  A common statistical situation concerns inferring an unknown distribution
Q(x) from a known distribution P(y), where X (dimension n), and Y (dimension m)
have a known functional relationship. Most commonly, n<m, and the task is
relatively straightforward. For example, if Y1 and Y2 are independent random
variables, each uniform on [0, 1], one can determine the distribution of X = Y1
+ Y2; here m=2 and n=1. However, biological and physical situations can arise
where n>m. In general, in the absence of additional information, there is no
unique solution to Q in those cases. Nevertheless, one may still want to draw
some inferences about Q. To this end, we propose a novel maximum entropy
(MaxEnt) approach that estimates Q(x) based only on the available data, namely,
P(y). The method has the additional advantage that one does not need to
explicitly calculate the Lagrange multipliers. In this paper we develop the
approach, for both discrete and continuous probability distributions, and
demonstrate its validity. We give an intuitive justification as well, and we
illustrate with examples.
