Show the following property of KL-divergence (i.e. convexity of KL-divergence). Let M1, M2, V1, V2 be distributions. Then for any λ = [0, 1] - - D(λµ₁ + (1 − λ)µ2||Av₁ + (1 − X)12) ≤ XD(µ1||11) + (1 − X)D(µ2||12) Hint Use Log-Sum Inequality.
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- It is known that a natural law obeys the quadratic relationship y = ax“. What is the best line of the form y = px + q that can be used to model data and minimize Mean-Squared-Error if all of the data points are drawn uniformly at random from the domain [0,1]? r* ur, a,A random variable X with two-sided exponential distribution given by has moment generating function given by M X (t)= e^ t +e^ -t -2 t^ 2 . f x (x)= x+1,&-1\\ 1-x,&0<= x<=1 - 1 <= x <= 0 (a) Using M_{X}(t) or otherwise, find the mean and variance of X. (b) Use Chebychev inequality to estimate the tail probability, P(X > delta) , for delta > 0 and compare your result with the exact tail probability.Let X be a random variable with density function 1) ={0. k(1 -x), if 0<*<1; otherwise. f(x) Find k, together with the expectation and the variance of the random variable Y defined as Y = 3X - 1.
- Consider the same house rent prediction problem where you are supposed to predict price of a house based on just its area. Suppose you have n samples with their respective areas, x(1), x(2), ... , x(n), their true house rents y(1), y(2),..., y(n). Let's say, you train a linear regres- sor that predicts f(x()) = 00 + 01x(e). The parameters 6o and 0, are scalars and are learned by minimizing mean-squared-error loss with L2-regularization through gradient descent with a learning rate a and the regularization strength constant A. Answer the following questions. 1. Express the loss function(L) in terms of x), y@), n, 0, 01, A. 2. Compute L 3. Compute 4. Write update rules for 6, and O1Prove that I(X; Y |Z) ≥ I(X; Y ) . Note: X, Y, and Z are random variables. X and Z are independent.Consider the same house rent prediction problem where you are supposed to predict price of a house based on just its area. Suppose you have n samples with their respective areas, x(¹), x(²),...,x(n), their true house rents y(¹), y(2),..., y(n). Let's say, you train a linear regres- sor that predicts f(x)) = 0 + 0₁x). The parameters, and 0₁ are scalars and are learned by minimizing mean-squared-error loss with L1-regularization through gradient descent with a learning rate a and the regularization strength constant A. Answer the following questions. 1. Express the loss function(L) in terms of x(i),y(i), n, 00, 01, X. 2. Compute L 200 ƏL 3. Compute 20₁ 4. Write update rules for 0o and 0₁ Hint: d|w| dw undefined -1 w>0 w=0 w <0
- What is the inverse CDF of the uniform distribution from 0 to 1? Write your answer in terms of u. E.g. the inverse of the CDF of the exponential distribution is -1/lambda*log(1-u)Let i, j, w, x ∈ Za with i, j different from each other and w, x different from each other. When an invertible function f : Za → Za is chosen at random, what is Pr[f(j) = x | f(i) = w]?Optimize the following Boolean function by finding all prime implicant andessential prime implicant f(W,X,Y,Z) =∑ m(0,2,3,5,7,8,10,11,12,13)
- In an Accept–Reject algorithm that generates a N (0, 1) random vari-able from a double-exponential distribution with density g(x|α) = (α/2) exp(−α|x|), compute the upper bound M over f /g and show that the choice α = 1 optimizes thecorresponding acceptance rate.Let X N(3,9) and consider Y = -5 X + 4. Which of the following describes the distribution of Y? Y-N(19, 225 ) OY-N(-11,45 ) OY-N(-11, 225 ) OY-N(-11,-41) Y N(-15, -45) Y-N(7,-45)Let X= (-3,-2,-1,0, 1}. Let f: X→ N by f(x) = [7z+11: The range of the function is