# Bayesian evidence estimation from posterior samples with normalizing flows

arXiv: https://arxiv.org/abs/2404.12294 · HTML: https://mlsanity.com/paper/2404.12294/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 32 references that are in the map.
- First posted on arXiv 18 Apr 2024.

## Builds on (most cited first, 5 of 5)

- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Complete parameter inference for GW150914 using deep learning](https://mlsanity.com/paper/2008.03312/) (2020)
- [Dynamic nested sampling: an improved algorithm for parameter estimation and evidence calculation](https://mlsanity.com/paper/1704.03459/) (2017)
- [Proximal nested sampling for high-dimensional Bayesian model selection](https://mlsanity.com/paper/2106.03646/) (2021)
- [Evidence Networks: simple losses for fast, amortized, neural Bayesian model comparison](https://mlsanity.com/paper/2305.11241/) (2023)

## Built on by (most cited first, 4 of 4)

- [Improving the Accuracy of Amortized Model Comparison with Self-Consistency](https://mlsanity.com/paper/2508.20614/) (2025)
- [mathttemuflow: Normalising Flows for Joint Cosmological Analysis](https://mlsanity.com/paper/2409.01407/) (2024)
- [FlowVAT: Normalizing Flow Variational Inference with Affine-Invariant Tempering](https://mlsanity.com/paper/2505.10466/) (2025)
- [Scalable Simulation-Based Model Inference with Test-Time Complexity Control](https://mlsanity.com/paper/2603.15292/) (2026)

## Abstract (arXiv)

We propose a novel method (floZ), based on normalizing flows, to estimate the Bayesian evidence (and its numerical uncertainty) from a pre-existing set of samples drawn from the unnormalized posterior distribution. We validate it on distributions whose evidence is known analytically, up to 15 parameter space dimensions, and compare with two state-of-the-art techniques for estimating the evidence: nested sampling (which computes the evidence as its main target) and a k-nearest-neighbors technique that produces evidence estimates from posterior samples. Provided representative samples from the target posterior are available, our method is more robust to posterior distributions with sharp features, especially in higher dimensions. For a simple multivariate Gaussian, we demonstrate its accuracy for up to 200 dimensions with 105 posterior samples. floZ has wide applicability, e.g., to estimate evidence from variational inference, Markov Chain Monte Carlo samples, or any other method that delivers samples and their likelihood from the unnormalized posterior density. As a physical application, we use floZ to compute the Bayes factor for the presence of the first overtone in the ringdown signal of the gravitational wave data of GW150914, finding good agreement with nested sampling.