Probably Approximately Knowing

TL;DR

This paper explores how agents should form probabilistic beliefs about conditions when performing actions under probabilistic constraints, showing that agents must probably approximately know the condition's truth to satisfy these constraints.

Contribution

It introduces the concept of probably approximately knowing (PAK-know) as a link between probabilistic beliefs and actions under probabilistic guarantees, a novel theoretical insight.

Findings

Expected belief in C equals the probability C holds when action a is performed.

Agents must assign high probabilistic belief to C when acting to meet probabilistic constraints.

Strong belief corresponds to probably approximately knowing C when acting.

Abstract

Whereas deterministic protocols are typically guaranteed to obtain particular goals of interest, probabilistic protocols typically provide only probabilistic guarantees. This paper initiates an investigation of the interdependence between actions and subjective beliefs of agents in a probabilistic setting. In particular, we study what probabilistic beliefs an agent should have when performing actions, in a protocol that satisfies a probabilistic constraint of the form: 'Condition C should hold with probability at least p when action a is performed'. Our main result is that the expected degree of an agent's belief in C when it performs a equals the probability that C holds when a is performed. Indeed, if the threshold of the probabilistic constraint should hold with probaility p=1-x^2 for some small value of x then, with probability 1-x, when the agent acts it will assign a probabilistic belief no smaller than 1-x to the possibility that C holds. In other words, viewing strong belief as, intuitively, approximate knowledge, the agent must probably approximately know (PAK-know) that C is true when it acts.