Implement logprob derivation for order statistics (original) (raw)

Description

The idea would be to derive the logprob for the following type of graphs

import aesara.tensor as at import pymc as pm

y_rv = at.max(at.random.uniform(0, 1, size=3)) # or min y_rv = at.sort(at.random.uniform(0, 1, size=3)) y_rv = at.sort(at.random.uniform(0, 1, size=3))[idx] # max / min correspond to idx==-1 or idx==0

y_vv = y_rv.clone() pm.logprob.factorized_joint_logprob({y_rv: y_vv})

https://en.wikipedia.org/wiki/Order_statistic#Probabilistic_analysis

This might be a bit far-fetched / difficult to find a good general solution that goes beyond a few simple cases (e.g, order statistics with non-i.i.d RVs):

y_rv = at.max(at.random.uniform([0, 1], [2, 3])) y_rv = at.max(at.stack([at.random.uniform(0, 1), at.random.normal(0, 1)]))

Probability for order statistics of IID variables are pretty straightforward to obtain, requiring expressions that depend on the CDFs. For non IID, things grow quickly in complexity: https://en.wikipedia.org/wiki/Bapat%E2%80%93Beg_theorem