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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 Determine the form of the approximate 0.95-credible interval of Section 7.3.1, for the Bernoulli... is a sample from an Determine the form of the approximate 0.95-credible interval of Section 7.3.1, for the Bernoulli... distribution The likelihood function is then given by

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

Suppose we take the prior distribution of Determine the form of the approximate 0.95-credible interval of Section 7.3.1, for the Bernoulli... for some specified Determine the form of the approximate 0.95-credible interval of Section 7.3.1, for the Bernoulli... The posterior density of µ is then proportional to

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

We immediately recognize this, as a function of µ, as being proportional to the densityof an

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

distribution.Notice that the posterior mean is a weighted average of the prior mean µ0 and thesample mean x, with weights

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

distribution.Notice that the posterior mean is a weighted average of the prior mean µ0 and thesample mean x, with weights

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

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 Determine the form of the approximate 0.95-credible interval of Section 7.3.1, for the Bernoulli... the less influence theprior has. For example, when Determine the form of the approximate 0.95-credible interval of Section 7.3.1, for the Bernoulli... then the ratio of the

posterior variance to the sample mean variance iS Determine the form of the approximate 0.95-credible interval of Section 7.3.1, for the Bernoulli... So there has been a5% improvement due to the use of prior information For example, suppose that Determine the form of the approximate 0.95-credible interval of Section 7.3.1, for the Bernoulli... and that for n = 10 we

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

 

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