Answer two multiple choice question about Linear Gaussian Systems & linear regression

1. This problem is regarding Linear Gaussian Systems. Consider the problem of locating the true location (y) of an airplane on a 2-D plane from a series of noisy radar observations. We assume that each radar observation (x) is coming from a sensor and can be modeled as a Gaussian with mean as y and the covariance matrix as [[3,0],[0,0.8]], which defines the error of the sensor. We assume that the prior estimate for the location of the airplane (y) is a Gaussian, centered at (0,0) and with covariance matrix as [[1,0],[0,1]]. Using the linear Gaussian system equations stated in the class (Part C1 slides and handouts) and the noisy observations provided below, compute the new estimates for the location of the airplane. Note that, the posterior estimate will also be a Gaussian distribution. Let the four noisy observations of the aircraft's locations be: 1.4,1.2 1.1,1.1 1.2,1.2 1.0,0.9 What will be posterior estimates for the true location of the aircraft. HINT: Instead of writing the code to compute the posterior estimates, you may use the code provided in the notebook -- LinearSystems. Please remember to use the correct values for each variable. a) After observing the first two observations, the posterior estimate will be a Gaussian with mean = [0.50,0.82] and covariance = [[0.60, 0.00],[0.00, 0.29]] b) After observing the first two observations, the posterior estimate will be a Gaussian with mean = [0.66667,0.35] and covariance = [[0.75, 0],[0.00, 0.4444]] c) After observing the first three observations, the posterior estimate will be a Gaussian with mean = [0.67,0.35] and covariance = [[0.75, 0],[0.00, 0.44]] d) After observing all four observations, the posterior estimate will be a Gaussian with mean = [0.66667,0.35] and covariance = [[0.75, 0],[0.00, 0.4444]] 2. This problem deals with linear regression. Consider the following data set, the input variable (first column) indicates the amount spent by an individual on internet service and the output variable (second column) indicates the degree of satisfaction with his/her internet service. Time Spent Satisfaction Degree 11 6 18 8 17 10 15 4 9 9 5 6 12 3 19 5 22 2 25 10 We are interested in choosing the best weight vector among four candidates. The fourth candidate (w4) uses higher-order features - <1,x,x2>. For all others we assume that first component of the weight vector corresponds to the bias/intercept term. We assume that the variance for each y is same (0.01) for all choices of the weights. We assume a Gaussian prior on the weight vectors with 0 mean and a diagonal covariance matrix with 0.01 at every diagonal entry. The candidates are: w1 = (0.30, 0.50) w2 = (5.75, 0.04) w3 = (3.20, 0.20) w4 = (8.75, -0.50, 0.02) Given the following statements: 1. Among the first three weights, w2 will be the best estimate in terms of likelihood of the given data. 2. Among the first three weights, w3 will be the best estimate in terms of likelihood of the given data. 3. Among all four weights, w2 will be the best estimate in terms of likelihood of the given data. 4. Among all four weights, w4 will be the best estimate in terms of likelihood of the given data. 5. w3 has the highest posterior 6. w4 has the highest posterior 7. Ridge regression will always choose a non-linear mapping on features over the raw linear feature Which of the following are true? a) Statements 2 and 5 are true b) Statements 2 and 3 are true c) Statements 1 and 5 are true and Statement 3 is false d) Statements 1 and 3 are true and 7 is false