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For the model discussed in Example 7.1.2, determine the posterior distribution of the third…

For the model discussed in Example 7.1.2, determine the posterior distribution of the third quartile For the model discussed in Example 7.1.2, determine the posterior distribution of the third... Determine the posterior mode and the posterior expectation of ψ

 

Example 7.1.2

Suppose that For the model discussed in Example 7.1.2, determine the posterior distribution of the third... is a sample from an For the model discussed in Example 7.1.2, determine the posterior distribution of the third... distribution The likelihood function is then given by

For the model discussed in Example 7.1.2, determine the posterior distribution of the third...

Suppose we take the prior distribution of For the model discussed in Example 7.1.2, determine the posterior distribution of the third... for some specified For the model discussed in Example 7.1.2, determine the posterior distribution of the third... The posterior density of µ is then proportional to

For the model discussed in Example 7.1.2, determine the posterior distribution of the third...

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

For the model discussed in Example 7.1.2, determine the posterior distribution of the third...

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

For the model discussed in Example 7.1.2, determine the posterior distribution of the third...

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

For the model discussed in Example 7.1.2, determine the posterior distribution of the third...

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 For the model discussed in Example 7.1.2, determine the posterior distribution of the third... the less influence theprior has. For example, when For the model discussed in Example 7.1.2, determine the posterior distribution of the third... then the ratio of the

posterior variance to the sample mean variance iS For the model discussed in Example 7.1.2, determine the posterior distribution of the third... So there has been a5% improvement due to the use of prior information For example, suppose that For the model discussed in Example 7.1.2, determine the posterior distribution of the third... and that for n = 10 we

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

 

For the model discussed in Example 7.1.2, determine the posterior distribution of the third...