Suppose that we have information that the normal mean is bounded between -m and m, for...

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Suppose that we have information that the normal mean is bounded between -m and m, for some known number m. In this case it is natural to elicit a prior on with the support on interval [-m, m]. A prior with interesting theoretical properties supported on [-m, m] is Bickel-Levit prior¹, -m<0<m. 2 TT (0) cos m TO 2m Assume that a sample [-2, -3, 5, -7,0, 4] is observed from normal distribution f(yl) x √T exp{-(y-0)²}, with a known precision 7 = = 1/3. Assume also that the prior on 0 is Bickel-Levit, with m = 2. This combination likelihood/prior does not result in an explicit posterior (in terms of elementary functions). Construct a Metropolis algorithm that will sample from the posterior of 0. (a) Simulate 20,000 observations from the posterior, after discarding first 500 observations (burn-in), and plot the histogram of the posterior. (b) Find Bayes estimator of 6, and 99% equitailed Credible Set based on the simulated observations. Suggestions: (i) Take uniform distribution on [-m, m] as a proposal distribution since it is easy to sample from. This is an independence proposal, the proposed 0 does not depend on the current value of the chain, 0. (ii) You will need to calculate 1(0)² for current 0 and 1(0)² for the proposed 0, prior to calculating the Metropolis ratio. Suppose that we have information that the normal mean is bounded between -m and m, for some known number m. In this case it is natural to elicit a prior on with the support on interval [-m, m]. A prior with interesting theoretical properties supported on [-m, m] is Bickel-Levit prior¹, -m<0<m. 2 TT (0) cos m TO 2m Assume that a sample [-2, -3, 5, -7,0, 4] is observed from normal distribution f(yl) x √T exp{-(y-0)²}, with a known precision 7 = = 1/3. Assume also that the prior on 0 is Bickel-Levit, with m = 2. This combination likelihood/prior does not result in an explicit posterior (in terms of elementary functions). Construct a Metropolis algorithm that will sample from the posterior of 0. (a) Simulate 20,000 observations from the posterior, after discarding first 500 observations (burn-in), and plot the histogram of the posterior. (b) Find Bayes estimator of 6, and 99% equitailed Credible Set based on the simulated observations. Suggestions: (i) Take uniform distribution on [-m, m] as a proposal distribution since it is easy to sample from. This is an independence proposal, the proposed 0 does not depend on the current value of the chain, 0. (ii) You will need to calculate 1(0)² for current 0 and 1(0)² for the proposed 0, prior to calculating the Metropolis ratio.

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Heres Python code that implements the metropolis algoritham and simulates 20000 observations from th... View the full answer