A Statistical Model for Euclidean Jordan Algebras

Erhard Neher Department of Mathematics and Statistics University of Ottawa Ottawa, Ontario K1N 6N5, Canada Email: neher@uottawa.ca

Abstract We give a review of the basic structure theory of Euclidean Jordan algebras and their relation to symmetric cones. We then describe the lattice conditional independence model on the symmetric cone of a Euclidean Jordan algebra.

1 Basic concepts

We start by reviewing our notation which mainly follows the recent book [3]. Let J be a Euclidean Jordan algebra, i.e., J is a finite-dimensional real unital Jordan algebra with a positive-definite symmetric bilinear form σ which is associative in the sense that σ(xy, z) = σ(x, yz). Equivalently [2, XI Satz 3.4], J is a finitedimensional real Jordan algebra satisfying the condition x^{2} + y^{2} = 0 ⇒ x = 0 = y. Note that Euclidean Jordan algebras are called formally real in [2] and in [4]. Euclidean Jordan algebras were the first class of Jordan algebras for which a structure theorem was found:

Structure Theorem for Euclidean Jordan Algebras [5]. Any Euclidean Jordan algebra is a direct sum of simple ideals which are itself Euclidean Jordan algebras. A simple Euclidean Jordan algebra is isomorphic to one of the following three types:

a) The Jordan algebra of hermitian matrices H_{r}(D) for r ≥ 1 and D = R, C or the real division quaternions H,

b) the Jordan algebra of a positive-definite symmetric bilinear form τ : X × X → R, dim X ≥ 3, defined on J = R1 ⊕ X by (a1 ⊕ x)(b1 ⊕ y) = (ab + τ(x, y))1 ⊕ (ay + bx), or

c) the exceptional 27-dimensional Jordan algebra H_{3}(O) of hermitian 3 × 3 matrices over the real division octonian algebra O.

Conversely, any of the algebras in (a) – (c) is a simple Euclidean algebra.

Let V be a finite-dimensional real vector space with a positive-definite symmetric bilinear form σ. A symmetric cone in V is an open convex cone Ω ⊂ V which has the following two properties: a) Ω is self-dual: Ω = {y ∈ V; σ(x, y) > 0 for all 0 ≠ x ∈ \overline{Ω}\}, and b) Ω is homogeneous: G(Ω) = \{g ∈ GL(V); gΩ = Ω\} operates transitively on Ω. The following fundamental theorem of M. Koecher gives the relation between Euclidean Jordan algebras and symmetric cones:

Structure Theorem for Symmetric Cones [6]. If J is a Euclidean Jordan algebra then

Ω(J) = {x^{2} ∈ J; x invertible}

is a symmetric cone in J. Every symmetric cone is obtained in this way: if Ω is a symmetric cone in V there exists a Euclidean Jordan algebra J on V such that Ω = Ω(J).

Several other descriptions of Ω can be found in [2, XI]. A symmetric cone is always a Riemannian symmetric space, namely a product of a Euclidean space and a Riemannian symmetric space of noncompact type [7, VIII Thm. 2.1]. In complex analysis symmetric cones turn up in the study of generalized half spaces which, by definition, are the spaces J ⊕iΩ(J) ⊂ J ⊗RC for J a Euclidean Jordan algebra, see for example [10].

2 Andersson-Perlman cones

We fix a Euclidean Jordan algebra J and a complete orthogonal system E = (e_{1}, …, e_{n}) of idempotents: e_{i}e_{j} = δ_{ij}e_{i} and e_{1} + ⋯ + e_{n} = identity element of J. It is not necessary that the e_{i} are primitive. For any L ⊂ {1, …, n} and y ∈ J we put

e_{L} = ∑_{i∈L} e_{i} and y_{L} = P (e_{L})y

where P is the quadratic representation of J. The element e_{L} is an idempotent; it is the identity element of the subalgebra J_{L} = P (e_{L})J of J, which is again a Euclidean Jordan algebra. One knows [2, §XI] that x_{L} ∈ Ω(J_{L}) for any x ∈ Ω(J). Hence x_{L} is invertible in J_{L}; we denote by x_{L}^{−1} = (x_{L})^{−1} the inverse of x_{L} in J_{L} viewed as an element of J (not to be confused with the J_{L}-component of x^{−1} !). By definition, the sum over the empty set is 0, hence J_{∅} = {0} and 0^{−1} = 0.

