Posterior-based decisions

Sometimes, describing the posterior is not enough. Sometimes, we need to make decisions based on our inferences. We have to reduce a continuous estimation to a dichotomous one: yes-no, health-sick, contaminated-safe, and so on. We may need to decide if the coin is fair or not. A fair coin is one with a  value of exactly 0.5. We can compare the value of 0.5 against the HPD interval. In Figure 2.2, we can see that the HPD goes from ≈0.02 to ≈0.71 and hence 0.5 is included in the HPD. According to our posterior, the coin seems to be tail-biased, but we cannot completely rule out the possibility that the coin is fair. If we want a sharper decision, we will need to collect more data to reduce the spread of the posterior or maybe we need to find out how to define a more informative prior.