Inductive Coherence

TL;DR

This paper introduces inductive coherence, a new framework for probabilistic reasoning about computational outputs, providing finite approximation guarantees and addressing limitations of traditional coherent distributions.

Contribution

It proposes inductive coherence, a strengthened form of coherence, along with an algorithm that ensures finite approximations satisfy this criterion.

Findings

The algorithm satisfies inductive coherence constraints.

Inductive coherence improves finite approximation quality.

Addresses limitations of traditional coherent distributions.

Abstract

While probability theory is normally applied to external environments, there has been some recent interest in probabilistic modeling of the outputs of computations that are too expensive to run. Since mathematical logic is a powerful tool for reasoning about computer programs, we consider this problem from the perspective of integrating probability and logic. Recent work on assigning probabilities to mathematical statements has used the concept of coherent distributions, which satisfy logical constraints such as the probability of a sentence and its negation summing to one. Although there are algorithms which converge to a coherent probability distribution in the limit, this yields only weak guarantees about finite approximations of these distributions. In our setting, this is a significant limitation: Coherent distributions assign probability one to all statements provable in a specific logical theory, such as Peano Arithmetic, which can prove what the output of any terminating computation is; thus, a coherent distribution must assign probability one to the output of any terminating computation. To model uncertainty about computations, we propose to work with approximations to coherent distributions. We introduce inductive coherence, a strengthening of coherence that provides appropriate constraints on finite approximations, and propose an algorithm which satisfies this criterion.