Probabilistic Deduction with Conditional Constraints over Basic Events

TL;DR

This paper investigates probabilistic deduction with conditional constraints, demonstrating NP-hardness in general and providing efficient algorithms for special cases involving trees with point or interval probabilities.

Contribution

It introduces efficient linear programming methods for probabilistic deduction in conditional constraint trees with interval probabilities, and a linear-time local approach for point probabilities.

Findings

NP-hardness of global probabilistic deduction with conditional constraints

Linear-time deduction method for point probability trees

Polynomial-time solution for interval probability trees using linear programming

Abstract

We study the problem of probabilistic deduction with conditional constraints over basic events. We show that globally complete probabilistic deduction with conditional constraints over basic events is NP-hard. We then concentrate on the special case of probabilistic deduction in conditional constraint trees. We elaborate very efficient techniques for globally complete probabilistic deduction. In detail, for conditional constraint trees with point probabilities, we present a local approach to globally complete probabilistic deduction, which runs in linear time in the size of the conditional constraint trees. For conditional constraint trees with interval probabilities, we show that globally complete probabilistic deduction can be done in a global approach by solving nonlinear programs. We show how these nonlinear programs can be transformed into equivalent linear programs, which are solvable in polynomial time in the size of the conditional constraint trees.