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The log-concave maximum likelihood estimator (MLE) problem answers: for a set of points $X_1,...X_n in mathbb R^d$, which log-concave density maximizes their likelihood? We present a characterization of the log-concave MLE that leads to an algorithm with runtime $poly(n,d, frac 1 epsilon,r)$ to compute a log-concave distribution whose log-likelihood is at most $epsilon$ less than that of the MLE, and $r$ is parameter of the problem that is bounded by the $ell_2$ norm of the vector of log-likelihoods the MLE evaluated at $X_1,...,X_n$.
We consider the problem of computing the maximum likelihood multivariate log-concave distribution for a set of points. Specifically, we present an algorithm which, given $n$ points in $mathbb{R}^d$ and an accuracy parameter $epsilon>0$, runs in time
We study the problem of computing the maximum likelihood estimator (MLE) of multivariate log-concave densities. Our main result is the first computationally efficient algorithm for this problem. In more detail, we give an algorithm that, on input a s
Let X_1, ..., X_n be independent and identically distributed random vectors with a log-concave (Lebesgue) density f. We first prove that, with probability one, there exists a unique maximum likelihood estimator of f. The use of this estimator is attr
We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=expvarphi_0$ where $varphi_0$ is a concave function on $mathbb{R}$. The pointwise limiting distributi
We introduce a notion called entropic independence for distributions $mu$ defined on pure simplicial complexes, i.e., subsets of size $k$ of a ground set of elements. Informally, we call a background measure $mu$ entropically independent if for any (