# Prior distributions and the Australia principle

May 20, 2018
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(This article was originally published at Statistical Modeling, Causal Inference, and Social Science, and syndicated at StatsBlogs.)

There’s an idea in philosophy called the Australia principle—I don’t know the original of this theory but here’s an example that turned up in a google search—that posits that Australia doesn’t exist; instead, they just build the parts that are needed when you visit: a little mock-up of the airport, a cityscape with a model of the Sydney Opera House in the background, some kangaroos, a bunch of desert in case you go into the outback, etc. The idea is that it would be ridiculously inefficient to build an entire continent and that it makes much more sense for them to just construct a sort of stage set for the few places you’ll ever go.

And this is the principle underlying the article, The prior can often only be understood in the context of the likelihood, by Dan Simpson, Mike Betancourt, and myself. The idea is that, for any given problem, for places in parameter space where the likelihood is strong, relative to the questions you’re asking, you won’t need to worry much about the prior; something vague will do. And in places where the likelihood is weak, relative to the questions you’re asking, you’ll need to construct more of a prior to make up the difference.

This implies:
1. The prior can often only be understood in the context of the likelihood.
2. What prior is needed can depend on the question being asked.

To follow up on item 2, consider a survey of 3000 people, each of whom is asked a binary survey response, and suppose this survey is a simple random sample of the general population. If this is a public opinion poll, N = 3000 is more than enough: the standard error of the sample proportion is something like 0.5/sqrt(3000) = 0.01; you can estimate a proportion to an accuracy of about 1 percentage point, which is fine for all practical purposes, especially considering that, realistically, nonsampling error will be likely be more than that anyway. On the other hand, if the question on this survey of 3000 people is whether your baby is a boy or a girl, and if the goal is to compare sex ratios of beautiful and ugly parents, then N = 3000 is way way too small to tell you anything (see, for example, the discussion on page 645 here), and if you want any kind of reasonable posterior distribution for the difference in sex ratios you’ll need a strong prior. You need to supply the relevant scenery yourself, as it’s not coming from the likelihood.

The same principle—that the prior you need depends on the other information you have and the question you’re asking—also applies to assumptions within the data model (which in turn determines the likelihood). But for simplicity here we’re following the usual convention and pretending that the likelihood is known exactly ahead of time so that all the modeling choices arise in the prior.

P.S. The funny thing is, Dan Simpson is from Australia himself. Just a coincidence, I’m sure.

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