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. # Determine the form of the approximate 0.95-credible interval of Section 7.3.1, for the Bernoulli…

Determine the form of the approximate 0.95-credible interval of Section 7.3.1, for the Bernoulli model with a Beta(α ß) prior, discussed in Example 7.2.2

Example 7.2.2

Suppose that is a sample from an distribution The likelihood function is then given by Suppose we take the prior distribution of for some specified The posterior density of µ is then proportional to We immediately recognize this, as a function of µ, as being proportional to the densityof an distribution.Notice that the posterior mean is a weighted average of the prior mean µ0 and thesample mean x, with weights distribution.Notice that the posterior mean is a weighted average of the prior mean µ0 and thesample mean x, with weights respectively. This implies that the posterior mean lies between the prior mean and the sample mean.Furthermore, the posterior variance is smaller than the variance of the sample mean.So if the information expressed by the prior is accurate, inferences about µ based on the posterior will be more accurate than those based on the sample mean alone. Note that the more diffuse the prior is — namely, the larger the less influence theprior has. For example, when then the ratio of the

posterior variance to the sample mean variance iS So there has been a5% improvement due to the use of prior information For example, suppose that and that for n = 10 we

observe x= 1.2 Then the prior is an N(0, 2) distribution, while the posterior is an 