WISHART EXPONENTIAL FAMILIES AND VARIANCE FUNCTION ON HOMOGENEOUS CONES

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WISHART EXPONENTIAL FAMILIES AND VARIANCE FUNCTION ON HOMOGENEOUS

CONES

Piotr Graczyk, Hideyuki Ishi, Bartosz Kolodziejek

To cite this version:

Piotr Graczyk, Hideyuki Ishi, Bartosz Kolodziejek. WISHART EXPONENTIAL FAMILIES AND

VARIANCE FUNCTION ON HOMOGENEOUS CONES. 2016. �hal-01358024�

HOMOGENEOUS CONES

PIOTR GRACZYK, HIDEYUKI ISHI, AND BARTOSZ KO LODZIEJEK

Abstract. We present a systematic study of Riesz measures and their natural exponential families of Wishart laws on a homogeneous cone. We compute explicitly the inverse of the mean map and the variance function of a Wishart exponential family.

1. Introduction

Modern statistics and multivariate analysis require use of models on subcones of the cone Sym₊(n,R) of positive definite symmetric matrices. Such subcones are obtained for example by prescribing some of the off-diagonal elements to be 0. Wishart laws on such subcones are laws of the Maximum Likeli- hood estimators of covariance and precision matrices in a multivariate normal sample, subject to condi- tional independence constraints [19, 21]. They are also important in Bayesian statistics, since they form Diaconis-Ylvisaker families of priors for the covariance in a covariance selection model [6, 21]

Many of such subcones are homogeneous, i.e. their automorphism group acts transitively. Recall that in the theory of graphical models [19], a decomposable graph generates a homogeneous cone if and only if it does not contain the graph • − • − • − •, denoted byA4, as an induced subgraph, cf. [14, 21]. The preponderant role of homogeneous cones among subcones of Sym₊(n,R) strongly motivates research on Wishart laws on general homogeneous cones.

Families of Wishart laws on homogeneous cones were studied by Andersson and Wojnar [1], Boutouria [3], Letac and Massam [21], Graczyk and Ishi [9], Ishi [15] and Ishi and Ko lodziejek [18].

This article is a continuation of [9, 15, 18], however here we identify the dual space using the trace inner product instead of the standard inner product. Even though the difference between standard and trace inner products is rather technical, it leads to new results on generalized power functions and on the variance function of Wishart exponential families. Moreover, the trace inner product will be indispensable in future applications to statistics. An objective of this paper is also to present our methods and results in a manner which is accessible to mathematical statisticians. For completeness and for convenience of the reader, we repeat and simplify some definitions and proofs from [9] and [15].

The variance function is an important characteristics of a natural exponential family, cf. [2]. Let us consider a classical Wishart exponential family on the cone Sym₊(n,R), i.e. the natural expo- nential family generated by a measure µp with the Laplace transform Lµ_p(θ) = (detθ)^−p, for θ ∈ Sym₊(n,R). Here p belongs to the Gindikin-Wallach set (in this context also called the Jørgensen set) Λ ={1/2,1,3/2, . . . ,(n−1)/2} ∪((n−1)/2,∞) andµp is a Riesz measure on Sym₊(n,R). It is well known and straightforward to check (see for example [20]) that the variance function of NEF generated byµp is given by

Vp(m) =1 pρ(m) (1)

Key words and phrases. natural exponential families; variance function; Wishart laws; Riesz measure; homogeneous cones; graphical cones.

1

where m∈(Sym₊(n,R))^∗≡Sym₊(n,R) andρ(m) is a linear map from Sym₊(n,R) to itself defined by ρ(m)Y = mY m^>, Y ∈ Sym₊(n,R). Generally, Wishart exponential family is the natural exponential family generated by some Riesz measure. In this paper we give explicit formulas for the variance function of Wishart exponential family on any homogeneous cone.

