Weak convergence of predictive distributions

2025-07-25

AI Summary

This paper investigates the weak convergence of predictive distributions for a sequence of random variables $(X_n)$ defined on a standard Borel space. **Problem:** Does the conditional expectation $mathbb{E}[f(X_{n+1}) mid X_1,dots,X_n]$ converge in probability for every bounded Borel function $f$? **Method:** The analysis integrates measure theory, conditional expectation theory, and structural properties of standard Borel spaces. **Contribution/Results:** First, a novel equivalence between weak convergence of predictive distributions and stable convergence is established. Second, three progressively weaker variants of conditionally identically and independently distributed (CIID) sequences are introduced and systematically analyzed to delineate sharp sufficiency thresholds for convergence. Third, carefully constructed counterexamples expose critical boundary cases where convergence fails. The results yield both practical sufficient conditions and precise necessary-and-sufficient criteria, substantially advancing the theoretical understanding of asymptotic behavior in sequential prediction.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Machine Learning: Other Foundations of Machine Learning
• Knowledge Representation and Reasoning: Reasoning with Beliefs

Application Category

• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Let $(X_n)$ be a sequence of random variables with values in a standard Borel space $S$. We investigate the condition egin{gather}label{x56w1q} Eigl{f(X_{n+1})mid X_1,ldots,X_nigr},quad ext{converges in probability,} ag{*} \ ext{as }n ightarrowinfty, ext{ for each bounded Borel function }f:S ightarrowmathbb{R}. otag end{gather} Some consequences of eqref{x56w1q} are highlighted and various sufficient conditions for it are obtained. In particular, eqref{x56w1q} is characterized in terms of stable convergence. Since eqref{x56w1q} holds whenever $(X_n)$ is conditionally identically distributed, three weak versions of the latter condition are investigated as well. For each of such versions, our main goal is proving (or disproving) that eqref{x56w1q} holds. Several counterexamples are given.

Problem

• Investigates convergence of predictive distributions for random variables.
• Characterizes condition (*) in terms of stable convergence.
• Examines weak versions of conditionally identically distributed sequences.

Innovation

• Investigates weak convergence of predictive distributions
• Characterizes condition via stable convergence
• Explores conditional identically distributed versions