Ordered bijector about probability HOT 6 CLOSED

Thanks for the clarification Bob and Junpeng! Given that order statistics is also something that can be a valid bijector, I was a little confused :).

@junpenglao, currently reviewing your PR. Apologies for any delays, since I am still familiarizing myself with the github review tools.

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Hi!
Just to clarify, given a vector [X1, ... X_n] this would be writing the order statistics X_(1), ... X_(n) in order, correct?

If I am understanding this correctly, this wouldn't be a bijection. Consider the vectors [2.1, 3., 4.], [3., 2.1, 4.] and [4., 3., 2.1]. All of them would map to [2.1, 3., 4.].

However, bijectors has been extended to a certain subclass of non-bijective transformations (e.g. there is an AbsoluteValue bijector), so I could see the same treatment being done here (because given a vector of length k, every preimage set is of cardinality k, up to multiplicity.).

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I misspoke and meant to say the preimage set is of size k! (every permutation).

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To generate an ordered vector y from an unconstrained vector x is set y[1] = x[1], and then y[2] = y[1] + exp(x[2]), and so on. The log Jacobian is just sum(x[2: ]). Then if you set y ~ normal(0, 1), you'll indeed see the order statistics just as if you'd taken an unconstrained vector x ~ normal(0, 1) and sorted.

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So I understand this is for the bijective "ordered transform" described in section 35.5 of the Stan reference manual v. 2.17.1? I might give this a try, though it might take a while.

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@mrosenkranz I actually just sent a PR implementing the ordered transformation tensorflow/tensorflow#18647.
And yes that's the same as in Stan (manual 2.17.1) and PyMC3 (WIP PR).

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