Let us describe shortly the plan of this paper. In Section 2 we recall the definition of a natural exponential family generated by a positive measure and we introduce their characteristics, mean and variance, used and studied throughout the whole paper.

Sections 3 and 4 are devoted to introducing the main tools for the analysis of Wishart exponential families on homogeneous cones. Like in [9], we consider two types of homogeneous cones, P_V which is in a matrix realization and its dual coneQ_V, and Riesz measures and Wishart families on them. This corresponds to the concept of Type I and Type II Wishart laws defined and studied in [21], as laws of MLEs of covariance and precision matrices of a normal vector. Any homogeneous cone is linearly isomorphic to some cone P_V [16]. It was observed in [9] that the matrix realization of a homogeneous cone makes analysis of Riesz and Wishart measures much easier. The present paper is also based on this technique, which is reviewed in Section 3.1.

In Section 3.2, we define the generalized power functionsδsand ∆son the cones Q_V andP_V, respec- tively. In Proposition 3.1(iii), we give a formula (9) for the power functionδs, which is new and useful.

In Definition 3.3 we introduce an important mapξ → ξˆbetween Q_V and Sym₊(N,R), inspired by an analogous map playing a fundamental role for decomposable graphical models [19].

In Section 5, we first prove Formula (11), which serves as a simple and useful tool in our argument.

Then we deduce from (11) an explicit evaluation of the inverse of the mean map (Theorem 5.1). It allows us to find the Lauritzen formula on any (not only graphical) homogeneous cone. We get in Section 6 the variance function formula for Wishart exponential families defined on the coneQ_V. The proof is based on Formula (11) again.

In Section 7 we give results on the variance function formula for Wishart exponential families onP_V. In particular, we propose a practical approach to the construction of a matrix realization of the dual cone QV, using basic quadratic maps. Note that if a homogeneous cone PV is in matrix realization, then, in general,QV is not, which makes analysis onQV harder.

The last Section 8 contains applications of the results and of the methods of this paper to symmetric cones and graphical homogeneous cones. We improve and complete in this way the results of [11] and [21].

2. Natural Exponential Families

In the following section we will give a short introduction to natural exponential families (NEFs). The standard reference book on exponential families is [2].

LetEbe a finite dimensional real linear space endowed with an inner producth·,·iand letE^∗ be the dual space of E. If ξ ∈ E^∗ is a linear functional on E, we will denote its action on x ∈ E by hξ, xi.

Let LS(E^∗,E) be the linear space of linear operators A: E^∗ → E such that for anyξ, η ∈E^∗, one has hξ, A(η)i=hη, A(ξ)i.

Letµbe a positive Radon measure onE. We define its Laplace transformLµ: E^∗→(0,∞] by Lµ(θ) :=

Z

E

e^−hθ,xiµ(dx).

Let Θ(µ) denote the interior of the set {θ ∈E^∗:Lµ(θ)<∞}. H¨older’s inequality implies that the set Θ(µ) is convex and the cumulant function

k_µ(θ) := logL_µ(θ)

is convex on Θ(µ) and it is strictly convex if and only ifµis not concentrated on any affine hyperplane of E. Let M(E) be the set of positive Radon measures onE such that Θ(µ) is not empty and µis not concentrated on any affine hyperplane ofE.

Forµ∈ M(E) we define thenatural exponential family (NEF) generated by µas the set of probability measures

F(µ) ={P(θ, µ)(dx) =e^{−hθ,xi−kµ(θ)}µ(dx) :θ∈Θ(µ)}.

Then, forθ∈Θ(µ),

m_µ(θ) :=−k⁰_µ(θ) = Z

E

x P(θ, µ)(dx),

−m⁰_µ(θ) =k_µ⁰⁰(θ) = Z

E

(x−m_µ(θ))⊗(x−m_µ(θ))P(θ, µ)(dx),

are respectively the mean and the covariance operator of the measureP(θ, µ). Herex⊗xis an element ofLS(E^∗,E) defined by (x⊗x)(ξ) =xhξ, xiforx∈Eandξ∈E^∗. The subsetM_F(µ):=mµ(Θ(µ)) ofE is called the domain of means ofF(µ). The map mµ: Θ(µ)→M_F(µ)is an analytic diffeomorphism, and its inverse is denoted byψµ:M_F(µ)→Θ(µ).

