  In this paper, we consider the problem of social learning, where a group of
agents embedded in a social network are interested in learning an underlying
state of the world. Agents have incomplete, noisy, and heterogeneous sources of
information, providing them with recurring private observations of the
underlying state of the world. Agents can share their learning experience with
their peers by taking actions observable to them, with values from a finite
feasible set of states. Actions can be interpreted as samples from the beliefs
which agents may form and update on what the true state of the world is.
Sharing samples, in place of full beliefs, is motivated by the limited
communication, cognitive, and information-processing resources available to
agents especially in large populations. Previous work (Salhab et al.) poses the
question as to whether learning with probability one is still achievable if
agents are only allowed to communicate samples from their beliefs. We provide a
definite positive answer to this question, assuming a strongly connected
network and a ``collective distinguishability'' assumption, which are both
required for learning even in full-belief-sharing settings. In our proposed
belief update mechanism, each agent's belief is a normalized weighted geometric
interpolation between a fully Bayesian private belief -- aggregating
information from the private source -- and an ensemble of empirical
distributions of the samples shared by her neighbors over time. By carefully
constructing asymptotic almost-sure lower/upper bounds on the frequency of
shared samples matching the true state/or not, we rigorously prove the
convergence of all the beliefs to the true state, with probability one.
