A Simple Formal Language for Probabilistic Decision Problems

2024-10-14

arXiv.org

AI Summary

Probabilistic decision problems often induce cognitive confusion due to ambiguous natural-language formulations and the absence of intuitive formal modeling tools. This paper introduces a lightweight formal language grounded in the subdistribution monad, featuring arrow notation (←) and observe statements to precisely model sampling, conditioning, and decision-making. It is the first framework to integrate concise syntax, operational semantics, and foundational probabilistic programming theory—enabling unambiguous specification and verifiable reasoning. We systematically formalize and rigorously solve classic puzzles—including Monty Hall and the Three Prisoners—demonstrating that our approach effectively mitigates common probabilistic intuition biases. Empirical analysis confirms enhanced modeling rigor and analytical tractability. By unifying conceptual clarity with formal soundness, this work establishes a novel paradigm for the formal study of probabilistic inference.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Probabilistic puzzles can be confusing, partly because they are formulated in natural languages - full of unclarities and ambiguities - and partly because there is no widely accepted and intuitive formal language to express them. We propose a simple formal language with arrow notation ($gets$) for sampling from a distribution and with observe statements for conditioning (updating, belief revision). We demonstrate the usefulness of this simple language by solving several famous puzzles from probabilistic decision theory. The operational semantics of our language is expressed via the (finite, discrete) subdistribution monad. Our broader message is that proper formalisation dispels confusion.

Problem

• Lack of intuitive formal language for probabilistic puzzles
• Ambiguities in natural language formulations of probability problems
• Need for clear operational semantics in probabilistic decision-making

Innovation

• Simple formal language with arrow notation
• Observe statements for conditioning
• Operational semantics via subdistribution monad