Let K be a lattice on {1, …, n}, i.e., K is a set of subsets of {1, …, n} closed under ∪ and ∩, and containing ∅ and {1, …, n}. We define Ω(K) to consist of those x ∈ Ω satisfying for all L, M ∈ K

x_{L∪M}^{−1} + x_{L∩M}^{−1} = x_{L}^{−1} + x_{M}^{−1}.  (1)

It is clear that Ω(K) is a subcone of Ω. We call it the Andersson-Perlman cone since for J = H_{r}(R) it was first considered by S. A. Andersson and M. D. Perlman [1]. It is easily seen that the condition (1) is fulfilled for a “diagonal” x ∈ Ω: for Ω_{i} = Ω(J_{{i}}) we have

Ω_{1} ⊕ ⋯ ⊕ Ω_{n} ⊂ Ω(K) ⊂ Ω.  (2)

Thus, one can think of Ω(K) as an interpolation between the full symmetric cone Ω and the direct sum of the symmetric cones Ω_{i}. Both upper and lower bounds are obtained: Ω(K) = Ω_{1} ⊕ ⋯ ⊕ Ω_{n} if K consists of all subsets of {1, …, n} while Ω(K) = Ω if K = {∅, {1, …, n}}.

Example 1. Let n = 3 and K = {∅, {1}, {1, 2}, {1, 3}, {1, 2, 3}}. Since the condition (1) is always fulfilled for L ⊂ M or M ⊂ L, the definition above yields

x ∈ Ω(K) ⇔ x^{−1} + x_{{1}}^{−1} = x_{{1,2}}^{−1} + x_{{1,3}}^{−1}.  (3)

One can show [8] that this is equivalent to either one of the following two conditions

the (23)-component of x^{−1} is 0;  (4)
x = ( \begin{matrix}a & b & av \\ b & c & bv \\ av & bv & d \end{matrix} ), a > 0, ac > b^{2}, d > av^{2},  (5)

for suitable a, b, c, d, v, ∈ R. Note that in this example Ω(K) is not convex.

The following theorems generalize the equivalences which hold in the example above. Theorem 1 says that Ω(K) can be defined by one equation, a generalisation of (3), or by the vanishing of certain components of x^{−1}, a generalisation of (4). To formulate this result we introduce, for any K ∈ K, its K-interior ⟨K⟩ = ∪{K′ ∈ K; K′ ⊂ K ≠ K′} ⊂ K and put J (K) = {K ∈ K; ⟨K⟩ ≠ K}, the set of join-irreducible subsets of K.

Theorem 1 [8]. For x ∈ Ω the following are equivalent:

a) x ∈ Ω(K);

b) for all L, M ∈ K we have P (e_{L\M} , e_{M\L})x_{L∪M}^{−1} = 0;

c) x^{−1} = ∑_{K∈J (K)} (x_{K}^{−1} − x_{⟨K⟩}^{−1}).

The proof of this theorem and other characterizations of Ω(K) are contained in [8]. For J = H_{n}(R) the theorem is proven in [1]. Our proof is different from the one in [1], even for the case of real symmetric matrices. Our techniques are Peirce decompositions and so-called Frobenius transformations, a special type of Bergman transformations. We illustrate this theorem with an example.

Example 2. (see also Example 2.7 of [1]). Let n = 5 and consider the following subsets of I = {1, …, 5}: A = {1, 2}, B = {1, 3}, C = {1, 2, 3, 4} and D = {1, 2, 3, 5}. Then

K = {∅, A ∩ B, A, B, A ∪ B = C ∩ D, C, D, C ∪ D}

is a lattice. For this K, condition a) of Theorem 1 means that x ∈ Ω has to satisfy the two equations

x^{−1} + x_{C∩D}^{−1} = x_{C}^{−1} + x_{D}^{−1} and x_{A∪B}^{−1} + x_{A∩B}^{−1} = x_{A}^{−1} + x_{B}^{−1}  (6)

while condition c) yields the equation

x^{−1} = x_{C}^{−1} + [x_{D}^{−1} − x_{C∩D}^{−1} + x_{A}^{−1} + x_{B}^{−1} − x_{A∪B}^{−1} − x_{A∩B}^{−1}]  (7)

using J (K) = {A ∩ B, A, B, C, D}. Clearly (6) implies (7). The essence of the proof of the reverse direction is to observe that the expression [...] in (7) lies in J_{D} and to show that this implies the first equation of (6). Then the second immediately follows.

