The following question What is a predictive distribution? was closed for being duplicate to this one:

Understanding Bayesian Predictive Distributions

But the last question is specifically about Bayesian predictive distributions, while the newly closed one was general, not specifically Bayesian. So it is not a duplicate, and I request that it to be reopened.

Predictive distribution is a concept that can also be treated by non-Bayesian methods!

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    $\begingroup$ Given the first carries the "bayesian" tag, talks about updating prior belief to posterior beiief and explicitly contains "$p(\theta|y_1,...,y_n)\propto p(y_1,...,y_n|\theta)p(\theta)$", I think it's unarguably a question about Bayesian predictive distributions. $\endgroup$ – Glen_b Aug 31 '15 at 8:41

I agree with @Glen_b there. But you've written a great answer on a rather neglected topic & I hope you've not been put off adding to it as promised. It's up to you of course, but if I were you I'd probably ask a new question on non-Bayesian predictive inference & put my answer there.

Some other references (just in case you haven't already seen them):

  • Lawless & Fredette (2005), "Frequentist prediction intervals and predictive distributions", Biometrika , 92, 3
  • Bjornstad, (1990), "Predictive Likelihood: A Review", Statist. Sci., 5, 2

† I decided to ask anyway: What non-Bayesian methods are there for predictive inference?

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