Lemma 2.1 ([15, Proposition IV.4]) Define Jµ(m) := sup_θ∈Θ(µ)^e−hθ,mi_L

µ(θ) for any m ∈ M_F(µ). Then ψµ=−(logJµ)⁰.

Proof. Since logJµ(−m) = sup_θ∈Θ(µ)(hθ, mi −kµ(θ)) is the Legendre-Fenchel transform of kµ(θ), the

statement follows from the Fenchel duality.

For anym∈M_F(µ)consider the covariance operatorVF(µ)(m) of the measureP(ψ_µ(m), µ). Then VF(µ)(m) =k⁰⁰_µ(ψµ(m)) =−[ψ_µ⁰(m)]⁻¹.

(2)

The map VF(µ): MF(µ) → LS(E^∗,E) is called the variance function of F(µ). Variance function is the central object of interest of natural exponential families, because it characterizes a NEF and the generating measure in the following way: if F(µ) andF(µ₀) are two natural exponential families such thatVF(µ) andVF(µ₀)coincide on a non-void open setJ ⊂M_F(µ)∩M_F(µ0), thenF(µ) =F(µ₀) and so µ0(dx) = exp{ha, xi+b}µ(dx) for somea∈E^∗ andb∈R. The variance function gives full knowledge of the NEF.

In the context of natural exponential families, often invariance properties under the action of a subgroup of general linear group or general affine group are considered. For the recent developments in this direction see [18].

Usually when one defines natural exponential family one starts with the moment generating function L_µ(θ) = R

Eexphθ, xiµ(dx). In such case, introducing the same concept of inverse of the mean map as above, the covariance operator has the form VF(µ)(m) = [ψ_µ⁰(m)]⁻¹. For our purposes however we find it more convenient to define NEF through the Laplace transform.

3. Basic facts on homogeneous cones

Let Mat(n, m;R), Sym(n,R) denote the linear spaces of real n×m matrices and symmetric real n×n matrices, respectively. Let Sym₊(n,R) be the cone of symmetric positive definite real n×n matrices. A^> denotes the transpose of a matrix A. For X ∈ Mat(N, N;R) define a linear operator ρ(X) : Sym(N,R)→Sym(N,R) by

ρ(X)Y =XY X^>, Y ∈Sym(N,R).

LetV be a real linear space and Ω a regular open convex cone inV. Open convex cone Ω isregular if Ω∩(−Ω) = {0}. The linear automorphism group preserving the cone is denoted by G(Ω) = {g ∈ GL(V) :gΩ = Ω}. The cone Ω is said to behomogeneous if G(Ω) acts transitively on Ω.

3.1. Homogeneous cones PV and QV. We recall from [9] a useful realization of any homogeneous cone. Let us take a partitionN =n1+. . .+nr of a positive integerN, and consider a system of vector spacesVlk ⊂Mat(nl, nk;R), 1≤k < l≤r, satisfying following three conditions:

(V1) A∈ Vlk,B∈ Vki =⇒ AB∈ Vli for any 1≤i < k < l≤r, (V2) A∈ Vli,B∈ Vki =⇒ AB^>∈ Vlk for any 1≤i < k < l≤r, (V3) A∈ Vlk =⇒ AA^>∈RIn_l for any 1≤k < l≤r.

LetZV be the subspace of Sym(N,R) defined by

Z_V:=









 x=











X11 X₂₁^> · · · X_r1^>

X21 X22 X_r2^>

... . .. Xr1 Xr2 Xrr











: Xlk∈ Vlk, 1≤k < l≤r Xkk=xkkIn_k, 1≤k≤r









 .

