Speaker: Francesca Zaffora Blando
Date and Time: Thursday, May 27th 2021, 16:30-18:00, Amsterdam time.
Title: Weak merging of opinions for computationally limited agents.
Abstract. A standard objection to subjective Bayesianism is that appealing to subjective probabilities threatens the objectivity of scientific inquiry. A standard Bayesian response to this charge relies on merging-of-opinions theorems: a family of results which establish that, as long as their respective priors are sufficiently compatible, two Bayesian agents with differing initial beliefs are guaranteed to almost surely reach a consensus with increasing evidence. So, objectivity can be recovered in the form of intersubjective agreement. One of the most well-known such results is the Blackwell-Dubins Theorem, which shows that Bayesian conditioning leads to a strong form of merging of opinions, provided that the agents agree on probability zero events to begin with—i.e., provided that their priors are mutually absolutely continuous. Since absolute continuity is a rather strong form of compatibility between priors, it is natural to wonder whether merging of opinions—and what type of merging of opinions—can be achieved with weaker assumptions. In this talk, I will address this question from the perspective of computationally limited Bayesian agents: agents whose priors are computable. I will argue that, for computable Bayesian learners, it is natural to appeal to the theory of algorithmic randomness—a branch of computability theory aimed at characterizing the concept of effective measure-theoretic typicality—to define notions of compatibility between priors. We will see that the proposed notions of compatibility induced by algorithmic randomness naturally correspond to restricted forms of absolute continuity. Then, I will show that some of these notions, while too weak to ensure merging of opinions in the strong sense of Blackwell and Dubins, nonetheless suffice to attain a weaker type of merging, first studied by Kalai and Lehrer, which only requires reaching a consensus over finite-horizon events.