Transcription of Posterior Predictive Distribution
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Example: confidence
Posterior Predictive Distribution
Posterior Predictive DistributionIRecall that for a fixed value of , our dataXfollow thedistributionp(X| ).IHowever, the true value of is uncertain, so we shouldaverage over the possible values of to get a better idea ofthe Distribution the sample, the uncertainty in is representedby the prior distributionp( ). So for some new data valuexnew, averaging overp( ) gives theprior predictivedistribution:p(xnew) = p(xnew, ) d = p(xnew| )p( ) d Posterior Predictive DistributionIAftertaking the sample, we have abetter representationofthe uncertainty in via our posteriorp( |x). So theposteriorpredictive distributionfor a new data pointxnewis:p(xnew|x) = p(xnew| ,x)p( |x) d = p(xnew| )p( |x) d (sincexnewis independent of the sample datax)IThis reflects how we would predict new data to behave / the data wedid observefollow this pattern closely, itindicates we have chosen our model and prior Predictive DistributionExample 2 again:X1,..,Xniid Poisson( ), Gamma( , ) |x= Gamma( xi+ ,n+ ) Posterior Predictive Distribution is:p(xnew|x) = 0p(xnew| )p( |x) d = 0[ xnewe (xnew)!]
Posterior Predictive Distribution I Recall that for a fixed value of θ, our data X follow the distribution p(X|θ). I However, the true value of θ is uncertain, so we should average over the possible values of θ to get a better idea of the distribution of X. I Before taking the sample, the uncertainty in θ is represented by the prior distribution p(θ).
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