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As my question states, I am wondering if there is any chance we use the prior predictive distribution. I am studying Bayesian Statistics and have understood what it is. It is a must to go through in Bayesian Statistics and seems very useful(at least to my understanding). My understanding about it is that it is used before seeing data to see what the distribution of observations might be. But I still cannot understand if we ever use it. Why would we use it? Is this for a simulation purpose? I wonder if you have ever used or if you are using it.

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The simplest answer is that it does not matter in the vast majority of cases. You can just assume you have a completely flat prior. Then let the data speak for itself.

In very rare circumstances it might matter. Usually that happens when you have very limited data. But, if so, then any statistical analysis used is suspicious anyway.

Why do you need the prior? Because by Bayes Theorem, P( hypothesis | data ) is proportional to P( data | hypothesis )P( hypothesis ). As you can see, you need the quantity P( hypothesis ) in the formula. Therefore, it is a necessary part of the formula.

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  • $\begingroup$ I know the prior is needed but my question is that I want to know if we ever use the prior predictive distribution. If you think I haven't understood your answer, please stretch it a bit further. $\endgroup$
    – mathccino
    Commented Jan 18 at 4:39
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    $\begingroup$ @mathlover The quick answer is "no". But, some people might tell you, that after you pick your prior distribution, it might be a good idea to run a random simulation using it to see if the data generated makes sense. For example, let us say your prior distribution results in people having heights of negative-inches, then that looks suspicious. So it might be a good idea to just run a quick simulation to see how sensible your prior is before you use it. $\endgroup$ Commented Jan 18 at 4:51
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    $\begingroup$ @NicolasBourbaki, your answer reads (at least to me) as though you discuss the prior and not the prior predictive distribution. As the links above discuss, they however are two different things. $\endgroup$ Commented Jan 18 at 9:29
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Do we ever use the prior predictive distributions of Bayesian Statistics?

Yes, frequently.

Why would we use it?

The prior predictive distribution is routinely used to develop Bayesian models in an activity termed prior predictive checking.

The primary use case is to help you understand your own model rather than actually making predictions without data (although technically allowed). This is especially helpful when you have a complex model with various priors or hyperpriors (which are just more parameters to give flexibility to your priors) that you don't have any explicit background information about. Often we are better at understanding and using background information about observable variables, so studying samples of observed variables from the prior helps us sanity check that the model is a priori predicting reasonable distributions of outcomes. You can read more about Bayesian workflow in Gelman et al 2020, which includes discussion of prior predictive simulation in section 2.4.

There are also tools to help you sample from the prior predictive distribution. See Prior and Posterior Predictive Checks for an example.

I wonder if you have ever used or if you are using it.

Yes, I try to look at prior predictive samples every time I practically can.

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