# Bayes Hilbert Spaces for Posterior Approximation

arXiv: https://arxiv.org/abs/2304.09053 · HTML: https://mlsanity.com/paper/2304.09053/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 75 references that are in the map.
- First posted on arXiv 18 Apr 2023.

## Builds on (most cited first, 26 of 26)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Coresets for Scalable Bayesian Logistic Regression](https://mlsanity.com/paper/1605.06423/) (2016)
- [Bayesian Coreset Construction via Greedy Iterative Geodesic Ascent](https://mlsanity.com/paper/1802.01737/) (2018)
- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [Automated Scalable Bayesian Inference via Hilbert Coresets](https://mlsanity.com/paper/1710.05053/) (2017)
- [Kernel Thinning](https://mlsanity.com/paper/2105.05842/) (2021)
- [Density Estimation in Infinite Dimensional Exponential Families](https://mlsanity.com/paper/1312.3516/) (2013)
- [Support points](https://mlsanity.com/paper/1609.01811/) (2016)
- [MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancy](https://mlsanity.com/paper/1909.13339/) (2019)
- [Stein's Method Meets Computational Statistics: A Review of Some Recent Developments](https://mlsanity.com/paper/2105.03481/) (2021)
- [Sparse Variational Inference: Bayesian Coresets from Scratch](https://mlsanity.com/paper/1906.03329/) (2019)
- [Optimal Thinning of MCMC Output](https://mlsanity.com/paper/2005.03952/) (2020)
- [Optimal quantisation of probability measures using maximum mean discrepancy](https://mlsanity.com/paper/2010.07064/) (2020)
- [PASS-GLM: polynomial approximate sufficient statistics for scalable Bayesian GLM inference](https://mlsanity.com/paper/1709.09216/) (2017)
- [Practical bounds on the error of Bayesian posterior approximations: A nonasymptotic approach](https://mlsanity.com/paper/1809.09505/) (2018)
- [Fast Bayesian Coresets via Subsampling and Quasi-Newton Refinement](https://mlsanity.com/paper/2203.09675/) (2022)
- [Hamiltonian Monte Carlo with Energy Conserving Subsampling](https://mlsanity.com/paper/1708.00955/) (2017)
- [Statistical Depth Meets Machine Learning: Kernel Mean Embeddings and Depth in Functional Data Analysis](https://mlsanity.com/paper/2105.12778/) (2021)
- [Aitchison's Compositional Data Analysis 40 Years On: A Reappraisal](https://mlsanity.com/paper/2201.05197/) (2022)
- [SwISS: A Scalable Markov chain Monte Carlo Divide-and-Conquer Strategy](https://mlsanity.com/paper/2208.04080/) (2022)
- [The Information-Geometric Perspective of Compositional Data Analysis](https://mlsanity.com/paper/2005.11510/) (2020)
- [Additive Density-on-Scalar Regression in Bayes Hilbert Spaces with an Application to Gender Economics](https://mlsanity.com/paper/2110.11771/) (2021)
- [Variational Inference as Iterative Projection in a Bayesian Hilbert Space with Application to Robotic State Estimation](https://mlsanity.com/paper/2005.07275/) (2020)

## Built on by (most cited first, 1 of 1)

- [On Reconstructing Training Data From Bayesian Posteriors and Trained Models](https://mlsanity.com/paper/2507.18372/) (2025)

## Abstract (arXiv)

Performing inference in Bayesian models requires sampling algorithms to draw samples from the posterior. This becomes prohibitively expensive as the size of data sets increase. Constructing approximations to the posterior which are cheap to evaluate is a popular approach to circumvent this issue. This begs the question of what is an appropriate space to perform approximation of Bayesian posterior measures. This manuscript studies the application of Bayes Hilbert spaces to the posterior approximation problem. Bayes Hilbert spaces are studied in functional data analysis in the context where observed functions are probability density functions and their application to computational Bayesian problems is in its infancy. This manuscript shall outline Bayes Hilbert spaces and their connection to Bayesian computation, in particular novel connections between Bayes Hilbert spaces, Bayesian coreset algorithms and kernel-based distances.