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Say we observe data $D$, which comes from a probability distribution $P[D|\theta]$, where $\theta$ are the unknown model parameters. Given this information, what is the probability distribution of the future data $D'$?

Further questions:

  1. Does this aim have a specific name in the literature?
  2. Can this question be addressed using Frequentist statistics?
  3. For Bayesian statistics I came up with the following procedure
    • Find posterior $P[\theta|D]$
    • Find $P[D'|D] = \int_\theta P[D'|\theta]P[\theta|D]d\theta$

Does this approach make sense? Is this what people typically do?

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    $\begingroup$ You found the posterior predictive distribution. $\endgroup$
    – Arya McCarthy
    Commented Apr 14, 2021 at 21:26
  • $\begingroup$ @AryaMcCarthy Thanks, that answers Q3 and Q1 $\endgroup$
    – Aleksejs Fomins
    Commented Apr 15, 2021 at 7:34
  • $\begingroup$ Wasn't trained in the frequentist arts, but I think their notion is to find $\theta_{\text{MLE}}$ based on $D$, then compute the probability of $D'$ under that model. $\endgroup$
    – Arya McCarthy
    Commented Apr 15, 2021 at 18:46
  • $\begingroup$ Yeah, that's the obvious solution, but it is of course wrong. Surely you don't need the notion of prior to comprehend the fact that picking the MLE parameter for prediction ignores the uncertainty of MLE. I was wondering if one can do something similar to predictive prior distribution, but instead integrate over the parameter estimator distribution $\endgroup$
    – Aleksejs Fomins
    Commented Apr 15, 2021 at 18:51
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    $\begingroup$ @Arya McCarthy: Well, we have frequentist prediction-intervals, see this post stats.stackexchange.com/questions/473512/…. They take into account parameter uncertainlt without integrating it out in Bayesian fashion, but typically using a pivot in the same way as in constructing a confidence interval. I will try to write an answer! $\endgroup$
    – kjetil b halvorsen ♦
    Commented Apr 15, 2021 at 23:53

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