  The joint distribution $P(X,Y)$ cannot be determined from its marginals
$P(X)$ and $P(Y)$ alone; one also needs one of the conditionals $P(X|Y)$ or
$P(Y|X)$. But is there a best guess, given only the marginals? Here we answer
this question in the affirmative, obtaining in closed form the function of the
marginals that has the lowest expected Kullbach-Liebler (KL) divergence between
the unknown "true" joint probability and the function value. The expectation is
taken with respect to Jeffreys' non-informative prior over the possible joint
probability values, given the marginals. This distribution can also be used to
obtain the expected information loss for any other "aggregation operator", as
such estimators are often called in fuzzy logic, for any given pair of marginal
input values. This enables such such operators, including ours, to be compared
according to their expected loss under the minimal knowledge conditions we
assume. We go on to develop a method for evaluating the expected accuracy of
any aggregation operator in the absence of knowledge of its inputs. This
requires averaging the expected loss over all possible input pairs, weighted by
an appropriate distribution. We obtain this distribution by marginalizing
Jeffreys' prior over the possible joint distributions (over the 3 functionally
independent coordinates of the space of joint distributions over two Boolean
variables) onto a joint distribution over the pair of marginal distributions, a
2-dimensional space with one parameter for each marginal. We report the
resulting input-averaged expected losses for a few commonly used operators, as
well as the optimal operator. Finally, we discuss the potential to develop our
methodology into a principled risk management approach to replace the often
rather arbitrary conditional-independence assumptions made for probabilistic
graphical models.
