On the Consistency of Approximate Multi-agent Probability Theory

Abstract

Bayesian models have proven to accurately predict many aspects of human cognition, but they generally lack the resources to describe higher-order reasoning about other people's knowledge. Recently, a number of suggestions have thus been made as to how these social aspects of cognition might be codified in computational reasoning systems. This paper examines one particularly ambitious attempt by Andreas Stuhlmüller and Noah Goodman, which was implemented in the stochastic programming language Church. This paper translates their proposal into a more conventional probabilistic language, comparing it to an alternative system which models subjective probabilities as random variables. Having spelled out their ideas in these more familiar and intuitive terms, I argue that the approximate reasoning methods used in their system have certain statistical consistency problems. Keywords Multi-agent probability theory Á Approximate inference Á Higher-order reasoning Probability theory is a useful and largely reasonable model of many aspects of human reasoning and decision-making. However, by its nature, it is geared towards reasoning about a static, inanimate environment, and it is thus not very well equipped to handle reasoning about mutually adapting agents in inherently social situations such as game-playing, war, bargaining, and conversation [3, 15]. Consider for instance the following example: Example 1 Ann and Bob both draw five cards from a standard deck of 52 cards with 4 aces.

On interactivity in probabilistic pragmatics: yet another rational analysis of scalar implicatures

“…An important step towards understanding such Bayesian reasoning about Bayesian reasoning has been made in the Bayesian modeling language Church that allows so-called nested queries for that purpose (Goodman et al 2008). However, many computational and conceptual problems remain (see e.g., Blokpoel 2015;Madsen 2015), especially when it comes to mutually recursive reasoning. Luckily, reasoning about reasoning has long been a topic for logicians, in particular in the context of multi-agent systems and epistemic logic, but the combination with reasoning under uncertainty using Bayesian beliefs is not very well understood yet in cognitive science.…”