Compare Bayesian and frequentist decisions

Suppose a binary KPI has Beta(1,1) priors, and each arm receives 10,000 users, with 515 conversions in one arm and 500 in the other. a) Derive each arm's posterior distribution and calculate the posterior probability that $\text{lift} > 0$. b) Specify a decision rule built on expected loss when the costs are asymmetric. c) Compare this decision to a frequentist test that allows optional stopping, and discuss the operational consequences, such as how to communicate a posterior versus a p-value and how alpha spending works. d) How would you choose or elicit priors so that they do not encourage over-optimism?

Overview: This item tests knowledge of Bayesian inference compared with frequentist hypothesis testing, decision theory under asymmetric loss, sequential monitoring or optional stopping, and prior elicitation in the context of A/B testing, within the Statistics & Math domain.