We set

PV :=ZV∩Sym₊(N,R).

Then PV is a regular open convex cone in the linear space ZV. Let HV be the group of real lower triangular matrices with positive diagonals defined by

H_V:=









 T =









 T11

T21 T22

... . .. Tr1 Tr2 Trr











: Tlk∈ Vlk, 1≤k < l≤r Tll=tllIn_l, tll>0, 1≤l≤r









 .

IfT ∈HV andx∈ ZV, thenρ(T)x=T x T^>∈ ZV thanks to (V1)−(V3). Moreover,ρ(HV) acts on the conePV (simply) transitively ([13, Proposition 3.2]), that is,PV is a homogeneous cone. Our interest in PV is motivated by the fact that any homogeneous cone is linearly isomorphic toPVdue to [13, Theorem D].

Condition (V3) allows us to define an inner product (·|·) onV_lk, 1≤k < l≤r, by (AB^>+BA^>)/2 = (A|B)I_nl, A, B∈ Vlk.

We definethe trace inner product onZ_V by hx, yi= tr(xy) =

r

X

k=1

n_kx_kky_kk+ 2 X

1≤k<l≤r

n_l(X_lk|Y_lk), x, y∈ Z_V.

Using the trace inner product we identify the dual spaceZ_V^∗ withZ_V. Define the dual coneQ_V by Q_V:={ξ∈ Z_V: hξ, xi>0 ∀x∈ P_V\ {0}},

whereP_V is the closure ofP_V. The dual coneQ_V is also homogeneous. It is easily seen that IN ∈ Q_V. ForT ∈H_V, we denote by ρ^∗(T) the adjoint operator of ρ(T)∈GL(Z_V) defined in such a way that hξ, ρ(T)xi = hρ^∗(T)ξ, xi for any ξ, x ∈ Z_V. For any ξ ∈ Q_V there exists a unique T ∈ H_V such that ξ=ρ^∗(T)IN ([24, Chapter 1, Proposition 9]).

3.2. Generalized power functions. Define a one-dimensional representationχsof the triangular group H_V by

χs(T) :=

r

Y

k=1

t^2s_kk^k,

wheres= (s₁, . . . , s_r)∈C^r. Note that any one-dimensional representationχofH_V is of the formχ_sfor somes∈C^r.

Definition 3.1 Let∆s: PV→C be the function given by

∆_s(ρ(T)I_N) :=χ_s(T), T ∈H_V. Let δs: QV →Cbe the function given by

δ_s(ρ^∗(T)I_N) :=χ_s(T), T ∈H_V. Functions ∆ andδare calledgeneralized power functions.

Let N_k = n₁+. . .+n_k, k = 1, . . . , r. For y ∈ Sym(N,R), by y_{1:k} ∈ Sym(N_k,R) we denote the submatrix (y_ij)_1≤i,j≤Nk. It is known that for any lower triangular matrix T one has

(T T^>)_{1:k}=T_{1:k}T_{1:k}^> .

Thus, for x=ρ(T)IN ∈ PV withT ∈HV one has detx_{1:k} = (detT_{1:k})² =Qk

i=1t²ⁿ_iiⁱ. This implies that for anyx∈ PV,

(3) ∆_s(x) = (detx)^nrsr

r−1

Y

k=1

(detx_{1:k})^sknk−

sk+1 nk+1.

We will expressδ_s(ξ) as a function ofξ∈ Q_V in the next Section (see Proposition 3.1).

By definition, ∆_s andδ_s are multiplicative in the following sense

∆s(ρ(T)x) = ∆s(ρ(T)IN) ∆s(x), (x, T)∈ PV×HV, (4)

δs(ρ^∗(T)ξ) =δs(ρ^∗(T)IN)δs(ξ), (ξ, T)∈ Q_V×H_V. (5)

Definition 3.2 Letπ: Sym(N;R)→ Z_V be the projection such that, for anyx∈Sym(N,R)the element π(x)∈ Z_V is uniquely determined by

tr(xa) =hπ(x), ai, ∀a∈ ZV. For anyx, y∈ ZV one has

hρ^∗(T)x, yi=hx, ρ(T)yi= tr(ρ(T^>)x·y) =

π(ρ(T^>)x), y , thus, for anyT ∈HV,

ρ^∗(T) =π◦ρ(T^>).

