🤖 AI Summary
This study investigates the construction of Bayesian predictive inference methods with favorable asymptotic properties and offers a novel Bayesian interpretation of kernel density estimation. Focusing on two classes of predictive rules—classical kernel density estimators and their recursive variants—the work systematically examines their weak almost sure convergence under sequential observations by integrating nonparametric kernel methods, stochastic process theory, and weak convergence analysis. The analysis reveals that the classical estimator converges weakly almost surely to a probability measure with compact support, whereas its recursive counterpart converges to one with non-compact support. Beyond establishing weak almost sure convergence for both schemes, this research extends the theoretical foundations of kernel methods within Bayesian predictive inference and provides a fresh Bayesian perspective on kernel density estimation.
📝 Abstract
Kernel density estimation is a widely used nonparametric approach to estimate an unknown distribution. Recent work in Bayesian predictive inference has considered stochastic processes formed by specifying the predictive distribution for the next data point given all observed data such that the resulting predictive distributions converge weakly almost surely. We study two kernel based prediction rules: the classic kernel density estimator, and a recursive version previously introduced for online problems. We show that both processes converge weakly almost surely, which opens the door for new Bayesian interpretations of kernel density estimation. Surprisingly, the process based on the classic kernel density estimates converges to a compactly supported measure, while the recursive version converges to a non-compactly supported measure.