Select Page

.

# (MV) In Example 7.2.1, write out ithe integral that you would need to evaluate if you wanted to…

(MV) In Example 7.2.1, write out ithe integral that you would need to evaluate if you wanted to compute the posterior density of the third quartile of the population distribution

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