9. Bayesian Econometric Methods (a) Deriving the Bayes Rule. The theoretical probability basics of the Bayes Rule, the central concept in Bayesian statistics, were laid down in the works of Reverend Thomas Bayes (1702-1761). Using the decomposition of joint probability into the product of conditional probability and marginal probability for two random variables (r.v.'s) A and B, derive the Bayes rule (as we did in class). (30 marks) (b) Bayesian versus Frequentist (or Classical) Econometrics. Then, appropriately replacing A and B with a vector (or matrix) of observed data y and a vector (or matrix) of parameters for a model that seeks to explain y, link the Bayes Rule (as we did in class) with the respective definitions of the likelihood function for the data, the prior density for the parameters and the posterior density for the parameters. Using this framework, distinguish the essential difference between Bayesian and frequentist (also known as classical) econometrics. (c) Bayesian Updating Rule. Define and discuss the Bayesian updating rule that arises from combining the three concepts introduced in part (b).
