  Collective perception is a foundational problem in swarm robotics, in which
the swarm must reach consensus on a coherent representation of the environment.
An important variant of collective perception casts it as a best-of-$n$
decision-making process, in which the swarm must identify the most likely
representation out of a set of alternatives. Past work on this variant
primarily focused on characterizing how different algorithms navigate the
speed-vs-accuracy tradeoff in a scenario where the swarm must decide on the
most frequent environmental feature. Crucially, past work on best-of-$n$
decision-making assumes the robot sensors to be perfect (noise- and
fault-less), limiting the real-world applicability of these algorithms. In this
paper, we derive from first principles an optimal, probabilistic framework for
minimalistic swarm robots equipped with flawed sensors. Then, we validate our
approach in a scenario where the swarm collectively decides the frequency of a
certain environmental feature. We study the speed and accuracy of the
decision-making process with respect to several parameters of interest. Our
approach can provide timely and accurate frequency estimates even in presence
of severe sensory noise.
