Symmetry in probability and statistics
Symmetry is powerful concept not just in geometry and physics but also in probability and statistics. Important manifestations of symmetry include exchangeability and invariant estimation .
Exchangeability
A random sequence is exchangeable if it is invariant in distribution under finite permutations. Pedagogical treatments of exchangeability include:
- Aldous, 1985: Exchangeability and related topics (doi, pdf)
- See also course notes by David Aldous with extensive bibliography
- Schervish, 1995: Theory of Statistics (doi)
- Sec. 1.2: Exchangeability & Sec. 1.4: De Finetti’s representation theorem
Symmetry and invariance
Exchangeability (permutation invariance) is not the only kind of invariance important in probability and statistics. Major references on group-theoretic and symmetry methods in probability are:
- Diaconis, 1988: Group representations in probability and statistics (online )
- Kallenberg, 2005: Probabilistic symmetries and invariance principles (doi)
And in statistics:
- Eaton, 1983: Multivariate statistics: A vector space approach (online )
- Eaton, 1989: Group invariance applications in statistics (online )
- Kondor, 2008, PhD thesis: Group theoretical methods in machine learning (pdf)
Several MathOverflow questions turn up other interesting references (1 ,2 ).