Understanding exchangeability

Question

Suppose you had information of a true success rate of passing an exam from different schools, and that the 'true' success rates are similar, can you assume exchangeability is applicable to these values, or is there not enough info to assume it?,if so, then what more info is needed to make a reasonable assumption?

Also, since the values are considered similar, what would be a logical way to estimate the success rate at a say 5-th school given you have values of the true success rate from 4 schools already, is it just a simple average? I'm not sure whether the concept of exchangeability can be applied here, and generally not getting how it can be applied to similar situations, involving a sequence of numbers.

I highly welcome questions about exchangeability. But could you please provide a clear and simple practical example. It could look something like: exam A on matter B in school C sees 90/100 students pass, etc. What exactly are you after? Comparing success rates? Between schools? What is your understanding of exchangeability? Link to the relevant wiki. It will greatly help your question. — Jim, Apr 16 '18 at 22:08
say the success rate in passing is 50%,75%,66% and 80% amongst 4 schools within a county in England, under what circumstances would you assume exchangeability? I am familiar with De fenetti's theorem and the general definition of exchangeability. I think exchangeability applies here as the rates in schools within a county can be assumed to be similar, and the rates seem somewhat close. — s.stats, Apr 16 '18 at 22:43
Also, to predict the 5-th school, would it be reasonable to assume a beta prior, calculate the params through mean and variance of rates, or is there an alternative to this, based on intuition? — s.stats, Apr 16 '18 at 22:44
Ah right, so if we assume independence that implies exchangeability. — s.stats, Apr 16 '18 at 22:57
Just so you know: I’m not commenting to critique. But simply to get your question into a form in which it is most likely to receive a good answer. :) — Jim, Apr 16 '18 at 23:00
@s.g.: Just ask yourself if you have any information about the schools (other than the pass rates) distinguishing between them. Are some in very poor neighborhoods, others not? ... independence is in a way much stronger than exchangeability, so is not a good way to argue. — kjetil b halvorsen, Dec 30 '18 at 20:59
More examples at https://stats.stackexchange.com/questions/353722/exchangeability-of-group-effects/353739#353739 — kjetil b halvorsen, Jan 03 '19 at 22:18

kjetil b halvorsen · Answer 1 · 2020-06-19T22:35:37.223

Exchangeability is a modeling concept, it cannot be applied to just "a sequence of numbers". Represent those numbers as realizations of some random variable and then we are playing. Your example is pass rate from (say) five different schools. Ask yourself: prior to seeing the actual numbers, do you expect more or less the same from the five schools? Or are there some distinguishing characteristics of some schools that make you have differing expectations? In the last case, you cannot assume exchangeability, at least not directly.

What kind of information could make you have different expectations for some schools? Some possibilities:

One schools is a selective science high school, the others ordinary schools.
some schools in rich neighborhoods, others in poor.
Some schools private, other public.
Most schools large, but one is small, so you expect larger variability simply from averaging over fewer students.
...

So, exchangeability is a form of symmetry: There is symmetry in the prior information that you have. This does not at all imply independence, exchangeability is much weaker than independence. When assuming exchangeability, you can model your random variables as IID, see details here. That should answer the question in your second paragraph. Do prediction as you would do in an IID model.

I agree with your characterisation of whether exchangeability can be assumed here. But surely the concept can be applied to a sequence of numbers, so long as there is a probability measure operating on that sequence --- indeed, isn't exchangeability defined by permutation invariance of a probability measure? (And for an infinite sequence of numbers, it is exchangeable if any finite vector within it is exchangeable.) — Ben, Jun 20 '20 at 04:40
One more useful thing to note here is that if the pass rate from the five schools comes to you in a random order (via SRSWOR) then the vector of values is exchangeable. — Ben, Jun 20 '20 at 04:41

Understanding exchangeability

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