  We consider the problem of distributed inference where agents in a network
observe a stream of private signals generated by an unknown state, and aim to
uniquely identify this state from a finite set of hypotheses. We focus on
scenarios where communication between agents is costly, and takes place over
channels with finite bandwidth. To reduce the frequency of communication, we
develop a novel event-triggered distributed learning rule that is based on the
principle of diffusing low beliefs on each false hypothesis. Building on this
principle, we design a trigger condition under which an agent broadcasts only
those components of its belief vector that have adequate innovation, to only
those neighbors that require such information. We prove that our rule
guarantees convergence to the true state exponentially fast almost surely
despite sparse communication, and that it has the potential to significantly
reduce information flow from uninformative agents to informative agents. Next,
to deal with finite-precision communication channels, we propose a distributed
learning rule that leverages the idea of adaptive quantization. We show that by
sequentially refining the range of the quantizers, every agent can learn the
truth exponentially fast almost surely, while using just $1$ bit to encode its
belief on each hypothesis. For both our proposed algorithms, we rigorously
characterize the trade-offs between communication-efficiency and the learning
rate.
