# What is a predictive distribution? [duplicate]

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?

• But this question do not ask for particularly Bayesian solutions, so I do not think it is a duplicate! – kjetil b halvorsen Aug 31 '15 at 7:38
• I think this (implicitly) is about the Bayesian concept since this asks to compare posterior (of parameter) and predictive. – Juho Kokkala Aug 31 '15 at 8:18

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.