Mathematical Statistics 2019/2020, Problem set 15 Introduction to Bayesian Statistics
1. A researcher measured the length of four insects of a newly discovered species; the measurements were (in cm): 0.38, 0.65, 0.72 and 1. The researcher assumes that the length of these insects follows a uniform distribution over the interval (0, θ], where the maximum length θ is an unknown parameter.
Prior to making the measurements, the researcher believed that the maximum length of the insect is a number between 12 and 2 (i.e., that θ ∼ U (12, 2)). Find the posterior distribution for the experiment.
Find three estimators of the parameter θ: the Bayesian Most Probable Estimator, the Bayesian estimator for a quadratic loss function (i.e., the average of the posterior distribution) and the Bayesian estimator for a modulus loss function (i.e., the median of the posterior distribution).
2. A pharmaceutical company conducts trials with a drug to cure a virus-inflicted illness, on a sample of 10 individuals whose reactions to the drug are independent. The results are that the administered drug alleviated the sickness symptoms in 3 cases. Prior to the experiment, one employee of the company was hesitant to make any predictions about the probability θ that the drug administered to a single individual will alleviate the illness symptoms of this individual, and worked with a uniform prior for θ: π1(θ) = 1[0,1](θ). Another employee was convinced that the drug would work in approximately two-thirds of the cases and worked with a Beta(4,2) prior for θ: π2(θ) = 20θ3(1 − θ)1[0,1](θ). Compare the Bayesian Most Probable estimates of the parameter θ for the two employees. Do the same with the Bayesian estimators for quadratic loss functions.
