  We analyze boundedly rational updating from aggregate statistics in a model
with binary actions and binary states. Agents each take an irreversible action
in sequence after observing the unordered set of previous actions. Each agent
first forms her prior based on the aggregate statistic, then incorporates her
signal with the prior based on Bayes rule, and finally applies a decision rule
that assigns a (mixed) action to each belief. If priors are formed according to
a discretized DeGroot rule, then actions converge to the state (in
probability), i.e., \emph{asymptotic learning}, in any informative information
structure if and only if the decision rule satisfies probability matching. This
result generalizes to unspecified information settings where information
structures differ across agents and agents know only the information structure
generating their own signal. Also, the main result extends to the case of $n$
states and $n$ actions.
