Select all correct statements The gradient descent solution to logistic regression model might converge to local optimums, and fail to find global optimal solution. The cost function of logistic regression is convex. In logistic regression, we model the odds ratio ( ) as a linear function р 1-p logistic regression is a regression method to estimate class posterior probabilities

Select all correct statements The gradient descent solution to logistic regression model might converge to local optimums, and fail to find global optimal solution. The cost function of logistic regression is convex. In logistic regression, we model the odds ratio ( ) as a linear function р 1-p logistic regression is a regression method to estimate class posterior probabilities

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A discrete random variable X with the range (i.e. the set of all possible values) Rx = {0, 1, 2, 3,... } has probability mass function f(x; 0) = -0.0 e- a! x = 0, 1, 2, 3,... where is an unknown parameter. We want to find maximum likelihood estimate MLE of the parameter based on a sample (x1, x2,...,xn)? To do that, we find the likelihood function n L(0) = L(0; x1, . . ., x n ) = II ƒ (xi ; 0) · = i=1 = i=1 n Oxi e 0x1 xi! e-o 0x2 e-o Өх x2! xn! xi! -по x₁! =en. ΑΣ 12 Then, the log-likelihood function is In L(0) = ln L(0; x1,...,xn) = lne¯no . n II i=1 In en+In+In ( 1 · II 1 xi ! n =-10+(Σ Σπι)· .In In i=1 1 ·II xi! To find Ô MLE (the maximum likelihood estimator of 0), we maximize In L(0) while thinking of it as a function of only, i.e. treating xi's as given constants, and thus, treating the last term In ( 1 I_! as a constant. Eventually, what do we get as the maximum likelihood estimator MLE of the parameter 0?
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Question 3. Regression need answer of part b Consider real-valued variables X and Y. The Y variable is generated, conditional on X, from the fol- lowing process: E~N(0,0²) YaX+e where every e is an independent variable, called a noise term, which is drawn from a Gaussian distri- bution with mean 0, and standard deviation σ. This is a one-feature linear regression model, where a is the only weight parameter. The conditional probability of Y has distribution p(YX, a) ~ N(aX, 0²), so it can be written as p(YX,a) = exp(- (-202 (Y-ax)²) 1 ν2πσ The following questions are all about this model. MLE estimation (a) Assume we have a training dataset of n pairs (X, Y) for i = 1..n, and σ is known. Which ones of the following equations correctly represent the maximum likelihood problem for estimating a? Say yes or no to each one. More than one of them should have the answer "yes." a 1 [Solution: no] arg max > 2πσ 1 [Solution: yes] arg max II a [Solution: no] arg max a [Solution: yes] arg max a 1…
2. Let D be a distribution over R where the mean is 5 and variance is 9. Suppose x1, ... , x10 are indepen- dent draws from D. Plot the possible positions of these random variables on the real line. 3. Let w E Rd be the variable, and let x E Rd and y E R be given. Calculate the gradient of the following functions with respect to w: • F(w) = (y – w · x)100; • F(w) = vrwsi 1 y+w.x' • F(w) = log(1+ yw · x); F(w) = e(w-x)².
You are provided with last year’s data showing which high school students chose standard or advanced coursework. The predictor variables include their writing score, math score, and science scores from previous years. Your task is to build a model that predicts if this year's incoming students are in advanced or standard coursework given the above predictor variables. Which model is suitable for this task? Linear regression k-means Clustering Logistic Regression Regression tree
Solve In R programmning language: Calculate the probability for each of the following events: (a) A standard normally distributed variable is less than -2.5. (b) A normally distributed variable with mean 35 and standard deviation 6 is larger than 42 but less than 45. (c) A normally distributed variable with mean 35 and standard deviation 6 is larger than 40 but less than 41. (d) X < 0.9 when X has the standard uniform distribution (min=0, max=1). (e) 1 < X < 3 in the exp distribution with rate λ = 2.
Assume the following simple regression model, Y = β0 + β1X + ϵ ϵ ∼ N(0, σ^2 ) Now run the following R-code to generate values of σ^2 = sig2, β1 = beta1 and β0 = beta0. Simulate the parameters using the following codes: Code: # Simulation ## set.seed("12345") beta0 <- rnorm(1, mean = 0, sd = 1) ## The true beta0 beta1 <- runif(n = 1, min = 1, max = 3) ## The true beta1 sig2 <- rchisq(n = 1, df = 25) ## The true value of the error variance sigmaˆ2 ## Multiple simulation will require loops ## nsample <- 10 ## Sample size n.sim <- 100 ## The number of simulations sigX <- 0.2 ## The variances of X # # Simulate the predictor variable ## X <- rnorm(nsample, mean = 0, sd = sqrt(sigX)) Q1 Fix the sample size nsample = 10 . Here, the values of X are fixed. You just need to generate ϵ and Y . Execute 100 simulations (i.e., n.sim = 100). For each simulation, estimate the regression coefficients (β0, β1) and the error variance (σ 2 ). Calculate the mean of…
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?