  The pattern maximum likelihood (PML) estimate, introduced by Orlitsky et al.,
is an estimate of the multiset of probabilities in an unknown probability
distribution $\mathbf{p}$, the estimate being obtained from $n$ i.i.d. samples
drawn from $\mathbf{p}$. The PML estimate involves solving a difficult
optimization problem over the set of all probability mass functions (pmfs) of
finite support. In this paper, we describe an interesting phase transition
phenomenon in the PML estimate: at a certain sharp threshold, the uniform
distribution goes from being a local maximum to being a local minimum for the
optimization problem in the estimate. We go on to consider the question of
whether a similar phase transition phenomenon also exists in the Bethe
approximation of the PML estimate, the latter being an approximation method
with origins in statistical physics. We show that the answer to this question
is a qualified "Yes". Our analysis involves the computation of the mean and
variance of the $(i,j)$th entry, $a_{i,j}$, in a random $k \times k$
non-negative integer matrix $A$ with row and column sums all equal to $M$,
drawn according to a distribution that assigns to $A$ a probability
proportional to $\prod_{i,j} \frac{(M-a_{i,j})!}{a_{i,j}!}$.
