Learning mixtures of separated nonspherical Gaussians

Mixtures of Gaussian (or normal) distributions arise in a variety of application areas. Many heuristics have been proposed for the task of finding the component Gaussians given samples from the mixture, such as the EM algorithm, a local-search heuristic from Dempster, Laird and Rubin [J. Roy. Statist. Soc. Ser. B 39 (1977) 1-38]. These do not provably run in polynomial time. We present the first algorithm that provably learns the component Gaussians in time that is polynomial in the dimension. The Gaussians may have arbitrary shape, but they must satisfy a ``separation condition'' which places a lower bound on the distance between the centers of any two component Gaussians. The mathematical results at the heart of our proof are ``distance concentration'' results--proved using isoperimetric inequalities--which establish bounds on the probability distribution of the distance between a pair of points generated according to the mixture. We also formalize the more general problem of max-likelihood fit of a Gaussian mixture to unstructured data.Comment: Published at http://dx.doi.org/10.1214/105051604000000512 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Computation of Counterfactual Equilibria in Homothetic Walrasian Economies

We propose a nonparametric test for multiple calibration of numerical general equilibrium models, and we present an effective algorithm for computing counterfactual equilibria in homothetic Walrasian economies, where counterfactual equilibria are solutions to the Walrasian inequalities.Applied general equilibrium analysis, Walrasian inequalities, Calibration

Two Algorithms for Solving the Walrasian Equilibrium Inequalities

We propose two algorithms for deciding if the Walrasian equilibrium inequalities are solvable. These algorithms may serve as nonparametric tests for multiple calibration of applied general equilibrium models or they can be used to compute counterfactual equilibria in applied general equilibrium models defined by the Walrasian equilibrium inequalities.Applied general equilibrium analysis, Walrasian equilibrium inequalities, Calibration

Indeterminacy, Nonparametric Calibration and Counterfactual Equilibria

We propose a nonparametric approach to multiple calibration of numerical general equilibrium models, where counterfactual equilibria are solutions to the Walrasian inequalities. We present efficient approximation schemes for deciding the solvability of Walrasian inequalities.Applied general equilibrium analysis, Walrasian inequalities, O-minimal structures, Monte Carlo algorithms

Decision Methods for Solving Systems of Walrasian Inequalities

We propose two algorithms for deciding if systems of Walrasian inequalities are solvable. These algorithms may serve as nonparametric tests for multiple calibration of applied general equilibrium models or they can be used to compute counterfactual equilibria in applied general equilibrium models defined by systems of Walrasian inequalities.Applied general equilibrium analysis, Walrasian inequalities, Calibration

Two Algorithms for Solving the Walrasian Equilibrium Inequalities

We propose two algorithms for deciding if the Walrasian equilibrium inequalities are solvable. These algorithms may serve as nonparametric tests for multiple calibration of applied general equilibrium models or they can be used to compute counterfactual equilibria in applied general equilibrium models defined by the Walrasian equilibrium inequalities.Applied general equilibrium analysis, Walrasian equilibrium inequalities, calaibration