A detector's efficiency is measured by counting successes out of test pulses
PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Data analysis and model fitting in the laboratory · Data analysis and model fitting in the laboratory
Error, Chance and the Machine, Ch Tests, intervals and Bayesian inference
Problem
A detector's efficiency θ is measured by counting k successes out of n test pulses. Two priors are used: the uniform prior Beta(1,1), giving posterior Beta(k+1,n−k+1), and Jeffreys' prior Beta(21,21), giving posterior Beta(k+21,n−k+21).
((a)) Compute the posterior mean under each prior for n=20, k=18.
((b)) Compute the posterior mean under each prior for n=200, k=180 (ten times the data, same fraction of successes).
((c)) Compute the posterior standard deviation under the uniform prior for each sample size (mean of a Beta(a,b) is a/(a+b) and its variance is ab/[(a+b)2(a+b+1)]).
((d)) Express the difference between the two posterior means, as a fraction of the uniform-prior standard deviation, for each sample size, and state in one sentence how the prior's influence changes as the data grow by a factor of ten.