Select Page

.

# Consider the sampling model and prior in Exercise 7.1.1. (a) Suppose we want to estimate A based…

Consider the sampling model and prior in Exercise 7.1.1.

(a) Suppose we want to estimate A based upon having observed s = 1Determine the posterior mode and posterior mean. Which would you prefer in this situation? Explain why.

(b) Determine a 0.8 HPD region for A based on having observed s = 1

(c) Suppose instead interest was in  dentify the prior distribution of ψ Identify the posterior distribution of ψ based on having observed s =1Determine a 0.5 HPD region for ψ

Example 7.1.1

Suppose that we observe a sample   from the Bernoulli(θ) distribution with  unknown. For the prior, we take π to be equal to a Beta(α,ß)     density  Then the posterior of π is proportional to the likelihood

This product is proportional to

We recognize this as the unnormalized density of a Beta  distribution.So in this example, we did not need to compute m  to obta α = ß i.e., we have a uniform prior on θ Then the posterior of θ is given by theBeta(11, 31) distribution. We plot the posterior density in Figure 7.1.3 as well as the

prior.

The spread of the posterior distribution gives us some idea of the precision of any probability statements we make about θ . Note how much information the data have added, as reflected in the graphs of the prior and posterior densities.