We present an axiomatic foundation for non-Bayesian social learning rules in networks, unifying and generalizing distributed learning updates that have been developed in the literature that combine Bayesian and DeGroot-style (consensus) updates. We show that any learning rule that satisfies general axioms of label neutrality, independence of irrelevance alternatives, history neglect, monotonicity, and unanimity will result in bounded rational updates that result in learning in strongly connected networks. More importantly, all such updates will be a hybrid of Bayesian and DeGrot-type updates. We characterize the rate of learning in these models and discuss the interplay of the network structure and discriminating power of agents’ observations even by their relative entropies. Joint work with Pooya Molavi (MIT Economics), Amin Rahimian (Penn) and Alireza Tahaz-Salehi (Columbia GSB)