(6)

Now we define a useful mapξ→ξˆbetweenQV and Sym₊(N,R), such that ( ˆξ)⁻¹∈ PV andπ( ˆξ) =ξ.

An analogous map is very important in statistics on decomposable graphical models [19].

Definition 3.3 Forξ=ρ^∗(T)IN ∈ QV withT ∈HV, we define ξˆ:=ρ(T^>)IN =T^>T ∈Sym₊(N,R).

Note that for anyξ∈ QV, one has ( ˆξ)⁻¹∈ PV (compare the definition of ˆξ in [21, Proposition 2.1]).

Indeed, ( ˆξ)⁻¹=ρ(T⁻¹)I_N ∈ P_V. Due to (6), we have π( ˆξ) =ξ.

Observe that forT ∈H_V,

∆s(ρ(T)IN) =χs(T) =χ_−s(T⁻¹) =δ_−s(ρ^∗(T⁻¹)IN) and, due to (6), ρ^∗(T⁻¹)I_N =π (ρ(T)I_N)⁻¹

. This implies that functions ∆ andδ are related by the following identity

(7) ∆s(x) =δ_−s(π(x⁻¹)), x∈ P_V,

or equivalently,

(8) ∆s( ˆξ⁻¹) =δ−s(ξ), ξ∈ QV.

In literature, functionδ_sis sometimes denoted by ∆^∗_s∗, wheres^∗= (s_r, . . . , s₁).

3.3. Basic quadratic mapsqi and associated maps φi. We recall from [9] the construction of basic quadratic maps. Let Wi, i = 1, . . . , r, be the subspace of Mat(N, ni;R) consisting of matrices xof the form

x=











0n₁+...+ni−1,n_i

xiiIn_i

... Xri









 ,

whereXli∈ Vli,l =i+ 1, . . . , r. Forx∈Wi, the symmetric matrixxx^> belongs toZV thanks to (V2) and (V3). We definethe basic quadratic map q_i: W_i3x7→xx^>∈ ZV.

Taking an orthonormal basis of each Vli with respect to (·|·), we identify the space W_i with R^mi, wherem_i= dimW_i= 1 + dimV_i+1,i+. . .+ dimV_ri. Forx∈W_i we write vec(x) for the element ofR^mi corresponding to x. It is convenient to choose a basis for W_i consistent with the block decomposition ofZ_V, that is, (v₁, . . . , v_mi), wherev₁ corresponds to V_ii 'R and (v₂, . . . , v_1+dimVi+1,i) corresponds to Vi+1,iand so on.

Definition 3.4 For the quadratic mapqiwe define the associated linear mapφi: ZV ≡ Z_V^∗ →Sym(mi,R) in such a way that forξ∈ ZV,

vec(x)^>φi(ξ) vec(x) =hξ, qi(x)i, ∀x∈Wi. Similarly we consider another subspace of Mat(N, ni;R), namely,

c Wi=









 x=











0n₁+...+n_i,n_i

Xi+1,i

... Xri











:Xli∈ Vli, l=i+ 1, . . . , r









 ,

the quadratic map b q_i:

c

Wi3x7→xx^>∈ Z_Vand its associated linear map b

φ_i: Z_V ≡ Z_V^∗ →Sym(mi−1,R).

