bv_cvxbook_extra_exercises

227 functions of a random variable with log concave

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Unformatted text preview: Y ∈ Sm , has the solution − T Y = X11 − X12 X221 X12 . (As usual, we take Sm as the domain of log det Y −1 .) ++ Hint. Use the Schur complement characterization of positive definite block matrices (page 651 of the book): if C ≻ 0 then AB 0 BT C if and only if A − BC −1 B T 0. (c) Combine the result in part (b) and the minimization property (page 3-19, lecture notes) to show that the function − T f (X ) = log det(X11 − X12 X221 X12 )−1 , with dom f = Sn , is convex. ++ − T (d) Show that (X11 − X12 X221 X12 )−1 is the leading m × m principal submatrix of X −1 , i.e., − T (X11 − X12 X221 X12 )−1 = P T X −1 P. Hence, the convex function f defined in part (c) can also be expressed as f (X ) = log det(P T X −1 P ). Hint. Use the formula for the inverse of a symmetric block matrix: A BT B C −1 = 0 0 0 C −1 + −I C −1 B T (A − BC −1 B T )−1 −I C −1 B T T if C and A − BC −1 B T are invertible. 2.27 Functions of a random variable with log-concave densit...
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