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I understand that we calculate a posterior belief by updating our prior belief with the information from given data, like

$$ p(\theta|y_1,...,y_n)\propto p(y_1,...,y_n|\theta)p(\theta)$$

But I dont fully understand the concept of a predictive distribution (the distribution of $\tilde{Y}$ having observed $\{Y_1=y_1,...,Y_n=y_n\}$ from the population.

What is the difference between a posterior distribution and a predictive distribution?

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  • $\begingroup$ But this question do not ask for particularly Bayesian solutions, so I do not think it is a duplicate! $\endgroup$ – kjetil b halvorsen Aug 31 '15 at 7:38
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    $\begingroup$ I think this (implicitly) is about the Bayesian concept since this asks to compare posterior (of parameter) and predictive. $\endgroup$ – Juho Kokkala Aug 31 '15 at 8:18
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Some want to close this question because they think Predictive Distributions is a Bayesian concept only, so already answered. But there are also frequentist tries on defining predictive distributions, one is the paper by A C Davison: "Approximate Predictive Likelihood" (Biometrika) http://biomet.oxfordjournals.org/content/73/2/323.abstract

The abstract sayes: "A predictive likelihood is given which approximates both Bayes and maximum likelihood predictive inference by expansion of a posterior likelihood. This synthesizes and extends previous results and is widely applicable. The approximation usually differs from exact Bayes posterior predictive density by Op(n–2), and from exact predictive likelihood by Op(n–2) but does not depend on the availability of prior information and is applicable when exact predictive likelihood cannot be found. The results are applied to the prediction of extremes using the generalized extreme-value distribution. "

So what is a predictive distribution, day for a (future) random variable $X$? It tries to approximate the conditional distribution $P(X \mid \text{data})$. In general such a conditional distribution will depend on unknown parameters, and a Bayesian solution will try to integrate out thse unknown parameers, over their posterior distribution. Frequentist solutions will try to eliminate those unknown parameters by other methods.

(I will come back and write more about those "other methods" later, now out of time).

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