Posterior Predictive


A note about this video section on Posterior Predictive.

I’m looking at the lectures in Statistical Rethinking 2023 by Richard McElreath and I was hit by this video section.

Initially, I could not make heads or tails with it. I mean, the process was clear enough after viewing it some three-four times, but why was too above my head.

So I headed to the book and to the previous version of the section (from 2022) and ideas started forming in my head… I hope I got them right and my dear future me will be able to remeber that I’m writing them here.

  1. This is an example. The basic question is “how about predicting the future if we want to draw 9 more values?”. There’s really nothing special in the number 9, but the interesting thing is that we put the Posterior distribution (which is about the water proportion $p$) into a prediction about something different although related, i.e. how many more waters we can expect if we want to do 9 more draws. Put in another way: the $x$ axis of the Posterior distribution is $p$, the $x$ axis of the Posterior Predictive in this example is number of Ws.

  2. The Posterior Predictive might have been about something else, even for the same Posterior distribution. E.g. about the number of land draws. Or about the number of water draws in a run of 20. So, again, it’s an example.

  3. The whole point of doing all the draws and accumulations is to remind the user that distributions matter. Summarizing them with very few numbers can be useful, but very often misleading. So even in the predictions, let’s build a distribution and get a real feel for the simulated stuff.

  4. Doing the calculation of the Posterior Predictive transfers the uncertainty in the Posterior on to the Prediction. Which also means: if we have a very tight Posterior, we will get a tight Posterior Predictive. If we have a spread Posterior, we will get a spread Posterior Predictive. So it’s important that we have a tool that allows us to do the transfer from the former to the latter.

And now… that’s all I understood and I want to write about it. Cheers!

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