For the binomial model, with success probability q and observed data k successes out of n trials, assume a Beta(α,β) prior distribution for q. Show that the posterior mean for q is always in between the prior mean for q and the maximum likelihood estimator qˆ. Show that if the prior is uniform, then the posterior variance for q is always less than the prior variance. Find an example of α,β,k and n where the posterior variance is greater than the prior variance.
For the binomial model, with success probability q and observed data k successes out of n trials, assume a Beta(α,β) prior distribution for q. Show that the posterior mean for q is always in between the prior mean for q and the maximum likelihood estimator qˆ. Show that if the prior is uniform, then the posterior variance for q is always less than the prior variance. Find an example of α,β,k and n where the posterior variance is greater than the prior variance.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 6CR
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For the binomial model, with success probability q and observed data k successes out of n trials, assume a Beta(α,β) prior distribution for q. Show that the posterior
Show that if the prior is uniform, then the posterior variance for q is always less than the prior variance. Find an example of α,β,k and n where the posterior variance is greater than the prior variance.
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VIEWStep 2: Show that posterior mean for q is always between prior mean for q and maximum likelihood estimators.
VIEWStep 3: Show that if the prior is uniform, then posterior variance for q is always less than prior variance.
VIEWStep 4: Determine the example of α, β, k and n where posterior variance is greater than prior variance.
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