  We study probability density functions that are log-concave. Despite the
space of all such densities being infinite-dimensional, the maximum likelihood
estimate is the exponential of a piecewise linear function determined by
finitely many quantities, namely the function values, or heights, at the data
points. We explore in what sense exact solutions to this problem are possible.
First, we show that the heights given by the maximum likelihood estimate are
generically transcendental. For a cell in one dimension, the maximum likelihood
estimator is expressed in closed form using the generalized W-Lambert function.
Even more, we show that finding the log-concave maximum likelihood estimate is
equivalent to solving a collection of polynomial-exponential systems of a
special form. Even in the case of two equations, very little is known about
solutions to these systems. As an alternative, we use Smale's alpha-theory to
refine approximate numerical solutions and to certify solutions to log-concave
density estimation.