Proposition 3.1 (i) For any ξ∈ ZV andi= 1, . . . , r−1, one has φi(ξ) =

niξii vi(ξ)^>

v_i(ξ) b φ_i(ξ)

,

wherev_i(ξ) :=







n_i+1vec(ξ_i+1,i) ... n_rvec(ξ_ri)





∈R^mi−1 andvec(ξ_ki)∈R^dimVki is the vectorization of ξ_ki∈ V_ki. Moreover, φ_r(ξ) =n_rξ_rr ∈R≡Sym(1,R).

(ii) Forξ=ρ^∗(T)I_N ∈ Q_V withT ∈H_V andi= 1, . . . , r−1one has detφi(ξ) =χm_i(T) detφi(IN), and

det b

φ_i(ξ) =χ_m^b

i(T) det b φ_i(I_N), wherem_i:= (0, . . . ,0,1, n_i+1,i, . . . , n_ri)∈Z^r and

b

m_i:= (0, . . . ,0,0, n_i+1,i, . . . , n_ri)∈Z^r. (iii) For any ξ∈ Q_V, one has

(9) δs(ξ) =Csφr(ξ)^sr

r−1

Y

i=1

detφi(ξ) det

b φ_i(ξ)

!si

, where the constant C_s does not depend onξ.

(iv) Forα, β∈ ZV andi= 1, . . . , r−1, one has trφi(α)φi(IN)⁻¹φi(β)φi(IN)⁻¹−tr

b φ_i(α)

b φ_i(IN)⁻¹

b φ_i(β)

b

φ_i(IN)⁻¹

=α_iiβ_ii+ 2

r

X

l=i+1

n_l ni

(α_li|βli).

Let us underline that the useful formula (9) for the power function δs is new and different from a formula given in [9] and [15].Precisely, it is just mentioned in [9, 15] thatδs(ξ) is a product of powers of detφi(ξ).

Proof. (i) It is a consequence of the choice of basis forWi and the fact thatWi 'R⊕ c Wi.

(ii) In [9] a very similar problem was considered, but there the dual spaceZ_V^∗ was identified withZV

using the, so-called, standard inner product, not the trace inner product. The only difference in the form ofφi in these two cases is that here block sizesniappear in (i, i) component and in the definition ofvi. The proof is virtually the same for both cases - see [9, Proposition 3.3].

(iii) From (ii) we see that ifξ=ρ^∗(T)IN, then t²_ii= ^χ_χ^mic(T)

mi(T) = ^det

b φ_i(I_N) detφi(IN)

detφ_i(ξ) det

b

φ_i(ξ) =n⁻¹_i ^detφi(ξ)

det b φ_i(ξ). (iv) Using the block decomposition given in (i), one has

trφ_i(α)φ_i(I_N)⁻¹φ_i(β)φ_i(I_N)⁻¹−tr b φ_i(α)

b φ_i(I_N)⁻¹

b φ_i(β)

b

φ_i(I_N)⁻¹

=αiiβii+vi(α)^>vi(β) +vi(β)^>vi(α) and the assertion follows from the definition of (·|·) andv_i.

4. Riesz measures and Wishart exponential families

Generalized power functions play a very important role and this is due to the following

Theorem 4.1 ([8, 12]) (i) There exists a positive measureR_s onZ_V with the Laplace transform LR_s(ξ) =δ−s(ξ), ξ∈ QV

if and only ifs∈Ξ :=F

ε∈{0,1}^rΞ(ε), where Ξ(ε) :=







sk >¹₂P

i<kεidimVki if εk= 1 s∈R^r;

sk =¹₂P

i<kεidimVki if εk= 0





 .

The support ofRsis contained in PV.

(ii) There exists a positive measure R^∗_s on Z_V^∗ ≡ ZV with the Laplace transform

(10) L_R^∗_s(θ) = ∆_−s(θ), θ∈ P_V

if and only ifs∈X:=F

ε∈{0,1}^rX(ε), where

X(ε) :=







sk> ¹₂P

l>kldimVlk ifεk = 1 s∈R^r;

sk= ¹₂P

l>kldimVlk ifεk = 0





 .

