Consider a linear regression problem in which training examples have differ- ent weights. Specifically, suppose we...

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Consider a linear regression problem in which training examples have differ- ent "weights". Specifically, suppose we want to minimize: J(0): = 1 - Σw(i) (0¹ x(i) - y(i)) ² 2m a) Show that J(0) can also be written m J(0) = (X0 – y)*W(X0−y) p(y (₁¹) |x (1¹); 0) for matrices X, y and W to be determined b) By taking the derivative VJ(0), and setting that to zero, give the value of 0 that minimizes J(0) in closed forms in terms of X, W, and y c) Suppose we have a training set {(r(¹), y¹)), i = 1,2,...,m} of m independent ex- amples, but in which the y(i)'s were observed with different variances. Specifically, suppose that = 1 √27010) exp (- (2πσ(i) (y(i) - Otx(i))2) 2((¹))2 Show that finding the maximum likelihood estimate of reduces to solving weighted linear problem. State clearly what the w(¹)'s are in terms of the o(¹)'s Activa Go to Se Consider a linear regression problem in which training examples have differ- ent "weights". Specifically, suppose we want to minimize: J(0): = 1 - Σw(i) (0¹ x(i) - y(i)) ² 2m a) Show that J(0) can also be written m J(0) = (X0 – y)*W(X0−y) p(y (₁¹) |x (1¹); 0) for matrices X, y and W to be determined b) By taking the derivative VJ(0), and setting that to zero, give the value of 0 that minimizes J(0) in closed forms in terms of X, W, and y c) Suppose we have a training set {(r(¹), y¹)), i = 1,2,...,m} of m independent ex- amples, but in which the y(i)'s were observed with different variances. Specifically, suppose that = 1 √27010) exp (- (2πσ(i) (y(i) - Otx(i))2) 2((¹))2 Show that finding the maximum likelihood estimate of reduces to solving weighted linear problem. State clearly what the w(¹)'s are in terms of the o(¹)'s Activa Go to Se

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To answer the question outlined in the provided image lets take it step by step a To show that Jtheta can also be written as Jtheta Xtheta yT W Xtheta y we have to understand the dimensions of the mat... View the full answer