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Register a new userMaximum likelihood estimation of a multidimensional log-concave density
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Added by
Madeleine Cule
Publication date
2008
fields
Mathematical Statistics
and research's language is
English
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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 attractive because, unlike kernel density estimation, the method is fully automatic, with no smoothing parameters to choose. Although the existence proof is non-constructive, we are able to reformulate the issue of computation in terms of a non-differentiable convex optimisation problem, and thus combine techniques of computational geometry with Shors r-algorithm to produce a sequence that converges to the maximum likelihood estimate. For the moderate or large sample sizes in our simulations, the maximum likelihood estimator is shown to provide an improvement in performance compared with kernel-based methods, even when we allow the use of a theoretical, optimal fixed bandwidth for the kernel estimator that would not be available in practice. We also present a real data clustering example, which shows that our methodology can be used in conjunction with the Expectation--Maximisation (EM) algorithm to fit finite mixtures of log-concave densities. An R version of the algorithm is available in the package LogConcDEAD -- Log-Concave Density Estimation in Arbitrary Dimensions.
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We present theoretical properties of the log-concave maximum likelihood estimator of a density based on an independent and identically distributed sample in $mathbb{R}^d$. Our study covers both the case where the true underlying density is log-concave, and where this model is misspecified. We begin by showing that for a sequence of log-concave densities, convergence in distribution implies much stronger types of convergence -- in particular, it implies convergence in Hellinger distance and even in certain exponentially weighted total variation norms. In our main result, we prove the existence and uniqueness of a log-concave density that minimises the Kullback--Leibler divergence from the true density over the class all log-concave densities, and also show that the log-concave maximum likelihood estimator converges almost surely in these exponentially weighted total variation norms to this minimiser. In the case of a correctly specified model, this demonstrates a strong type of consistency for the estimator; in a misspecified model, it shows that the estimator converges to the log-concave density that is closest in the Kullback--Leibler sense to the true density.
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 distributions depend on the second and third derivatives at 0 of $H_k$, the lower invelope of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of $varphi_0=log f_0$ at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode $M(f_0)$ and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.
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 $X_1,...,X_n$.
Shape-constrained density estimation is an important topic in mathematical statistics. We focus on densities on $mathbb{R}^d$ that are log-concave, and we study geometric properties of the maximum likelihood estimator (MLE) for weighted samples. Cule, Samworth, and Stewart showed that the logarithm of the optimal log-concave density is piecewise linear and supported on a regular subdivision of the samples. This defines a map from the space of weights to the set of regular subdivisions of the samples, i.e. the face poset of their secondary polytope. We prove that this map is surjective. In fact, every regular subdivision arises in the MLE for some set of weights with positive probability, but coarser subdivisions appear to be more likely to arise than finer ones. To quantify these results, we introduce a continuous version of the secondary polytope, whose dual we name the Samworth body. This article establishes a new link between geometric combinatorics and nonparametric statistics, and it suggests numerous open problems.
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 $poly(n,d,1/epsilon),$ and returns a log-concave distribution which, with high probability, has the property that the likelihood of the $n$ points under the returned distribution is at most an additive $epsilon$ less than the maximum likelihood that could be achieved via any log-concave distribution. This is the first computationally efficient (polynomial time) algorithm for this fundamental and practically important task. Our algorithm rests on a novel connection with exponential families: the maximum likelihood log-concave distribution belongs to a class of structured distributions which, while not an exponential family, locally possesses key properties of exponential families. This connection then allows the problem of computing the log-concave maximum likelihood distribution to be formulated as a convex optimization problem, and solved via an approximate first-order method. Efficiently approximating the (sub) gradients of the objective function of this optimization problem is quite delicate, and is the main technical challenge in this work.
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