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Unformatted text preview: d density. The parameters µ and σ are the mean and standard deviation of the distribution p, and are not known. The maximum likelihood estimates of x, µ, σ are the maximizers of the log-likelihood function m m i=1 log p(yi − aT x) i = −m log σ + log f ( i=1 yi − a T x − µ i ), σ where y is the observed value. Show that if f is log-concave, then the maximum likelihood estimates of x, µ, σ can be determined by solving a convex optimization problem. 6.2 Mean and covariance estimation with conditional independence constraints. Let X ∈ Rn be a Gaussian random variable with density p( x ) = 1 (2π )n/2 (det S )1/2 exp(−(x − a)T S −1 (x − a)/2). The conditional density of a subvector (Xi , Xj ) ∈ R2 of X , given the remaining variables, is also Gaussian, and its covariance matrix Rij is equal to the Schur complement of the 2 × 2 submatrix Sii Sij Sij Sjj in the covariance matrix S . The variables Xi , Xj are called conditionally independent if the covariance matrix Rij of their conditional distribution is diagonal. Formulate the following problem as a convex optimization problem. We are given N independent samples y1 , . . . , yN ∈ Rn of X . We are also given a li...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.

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