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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![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]
-
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D(λµ₁ + (1 − λ)µ2||Av₁ + (1 − X)12) ≤ XD(µ1||11) + (1 − X)D(µ2||12)
Hint Use Log-Sum Inequality.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1822b493-2c18-4a65-99e5-18fca75fe15d%2F78b70890-ea6b-433f-89c5-3c9f6f0a2e23%2Fmvir48ro_processed.png&w=3840&q=75)
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- Given a two-category classification problem under the univariate case, where there are two training sets (one for each category) as follows: D₁ = (-3,-1,0,4} D₂ = {-2,1,2,3,6,8} Given the test example x = 5, please answer the following questions: have and a) Assume that the likelihood function of each category has certain paramétric form. Specifically, we p(x | w₁) N, 07) p(x₂)~ N(μ₂, 02). Which category should we decide on when maximum-likelihood estimation is employed to make the prediction?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.Consider a real random variable X with zero mean and variance σ2X . Suppose that we cannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and {Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe…
- it is known that a natural. law obeys the quadratic relationship y=ax^2.what is the best line of form y=px+q that can be used to model data and minimise Mean-squared-error, if all of the data points are drawn uniformly at random from the domain [0,1]?For following observations, fit a line y= a+bx by the method of least squares. Estimate the coefficients, assume that the observations are gathered randomly and independent from populations of normal distributions with constant Variance: x 0.34. 1.38. -0.65. 0.68. 1.40 y. 0.27. 1.34. -0.53. 0.35. 1.28 Continue… x -0.88. -0.3. -1.18. 0.5. -1.75 y -0.98. 0.72. -0.81. 0.64. -1.59Consider a logistic regression classifier that implements the 2-input OR gate. At iteration t, the parameters are given by w0=0, w1=0, w2=0. Given binary input (x1,x2), output of logistic regression is given by 1/(1+exp(-w0-w1*x1-w2*x2)). What will be value of the loss function at t? What will be the values of w0, w1 and w2 at (t+1) with learning rate ɳ=1?
- GD algorithm Consider Linear Regression with single variable (univariate) problem. What will be the (approximate if can’t say accurately) values of derivatives of cost/loss function ‘J’ w.r.t. all the parameters by considering one at a time, and why? What is the significance and/or usage of these θj* for the cost function ‘J’ and hypothesis ‘h’? Given a dataset where first column is the label ‘y’ while other columns represent factors ‘xi’ as follows: X = [ 1 0 1 0 1 0 ] Using GD algorithm, find the linear model. Show all the calculationsConsider a text corpus consisting of N tokens of d distinct words and the number of times each distinct word w appears is given by . We want to apply a version of Laplace smoothing that estimates a word's probability as: xw+a/ N+ad for some constant a (Laplace recommended a = 1, but other values are possible.) In the following problems, assume N is 100,000, d is 10,000 and a is 2. A. Give both the unsmoothed maximum likelihood probability estimate and the Laplace smoothed estimate of a word that appears 1,000 times in the corpus. B. Do the same for a word that does not appear at all. C. You are running a Naive Bayes text classifier with Laplace Smoothing, and you suspect that you are overfitting the data. How would you increase or decrease the parameter a? D. Could increasing a increase or decrease training set error? Increase or decrease validation set error?Feature X has a uniform distribution X∼U([a,b])) where 0≤a<b≤1. Find maximum likelihood estimation and Bayesian estimation of X given a sample x1=.9,x2=.2,x3=.8,x4=.1
- Conditional Probability How do I find P(A) and also P(B) given P(C) P(C') P(AlC) P(BlC) P(AlC') and P(BlC'). And how do I find out if P(A) and P(B) are independent?ANY help would be greatly appreciated. From 1965 to 1974, in U.S. there were M= 17,857,857 male livebirths and F= 16,974,194 female livebirths. We model the number of male livebirth as a binomial distribution withparameterssize = M+F and prob = p. The following code computes the maximum likelihood estimator for p. male = 17857857 female = 16974194 ll <-function(p){dbinom(male, size = male+female, prob=p, log=TRUE) } ps <-seq(0.01, 0.99, by = 0.001) ll.ps <-ll(ps) plot(ps, ll.ps, type='l') phat <- ps[which.max(ll.ps)] abline(v = phat, col='blue') QUESTION: For this problem, can you give a theoretical formula for the maximum likelihood estimator,ˆp, usingMandF? (No need to compute the numerical value.)Generate 100 synthetic data points (x,y) as follows: x is uniform over [0,1]10 and y = P10 i=1 i ∗ xi + 0.1 ∗ N(0,1) where N(0,1) is the standard normal distribution. Implement full gradient descent and stochastic gradient descent, and test them on linear regression over the synthetic data points. Subject: Python Programming