An Introduction to Multi-Agent Statistics

Building on ideas from probabilistic dynamic epistemic logic, the aim of this workshop contribution is to present the practical details of how to build and use Bayesian models that can represent multi-agent uncertainty, including probability distributions over (other people’s) probability distributions. The key insight that underlies the approach taken here is that probabilities like PA(φ) are random variables and can be treated as such. By keeping carefully track of what various agents know under various circumstances, this way of looking at the issue provides enough structure to support queries about higher-order probabilistic belief statements. Recent decades have seen a surge in two kinds of computational reasoning, sophistcated dynamic logics for multi-agent and higher-order uncertainty [1, 4], and probabilistic reasoning using causal Bayesian networks, often in conjunction with approximation methods like Markov Chain Monte Carlo or Gibbs sampling [2]. The purpose of my contribution to the workshop is to explain and strengthen the connection between these two styles of computational reasoning by presenting a system that integrates the core insights from dynamic logic into an explicitly statistical framework. In this system, a series of statements about uncertain events can be translated systematically into a statistical model that includes information about which agents possess what pieces of information. This posterior model can be queried for higher-order questions such as the probability that a specific agent assigns to the beliefs of another agent. The principles underlying this dynamic style of model construction come almost directly out of the literature on dynamic epistemic logic, especially its probabilistic incarnations [4]. My strategy throughout has been to use the simplest possible version of these ideas but to augment them with a useful statistical interpretation. Because of space constraints, this note is not intended to completely specify the nature of the system, but rather to provide a suggestive and representative overview of its features. This makes the current abstract less formal than some (including perhaps myself) may prefer, but I hope the loss of precision is made up by the gain in intuitive clarity.