The support ofR^∗_s is contained inQV.

The measureRs(resp. R^∗_s) is calledthe Riesz measure on the conePV(resp. QV). Ξ andXare called the Gindikin-Wallach sets.

Riesz measures were described explicitly in [12]. Rs (resp. R^∗_s) is a singular measure unless s ∈ Ξ(1, . . . ,1) (resp. s∈X(1, . . . ,1)). If s∈Ξ(1, . . . ,1) (resp. s∈X(1, . . . ,1)), then the Riesz measure is

an absolutely continuous measure with respect to the Lebesgue measure. In such case, the support ofRs

(resp. R^∗_s) equalsP_V (resp. Q_V).

We are interested in the description of natural exponential families generated byR_sandR^∗_s. Members of F(R_s) and F(R^∗_s) are called Wishart distributions on P_V and Q_V, respectively. In order to define NEFs generated byRs andR^∗_s we have to ensure thatRs∈ M(Z_V) andR^∗_s ∈ M(Z_V^∗) at least for some s. We have the following

Theorem 4.2 ([18, Theorem 6]) (i) Lets∈Ξ. The support ofR_sis not concentrated on any affine hyperplane inZ_V if and only if s_k>0 for allk= 1, . . . , r.

(ii) Let s∈X. The support of R^∗_s is not concentrated on any affine hyperplane in Z_V^∗ if and only if s_k>0 for allk= 1, . . . , r.

4.1. Group equivariance of the Wishart exponential families. We say that a measure µonEis relatively invariant under a subgroupGofGL(E), if for allg∈Gthere exists a constantcg>0 for which µ(gA) =cgµ(A) for any measurableA⊂E. This condition is equivalent to

Lµ(g^∗θ) =c⁻¹_g Lµ(θ), θ∈Θ(µ), whereg^∗ is the adjoint ofg.

Formulas (4) and (5) imply that Riesz measureRsis invariant under the groupρ(HV), while the dual Riesz measure R^∗_s is invariant underρ^∗(H_V). It follows that the Wishart exponential family F(R^∗_s) is invariant underρ(H_V) and, analogously,F(Rs) is invariant underρ^∗(H_V).

5. The inverse of the mean map and the Lauritzen formula onQV

Lets∈X∩R^r>0. Then we haveR^∗_s ∈ M(Z_V^∗) by Theorem 4.2. Denote byψ_s:=ψ_R^∗

s the inverse of the mean map fromM_F(R∗

s)to Θ(R^∗_s). In this section, we give an explicit formula for ψs(m), m∈M_F(R∗ s). Thanks to Theorem 4.1, we have P_V ⊂Θ(R^∗_s) and M_F(R∗

s)⊂ Q_V. Applying [15, Proposition IV.3], we can show that PV = Θ(R^∗_s) and M_F(R∗

s) = QV. For this, it suffices to check that, for any sequence {yk}k∈N inPV converging to a point in∂PV, we have limk→∞∆−s(yk) = +∞becauses∈R^r>0.

Proposition 5.1 The inverse of the mean map on QV is expressed by ψs(m) =−(logδ−s)⁰(m).

(11)

Proof. By Lemma 2.1 we obtainψs=−(logJR^∗_s)⁰. ForT ∈HV, we have JR^∗_s(ρ^∗(T)IN) = sup

θ∈PV

e^{−hρ∗(T)IN,θi} L_R^∗_s(θ) = sup

θ∈PV

χ−s(T)e^{−hIN,ρ(T)θi} L_R^∗_s(ρ(T)θ), where the last equality follows from (4). Sinceρ(T)PV =PV, we get

J_R^∗_s(ρ^∗(T)IN) =χ_−s(T)· sup

θ∈PV

e^−hIN,θi L_R^∗

s(θ) =δ_−s(ρ^∗(T)IN)J_R^∗_s(IN).

