  Assuming distributions are Gaussian often facilitates computations that are
otherwise intractable. We study the performance of an agent that attains a
bounded information ratio with respect to a bandit environment with a Gaussian
prior distribution and a Gaussian likelihood function when applied instead to a
Bernoulli bandit. Relative to an information-theoretic bound on the Bayesian
regret the agent would incur when interacting with the Gaussian bandit, we
bound the increase in regret when the agent interacts with the Bernoulli
bandit. If the Gaussian prior distribution and likelihood function are
sufficiently diffuse, this increase grows at a rate which is at most linear in
the square-root of the time horizon, and thus the per-timestep increase
vanishes. Our results formalize the folklore that so-called Bayesian agents
remain effective when instantiated with diffuse misspecified distributions.
