  Statistical modeling often involves identifying an optimal estimate to some
underlying probability distribution known to satisfy some given constraints. I
show here that choosing as estimate the centroid, or center of mass, of the set
consistent with the constraints formally minimizes an objective measure of the
expected error. Further, I obtain a useful approximation to this point, valid
in the thermodynamic limit, that immediately provides much information relating
to the full solution set's geometry. For weak constraints, the centroid is
close to the popular maximum entropy solution, whereas for strong constraints
the two are far apart. Because of this, centroid inference is often
substantially more accurate. The results I present allow for its
straightforward application.
