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Abstract Agents learn about a changing state using private signals and their neighbours’ past estimates of the state. We present a model in which Bayesian agents in equilibrium use neighbours’ estimates simply by taking weighted sums with time-invariant weights. The dynamics thus parallel those of the tractable DeGroot model of learning in networks, but arise as an equilibrium outcome rather than a behavioural assumption. We examine whether information aggregation is nearly optimal as neighbourhoods grow large. A key condition for this is signal diversity: each individual’s neighbours have private signals that not only contain independent information, but also have sufficiently different distributions. Without signal diversity—e.g. if private signals are i.i.d.—learning is suboptimal in all networks and highly inefficient in some. Turning to social influence, we find it is much more sensitive to one’s signal quality than to one’s number of neighbours, in contrast to standard models with exogenous updating rules.