# Bayesian Consistency with the Supremum Metric

arXiv: https://arxiv.org/abs/2201.03447 · HTML: https://mlsanity.com/paper/2201.03447/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 14 references that are in the map.
- First posted on arXiv 10 Jan 2022.

## Builds on (most cited first, 1 of 1)

- [Multivariate Smoothing via the Fourier Integral Theorem and Fourier Kernel](https://mlsanity.com/paper/2012.14482/) (2020)

## Abstract (arXiv)

We present simple conditions for Bayesian consistency in the supremum metric. The key to the technique is a triangle inequality which allows us to explicitly use weak convergence, a consequence of the standard Kullback--Leibler support condition for the prior. A further condition is to ensure that smoothed versions of densities are not too far from the original density, thus dealing with densities which could track the data too closely. A key result of the paper is that we demonstrate supremum consistency using weaker conditions compared to those currently used to secure mathbbL1 consistency.