximum Likelihood for Regular Non-Smooth Models: Measure-Theoretic Foundations and Geometric Sampling](https://mlsanity.com/paper/2605.24477/) (2026)
- [Exact Schur-Sylvester Dimensionality Reductions for Non-Smooth Stochastic Complexity and Manifold Sampling](https://mlsanity.com/paper/2606.23867/) (2026)

## Abstract (arXiv)

The normalized maximum likelihood (NML) code length is widely used as a model selection criterion based on the minimum description length principle, where the model with the shortest NML code length is selected. A common method to calculate the NML code length is to use the sum (for a discrete model) or integral (for a continuous model) of a function defined by the distribution of the maximum likelihood estimator. While this method has been proven to correctly calculate the NML code length of discrete models, no proof has been provided for continuous cases. Consequently, it has remained unclear whether the method can accurately calculate the NML code length of continuous models. In this paper, we solve this problem affirmatively, proving that the method is also correct for continuous cases. Remarkably, completing the proof for continuous cases is non-trivial in that it cannot be achieved by merely replacing the sums in discrete cases with integrals, as the decomposition trick applied to sums in the discrete model case proof is not applicable to integrals in the continuous model case proof. To overcome this, we introduce a novel decomposition approach based on the coarea formula from geometric measure theory, which is essential to establishing our proof for continuous cases.