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Résumé |
Large empirical sample covariance matrices exhibit a distribution of eigenvalues well described by the work of Marcenko and Pastur. As a consequence, mean-variance optimization using the sample-covariance matrix gives disastrous out-of-sample results as the methods overweighs modes of spuriously low variance. We propose here a Bayesian framework for the true covariance matrix whose conditional expectation should be used for mean-variance optimization. While the framework is quite general, we explore a few rotationally invariant priors, namely shifted-Wigner, Wishart and inverse-Wishart. While the latter is easily solved and leads to linear shrinkage the other two give rise to interesting matrix integrals. A naïve matrix saddle point gives an approximate solution, the zeroth order term in a systematic expansion in planar Feynman diagrams. The first few terms of this expansion can be computed. The eigenvalue saddle point involves the orthogonal version of the Itzykson-Zuber integral that we solve exactly only in a very special case. We also discuss a numerical method that allows the computation of the conditional mean by Monte Carlo simulation. The methodology presented here has direct applications in finance (portfolio optimization) and should be applicable to other fields of ``big data''. \\ \\ (LPTMS, salle 201, 2ème étage, Bât 100, Campus d'Orsay) \\ (http://lptms.u-psud.fr/seminars/tri-semianaire-de-physique-marc-potters/) |
