[Paper Review] Why Jordan algebras are natural in statistics:quadratic regression implies Wishart distributions

Gérard Letac, Jacek Wesołowski|arXiv (Cornell University)|Apr 19, 2010

Bayesian Methods and Mixture Models20 references4 citations

TL;DR

This paper establishes that quadratic regression structures on a finite-dimensional real vector space uniquely imply Wishart distributions when the space of quadratic forms decomposes into orthogonal subspaces with specific conditional expectation properties. The key result shows that such structures only arise when the space admits a Euclidean Jordan algebra structure, and the random variables must be Wishart-distributed on the associated symmetric cone, with k=2 being the only possible decomposition.

ABSTRACT

If the space $\\mathcal{Q}$ of quadratic forms in $\\mathbb{R}^n$ is splitted in a direct sum $\\mathcal{Q}_1\\oplus...\\oplus \\mathcal{Q}_k$ and if $X$ and $Y$ are independent random variables of $\\mathbb{R}^n$, assume that there exist a real number $a$ such that $E(X|X+Y)=a(X+Y)$ and real distinct numbers $b_1,...,b_k$ such that $E(q(X)|X+Y)=b_iq(X+Y)$ for any $q$ in $\\mathcal{Q}_i.$ We prove that this happens only when $k=2$, when $\\mathbb{R}^n$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.

Motivation & Objective

• To characterize when quadratic regression on a real vector space implies Wishart distributions.
• To investigate the conditions under which the conditional expectation of quadratic forms given X+Y is proportional to the quadratic form of X+Y.
• To prove that such structures can only exist when the underlying space is a Euclidean Jordan algebra and the decomposition of quadratic forms has exactly two components.
• To establish a reciprocal to the Lukacs-Olkin-Rubin theorem in the context of symmetric cones and Wishart laws.

Proposed method

• The authors analyze the conditional expectation structure E(q(X)|X+Y) = b_i q(X+Y) for q in subspaces Q_i of quadratic forms.
• They decompose the space of quadratic forms Q into a direct sum Q_1 ⊕ ... ⊕ Q_k and impose linear constraints on the conditional expectations.
• Using properties of exponential moments and invariance under orthogonal transformations, they derive functional equations on the characteristic functionals of X and Y.
• They apply results from Jordan algebra theory, particularly the structure of symmetric cones and the spectral decomposition, to classify the possible spaces V admitting such decompositions.
• They prove that the only possible k is 2, and that V must carry a Euclidean Jordan algebra structure for the conditions to hold.
• The proof relies on trace computations of endomorphisms on the space of symmetric endomorphisms, using orthonormal bases and spectral properties of the operators involved.

Experimental results

Research questions

• RQ1 Under what conditions on the decomposition of quadratic forms does the conditional expectation E(q(X)|X+Y) = b_i q(X+Y) hold for all q in Q_i and distinct b_i?
• RQ2 Can such a conditional expectation structure on a general real vector space Q imply that X and Y are Wishart-distributed?
• RQ3 Is k=2 the only possible number of components in the decomposition of Q for which such a structure exists?
• RQ4 What algebraic structure on the underlying space V is necessary for the existence of such a decomposition and conditional expectation property?
• RQ5 How does the Wishart distribution on a symmetric cone relate to the spectral decomposition and Jordan algebra structure?

Key findings

• The only possible value of k for which the conditional expectation structure holds is k=2.
• The underlying space V must be equipped with a Euclidean Jordan algebra structure for the conditions to be satisfied.
• The random variables X and Y are necessarily Wishart-distributed with respect to the symmetric cone associated with the Jordan algebra.
• The conditional expectation E(X|X+Y) = a(X+Y) holds if and only if X and Y are Wishart-distributed with the same scale parameter.
• The decomposition of the space of quadratic forms Q into Q_1 ⊕ Q_2 is canonical and corresponds to the symmetric and antisymmetric parts of the quadratic form in the Jordan algebra framework.
• The trace computation of the operator Ψ shows that only specific components (A1, B2, B4) contribute to the trace with value 1, confirming the uniqueness of the decomposition.