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4.29 Maximizing probability of satisfying a linear inequality. Let c be a random variable in Rn, normally distributed with mean ¯c and covariance matrix R. Consider the problem maximize prob(cTx ≥ α) subject to Fx g, Ax = b. Find the conditions under which this is equivalent to a convex or quasiconvex optimiza-
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Probability Theory Review for Machine Learning
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see.stanford.eduLeast-squares (approximate) solution • assume A is full rank, skinny • to find xls, we’ll minimize norm of residual squared, krk2 = xTATAx−2yTAx+yTy • set gradient w.r.t. x to zero: ∇xkrk2 = 2ATAx−2ATy = 0 • yields the normal equations: ATAx = ATy • assumptions imply ATA invertible, so we have xls = (ATA)−1ATy. . . a very famous formula
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