  One aspect of the algorithmic lens in theoretical computer science is a view
on other scientific disciplines that focuses on satisfactory solutions that
adhere to real-world constraints, as opposed to solutions that would be optimal
ignoring such constraints. The algorithmic lens has provided a unique and
important perspective on many academic fields, including molecular biology,
ecology, neuroscience, quantum physics, economics, and social science.
  This thesis applies the algorithmic lens to Bayesian epistemology.
Traditional Bayesian epistemology provides a comprehensive framework for how an
individual's beliefs should evolve upon receiving new information. However,
these methods typically assume an exhaustive model of such information,
including the correlation structure between different pieces of evidence. In
reality, individuals might lack such an exhaustive model, while still needing
to form beliefs. Beyond such informational constraints, an individual may be
bounded by limited computation, or by limited communication with agents that
have access to information, or by the strategic behavior of such agents. Even
when these restrictions prevent the formation of a *perfectly* accurate belief,
arriving at a *reasonably* accurate belief remains crucial. In this thesis, we
establish fundamental possibility and impossibility results about belief
formation under a variety of restrictions, and lay the groundwork for further
exploration.
