(a)The following questions refer to Bayesian Statistical Inference and Bayes' Theorem:

(i)What is a normalising constant and what role does it play in Bayesian Inference?

(ii)Does a prior distribution have to always be a subjective statement about uncertainty? Explain your answer and use examples to back up your argument.

(iii)What is a vague prior? Under what circumstances might you use one? Give an example of such a circumstance.

(iv)Doctors have developed a test for gluten allergy. They know that if a person has an allergy to gluten, their test will show a positive result 96% of the time. However, they also know that the false positive rate of their test is 15%. It is known that the prevalence of gluten allergies in the population is 3%. What is the probability that a person is allergic to gluten if the Doctors' test returns a positive result?