One can also generalize the parametrization (5) of Ω(K). For the technical details of this so-called K-parametrization the reader is referred to [8]. In particular, it shows that Ω(K) is a submanifold of Ω. This also follows from the next theorem whose proof will be contained in [9]. For J = H_{n}(R) it is proven in [1].

Theorem 2. The Lie group {g ∈ GL(J); gΩ = Ω, gΩ(K) = Ω(K)} operates transitively on Ω(K).

For Example 1 this result can be explained in the following way. We have x = AyA^{t} for

A = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ v & 0 & 1 \end{matrix} ) and y = ( \begin{matrix}a & b & 0 \\ b & c & 0 \\ 0 & 0 & d′ \end{matrix} )

for a suitable d′. The transformation x ↦ AxA^{t} leaves the cones Ω as well as Ω(K) invariant. The same is true for the product of the automorphism groups of the cones Ω_{1} and Ω_{2} where Ω_{1} is the cone of positive-definite 2 × 2 matrices imbedded in the upper left corner while Ω_{2} is the cone of positive real numbers imbedded in the (33)-component.

3 Statistics

The motivation for studying Ω(K) comes from statistics. For J = H_{n}(R) the following interpretation is given in [1].

For any vector z = (z_{1}, …, z_{n}) ∈ R^{n} and L ⊂ {1, …, n} the vector z_{L} arises from z by putting z_{i} = 0 if i ∉ L. Then x ∈ Ω(K) if and only if for any n-variate random variable z which has central normal distribution with covariance x we have that for all L, M ∈ K the vectors z_{L} and z_{M} are conditionally independent given z_{L∩M}. A similar interpretation holds for the Jordan algebras H_{n}(D), D = C or H.

The techniques developed for proving Theorem 1 can be used to show the following result:

Theorem 3 [8]. Let Z_{1}, …, Z_{N} be random variables with values in D^{n} and assume that the Z_{i} have a normal distribution with mean 0 and covariance x ∈ Ω(K). Form the matrices Z = (Z_{1}, …, Z_{N}) ∈ Mat_{nN}(D) and s = \frac{1}{N} ZZ̄^{t} ∈ H_{n}(D). Then the maximum likelihood estimate x̂ of x is unique and exists for almost every Z if and only if |K| ≤ N for all K ∈ J (K). If x̂ exists, it can be calculated from the following closed-form formula:

x^{−1} = ∑_{K∈J (K)} (s_{K}^{−1} − s_{⟨K⟩}^{−1}).

References

[1] S. A. Andersson and M. D. Perlman, Lattice models for conditional independence in a multivariate normal distribution. Ann. Stat. 21, 1318–1358 (1993).

[2] H. Braun und M. Koecher, Jordan-Algebren. Grundlehren der mathematischen Wissenschaften 128, Springer-Verlag 1967.

[3] J. Faraut and A. Koranyi, Analysis on symmetric cones. Clarendon Press, Oxford 1994.

[4] N. Jacobson, Structure and representation of Jordan algebras. Amer. Math. Soc. Coll. Publ. 39 1968.

[5] P. Jordan, J. von Neumann and E. Wigner, On an algebraic generalization of the quantum mechanic formalism. Ann. Math. 36, 29–64 (1934).

[6] M. Koecher, Jordan algebras and their applications. Lecture Notes, University of Minnesota, Minneapolis 1962.

[7] O. Loos, Symmetric Spaces, vol. I and II, Mathematics Lecture Notes Series, Benjamin 1969.

[8] H. Massam and E. Neher, Estimation and testing for lattice conditional independence models on Euclidean Jordan algebras. Ann. Stat. (to appear).

[9] E. Neher, Transformation groups of the Andersson-Perlman cone. Preprint 1997.

[10] I. Satake, Algebraic structures of symmetric domains. Princeton University Press 1980.