2.3 Posterior-based decisions

Sometimes, describing the posterior is not enough. We may need to make decisions based on our inferences and reduce a continuous estimation to a dichotomous one: yes-no, healthy-sick, contaminated-safe, and so on. For instance, is the coin fair? A fair coin is one with a θ value of exactly 0.5. We can compare the value of 0.5 against the HDI interval. From Figure 2.3, we can see that the HDI goes from 0.03 to 0.7 and hence 0.5 is included in the HDI. We can interpret this as an indication that the coin may be tail-biased, but we cannot completely rule out the possibility that the coin is actually 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.

2.3.1 Savage-Dickey density ratio

One way to evaluate how much support the posterior provides for a given value is to compare the ratio of the posterior and prior densities at...