# Constructive Disintegration and Conditional Modes

arXiv: https://arxiv.org/abs/2508.00617 · HTML: https://mlsanity.com/paper/2508.00617/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 28 references that are in the map.
- First posted on arXiv 1 Aug 2025.

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

- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective](https://mlsanity.com/paper/1810.03440/) (2018)
- [Towards Efficient MCMC Sampling in Bayesian Neural Networks by Exploiting Symmetry](https://mlsanity.com/paper/2304.02902/) (2023)
- [Bayesian ODE Solvers: The Maximum A Posteriori Estimate](https://mlsanity.com/paper/2004.00623/) (2020)
- [A Symmetry-Aware Exploration of Bayesian Neural Network Posteriors](https://mlsanity.com/paper/2310.08287/) (2023)
- [FSP-Laplace: Function-Space Priors for the Laplace Approximation in Bayesian Deep Learning](https://mlsanity.com/paper/2407.13711/) (2024)
- [Generalized modes in Bayesian inverse problems](https://mlsanity.com/paper/1806.00519/) (2018)
- [Equivalence of weak and strong modes of measures on topological vector spaces](https://mlsanity.com/paper/1708.02516/) (2017)

## Abstract (arXiv)

Conditioning, the central operation in Bayesian statistics, is formalised by the notion of disintegration of measures. However, due to the implicit nature of their definition, constructing disintegrations is often difficult. A folklore result in machine learning conflates the construction of a disintegration with the restriction of probability density functions onto the subset of events that are consistent with a given observation. We provide a comprehensive set of mathematical tools which can be used to construct disintegrations and apply these to find densities of disintegrations on differentiable manifolds. Using our results, we provide a disturbingly simple example in which the restricted density and the disintegration density drastically disagree. Motivated by applications in approximate Bayesian inference and Bayesian inverse problems, we further study the modes of disintegrations. We show that the recently introduced notion of a "conditional mode" does not coincide in general with the modes of the conditional measure obtained through disintegration, but rather the modes of the restricted measure. We also discuss the implications of the discrepancy between the two measures in practice, advocating for the utility of both approaches depending on the modelling context.