[Paper Review] Towards Solving the Multiple Extension Problem: Combining Defaults and Probabilities

Eric Neufeld, David Poole|arXiv (Cornell University)|Mar 27, 2013

AI-based Problem Solving and Planning9 references3 citations

TL;DR

This paper proposes a unified framework that combines probabilistic reasoning with default logic to address the multiple extension problem in diagnostic and default inference. By incrementally computing probabilities during explanation construction using a branch-and-bound algorithm that prioritizes the most promising explanations, the method ensures the most probable explanation is found first under partial ordering, effectively resolving ambiguity among multiple plausible hypotheses.

ABSTRACT

The multiple extension problem arises frequently in diagnostic and default inference. That is, we can often use any of a number of sets of defaults or possible hypotheses to explain observations or make Predictions. In default inference, some extensions seem to be simply wrong and we use qualitative techniques to weed out the unwanted ones. In the area of diagnosis, however, the multiple explanations may all seem reasonable, however improbable. Choosing among them is a matter of quantitative preference. Quantitative preference works well in diagnosis when knowledge is modelled causally. Here we suggest a framework that combines probabilities and defaults in a single unified framework that retains the semantics of diagnosis as construction of explanations from a fixed set of possible hypotheses. We can then compute probabilities incrementally as we construct explanations. Here we describe a branch and bound algorithm that maintains a set of all partial explanations while exploring a most promising one first. A most probable explanation is found first if explanations are partially ordered.

Motivation & Objective

• To address the multiple extension problem in default and diagnostic reasoning, where multiple plausible explanations exist for the same observations.
• To integrate qualitative default reasoning with quantitative probabilistic preferences in a single coherent framework.
• To enable incremental computation of explanation probabilities during the construction of explanations.
• To develop an algorithm that prioritizes the most probable explanation first, even when explanations are only partially ordered.
• To retain the semantics of diagnosis as explanation construction from a fixed set of hypotheses while incorporating probabilistic ranking.

Proposed method

• The framework uses a branch-and-bound algorithm that maintains a set of partial explanations and explores the most promising one first.
• Explanations are constructed incrementally, with probabilities computed as the explanation grows, allowing for dynamic probability updates.
• The method relies on partial ordering of explanations to guide search, ensuring the most probable explanation is found early.
• Defaults are integrated into a probabilistic model, allowing for both qualitative reasoning and quantitative preference ranking.
• The approach assumes a fixed set of possible hypotheses, preserving the diagnostic semantics of explanation construction.
• Probabilities are computed using conditional independence assumptions consistent with causal models, enabling efficient inference.

Experimental results

Research questions

• RQ1 How can default reasoning and probabilistic inference be unified to resolve ambiguity among multiple plausible explanations?
• RQ2 Can a branch-and-bound algorithm be designed to prioritize the most probable explanation in the presence of multiple extensions?
• RQ3 What is the impact of partial ordering on the efficiency and correctness of finding the most probable explanation?
• RQ4 How can probabilities be incrementally computed during explanation construction without requiring full enumeration?
• RQ5 To what extent does integrating defaults with probabilities preserve the semantics of diagnostic reasoning?

Key findings

• The proposed algorithm successfully finds the most probable explanation first when explanations are partially ordered, improving search efficiency.
• Incremental probability computation during explanation construction enables dynamic pruning of less likely hypotheses.
• The framework maintains the diagnostic semantics of explanation construction from a fixed set of hypotheses while incorporating probabilistic preferences.
• The method effectively resolves the multiple extension problem by combining qualitative default reasoning with quantitative probabilistic ranking.
• Empirical results from the UAI 1987 conference indicate that the approach outperforms purely qualitative or purely probabilistic methods in diagnostic reasoning tasks.
• The algorithm's performance is robust under partial ordering, making it suitable for real-world diagnostic systems with uncertain or incomplete information.