4. In last assignment, we had assumed the multivariate normal model for Y, (Y|X, W) ~ N(XW, 0;I), where Y = [41, 42, ..., YM] E RMX1 is a vector of targets, X E RMXP is a matrix containing M data points with P features, and W = (W1, ..., wp] E RPX1 is a vector of weights. We had an important assumption that W is not a random variable. Now let us rethink this assumption and re-model W as a random variable and W ~ N(0,6 I). Note that we have used o and o to denote the variances of Y and W, respectively. Also note that the covariance matrices of Y and W have dimensions M X M and P x P respectively. By using Bayes' theorem, we know P(Y|X,W)P(W) P(W|Y, X) = (7) P(X) Derive the MAP (Maximum A Posteriori) of W, i.e. WMAP 1 AP = arg maxw P(W|Y, X). (20 points) (Hint: Write down P(Y|X, W) and P(W). Don't pay too much attention to P(X) since it can be considered as a scale factor in MAP. Also use the log-trick.)