  There is a long history in game theory on the topic of Bayesian or "rational"
learning, in which each player maintains beliefs over a set of alternative
behaviours, or types, for the other players. This idea has gained increasing
interest in the artificial intelligence (AI) community, where it is used as a
method to control a single agent in a system composed of multiple agents with
unknown behaviours. The idea is to hypothesise a set of types, each specifying
a possible behaviour for the other agents, and to plan our own actions with
respect to those types which we believe are most likely, given the observed
actions of the agents. The game theory literature studies this idea primarily
in the context of equilibrium attainment. In contrast, many AI applications
have a focus on task completion and payoff maximisation. With this perspective
in mind, we identify and address a spectrum of questions pertaining to belief
and truth in hypothesised types. We formulate three basic ways to incorporate
evidence into posterior beliefs and show when the resulting beliefs are
correct, and when they may fail to be correct. Moreover, we demonstrate that
prior beliefs can have a significant impact on our ability to maximise payoffs
in the long-term, and that they can be computed automatically with consistent
performance effects. Furthermore, we analyse the conditions under which we are
able complete our task optimally, despite inaccuracies in the hypothesised
types. Finally, we show how the correctness of hypothesised types can be
ascertained during the interaction via an automated statistical analysis.
