  Recommendations based on behavioral data may be faced with ambiguous
statistical evidence. We consider the case of association rules, relevant
e.g.~for query and product recommendations. For example: Suppose that a
customer belongs to categories A and B, each of which is known to have positive
correlation with buying product C, how do we estimate the probability that she
will buy product C?
  For rare terms or products there may not be enough data to directly produce
such an estimate --- perhaps we never directly observed a connection between A,
B, and C. What can we do when there is no support for estimating the
probability by simply computing the observed frequency? In particular, what is
the right thing to do when A and B give rise to very different estimates of the
probability of C?
  We consider the use of maximum entropy probability estimates, which give a
principled way of extrapolating probabilities of events that do not even occur
in the data set! Focusing on the basic case of three variables, our main
technical contributions are that (under mild assumptions): 1) There exists a
simple, explicit formula that gives a good approximation of maximum entropy
estimates, and 2) Maximum entropy estimates based on a small number of samples
are provably tightly concentrated around the true maximum entropy frequency
that arises if we let the number of samples go to infinity.
  Our empirical work demonstrates the surprising precision of maximum entropy
estimates, across a range of real-life transaction data sets. In particular we
observe the average absolute error on maximum entropy estimates is a factor
$3$--$14$ less compared to using independence or extrapolation estimates, when
the data used to make the estimates has low support. We believe that the same
principle can be used to synthesize probability estimates in many settings.
