# Asymptotics of predictive distributions driven by sample means and variances

arXiv: https://arxiv.org/abs/2403.16828 · HTML: https://mlsanity.com/paper/2403.16828/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Mar 2024.

## Built on by (most cited first, 2 of 2)

- [Quasi-Bayes meets Vines](https://mlsanity.com/paper/2406.12764/) (2024)
- [Concentration and Calibration in Predictive Bayesian Inference](https://mlsanity.com/paper/2605.00455/) (2026)

## Abstract (arXiv)

Let αn(·)=Pbigl(Xn+1∈·mid X1,ldots,Xnbigr) be the predictive distributions of a sequence (X1,X2,ldots) of p-dimensional random vectors. Suppose αn= N p (Mn,Qn) where Mn=frac1nsumi=1nXi and Qn=frac1nsumi=1n(Xi-Mn)(Xi-Mn)t. Then, there is a random probability measure α on the Borel subsets of ℝp such that lVertαn-αrVertoverseta.s.longrightarrow 0 where lVert·rVert is total variation distance. An explicit expression for α is provided and the convergence rate of lVertαn-αrVert is shown to be arbitrarily close to n-1/2. Moreover, it is still true that lVertαn-αrVertoverseta.s.longrightarrow 0 even if αn=L(Mn,Qn) where L belongs to a class of distributions much larger than the normal. The predictives αn are useful in various frameworks, including Bayesian predictive inference and predictive resampling. Finally, the asymptotic behavior of copula-based predictive distributions (introduced in [13]) is investigated and a numerical experiment is performed.