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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 Consider the sampling model and prior in Exercise 7.1.1. (a) Suppose we want to estimate A based... 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 Consider the sampling model and prior in Exercise 7.1.1. (a) Suppose we want to estimate A based...  from the Bernoulli(θ) distribution with Consider the sampling model and prior in Exercise 7.1.1. (a) Suppose we want to estimate A based... 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

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

This product is proportional to

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

We recognize this as the unnormalized density of a Beta Consider the sampling model and prior in Exercise 7.1.1. (a) Suppose we want to estimate A based... distribution.So in this example, we did not need to compute m Consider the sampling model and prior in Exercise 7.1.1. (a) Suppose we want to estimate A based... 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.

 

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

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.