Comparing ICLR to AISTATS/UAI is like comparing apple to orange.

ICLR focuses on deep learning with more architecture stuffs, while AISTATS/UAI focuses more on statistical machine learning (e.g. kernel methods, Bayesian statistics, causal inference, optimization) with more theoretical results. I would argue that NeurIPS/ICML has a combination of both. NeurIPS seems to have more application papers in deep learning and architecture stuffs nowadays.

Thanks to the recent popularity in deep learning, ICLR quickly arises to the "Big 3" machine learning conference. This is just because deep learning has become a major part of machine learning nowadays.