Random Worlds and Maximum Entropy
  Given a knowledge base KB containing first-order and statistical facts, we
consider a principled method, called the random-worlds method, for computing a
degree of belief that some formula Phi holds given KB. If we are reasoning
about a world or system consisting of N individuals, then we can consider all
possible worlds, or first-order models, with domain {1,...,N} that satisfy KB,
and compute the fraction of them in which Phi is true. We define the degree of
belief to be the asymptotic value of this fraction as N grows large. We show
that when the vocabulary underlying Phi and KB uses constants and unary
predicates only, we can naturally associate an entropy with each world. As N
grows larger, there are many more worlds with higher entropy. Therefore, we can
use a maximum-entropy computation to compute the degree of belief. This result
is in a similar spirit to previous work in physics and artificial intelligence,
but is far more general. Of equal interest to the result itself are the
limitations on its scope. Most importantly, the restriction to unary predicates
seems necessary. Although the random-worlds method makes sense in general, the
connection to maximum entropy seems to disappear in the non-unary case. These
observations suggest unexpected limitations to the applicability of
maximum-entropy methods.
