Bayesian Consistency with the Supremum Metric
This work addresses a foundational statistical problem for Bayesian inference, providing incremental improvements in theoretical conditions for consistency.
The paper tackles the problem of establishing Bayesian consistency in the supremum metric, presenting simpler conditions than those required for L1 consistency, with a key result showing that weaker conditions suffice for this stronger form of consistency.
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 $\mathbb{L}_1$ consistency.