Updating with incomplete observations

TL;DR

This paper introduces a conservative, imprecise-probability-based updating rule for incomplete observations, addressing limitations of existing methods and applicable to Bayesian networks and decision problems.

Contribution

It proposes a new, weak-assumption updating rule using lower previsions, ensuring coherence and handling ignorance about the incompleteness mechanism.

Findings

The new rule produces lower posterior probabilities and partial decisions.

It addresses the Monty Hall paradox effectively.

Provides a linear-time exact algorithm for Bayesian network classification.

Abstract

Currently, there is renewed interest in the problem, raised by Shafer in 1985, of updating probabilities when observations are incomplete (or set-valued). This is a fundamental problem, and of particular interest for Bayesian networks. Recently, Grunwald and Halpern have shown that commonly used updating strategies fail here, except under very special assumptions. We propose a new rule for updating probabilities with incomplete observations. Our approach is deliberately conservative: we make no or weak assumptions about the so-called incompleteness mechanism that produces incomplete observations. We model our ignorance about this mechanism by a vacuous lower prevision, a tool from the theory of imprecise probabilities, and we derive a new updating rule using coherence arguments. In general, our rule produces lower posterior probabilities, as well as partially determinate decisions. This is a logical consequence of the ignorance about the incompleteness mechanism. We show how the new rule can properly address the apparent paradox in the 'Monty Hall' puzzle. In addition, we apply it to the classification of new evidence in Bayesian networks constructed using expert knowledge. We provide an exact algorithm for this task with linear-time complexity, also for multiply connected nets.