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.

翻译：我们研究具有对数凹性质的概率密度函数。尽管这类密度函数的空间是无限维的，但其最大似然估计量却是由有限个量（即数据点处的函数值（或高度））决定的逐段线性函数的指数形式。本文探索此问题在何种意义上可得到精确解。首先，我们证明由最大似然估计给出的高度值通常是超越数。在一维胞腔中，最大似然估计量可通过广义W-Lambert函数以封闭形式表达。更关键的是，我们证明求解对数凹最大似然估计等价于求解一类特殊形式的多项式-指数方程组。即便在仅包含两个方程的情形下，该方程组的解在数学上仍知之甚少。作为替代方案，我们利用Smale的α理论对近似数值解进行精化，并验证对数凹密度估计问题的解。