Math 50 Stat Inf Homework 7 due Wed Feb 22 First few are exam review so focus on them before Monday 5 3 11 12 both review but very important 5 5 1 review A Samples yi i 1 n are taken from a normal with parameters and Let s do Bayesian inference on the variance This is also relevant practise for the exam No matlab i Write down the likelihood function L y This is exactly what you plotted in HW5 Replace any constant parameter independent terms by an overall constant we will not bother keeping track of such constants ii Assuming a uniform improper prior on write the posterior g y R iii Find the marginal posterior pdf on alone g y g y d This corresponds to inference on with unknown Hint This is the trickiest part You may leave an unknown R 2 2 constant independent factor out front You ll need the Gaussian integral e a x 2 bx dx 2 2 2 a eb 2a which can be proven simply by completing the square iv Take d d ln g y to find the maximum a posteriori estimate i e peak of the posterior pdf for It should be familiar where have you seen it Note how the skewness of the L peak you plotted in HW5 made the peak of the marginal on move from the ML joint estimate of B Bayesian prediction for bus waiting time No matlab i Solve part D of Worksheet 2 13 06 to get the predictive pdf h y y 1 of a future sample drawn from the bus wait time pdf As you can see the Bayesian method is to follow the laws of probability by averaging marginalizing over all the possible biases weighted according to their posterior pdf deduced from the data y1 Have you seen the resulting pdf recently ii Write down the predictive pdf for y following the frequentist non Bayesian approach of finding a single ML estimate then using that to predict iii Practical consequences You start taking a new bus route with unknown arrival rate use a uninformative prior Today day 1 you wait y1 10 mins Let Y be tomorrow s wait time Compute the Bayesian s and frequentist s predictions for p Y 60mins Hint you should be able to integrate both on paper think about tan 1 iv Discuss why they differ Which would you trust 6 2 1 careful whether one or two sided 2 6 3 1 6 1 6 4 1 3 7