We see that the functionδ_−sequalsJ_R^∗_s up to a constant multiple. Therefore−(logδ_−s)⁰ coincides with

−(logJ_R^∗_s)⁰.

Remark 5.1 It is shown in [22, Proposition 3.16] that−(log ∆_−s)⁰ gives a diffeomorphism from the ho- mogeneous coneP_V ontoQ_V for anys∈R^r>0, and that−(logδ_−s)⁰ gives the inverse map of−(log ∆_−s)⁰. Proposition 5.1 follows from this fact because the mean map mR^∗_s equals−(logLR^∗_s)⁰ =−(log ∆−s)⁰ for s∈Ξ∩R^r>0by (10). Nevertheless, we have given a short simple proof of Proposition 5.1 for completeness.

Let us evaluate ψs(m) ∈ PV for m ∈ QV. In general, for a positive integer M, we regard the set Sym(M,R) of M ×M symmetric matrices as a Euclidean vector space with the trace inner product trXY (X, Y ∈ Sym(M,R)). Then we consider the linear map φ^∗_i : Sym(mi,R) → ZV, i = 1, . . . , r, adjoint toφ_i:ZV→Sym(m_i,R) defined in such a way that

hξ, φ^∗_i(X)i= trφi(ξ)X, X ∈Sym(mi,R), ξ∈ Z_V. The linear map

b

φ^∗_i : Sym(mi−1,R)→ Z_V, i= 1, . . . , r,adjoint to b

φ_i:Z_V→Sym(mi−1,R) is defined similarly.

Theorem 5.1 The inverse of the mean map on Q_V is given by the formula (12) ψs(m) =srφ^∗_r(φr(m)⁻¹) +

r−1

X

i=1

si

φ^∗_i(φi(m)⁻¹)− b φ^∗_i(

b

φ_i(m)⁻¹) . Proof. For anyα∈ ZV, we see from (11) and Proposition 3.1 (iii) that

α, ψ_s(m)

=−Dαlogδ_−s(m) =s_rD_αlogφ_r(m) +

r−1

X

i=1

s_iD_α

log detφ_i(m)−log det b φ_i(m)

, where Dα denotes the directional derivative in the direction of α. By the well-known formula for the derivative of log-determinant, we get

(13)

α, ψs(m)

=sr

φr(α) φr(m)+

r−1

X

i=1

si

trφi(α)φi(m)⁻¹−tr b φ_i(α)

b

φ_i(m)⁻¹ . Using the adjoint maps, we rewrite (13) as

α, ψs(m)

=sr

α, φ^∗_r(φr(m)⁻¹) +

r−1

X

i=1

si

α, φ^∗_i(φi(m)⁻¹)

−D α,

b φ^∗_i(

b

φ_i(m)⁻¹)E , so that we obtain formula (12).

Letn:= (n₁, . . . , n_r). Noting that ˆm⁻¹∈ P_V, we have det ˆm⁻¹= ∆_n( ˆm⁻¹) =δ_−n(m) by (3) and (8).

Thus, forα∈ Z_V we observe α, ψn(m)

=−Dαlogδ−n(m) =−Dαlog det ˆm⁻¹= trαmˆ⁻¹=

α,mˆ⁻¹ . Therefore, by (12) we get

Corollary 5.1 The inverse of the bijectiony→π(y⁻¹),P_V→ Q_V is given explicitly by (14) m7→mˆ⁻¹=ψn(m) =nrφ^∗_r(φr(m)⁻¹) +

r−1

X

i=1

ni

φ^∗_i(φi(m)⁻¹)− b φ^∗_i(

b

φ_i(m)⁻¹) .

If n1 =n2 =. . . =nr = 1, then (14) yields the Lauritzen formula for homogeneous graphical cones (cf. Example 6.1). Formula (14) generalizes the Lauritzen formula to all homogeneous cones.

6. Variance function of Wishart exponential families on QV

As in the previous section, lets∈X∩R^r>0.

Lemma 6.1 The variance functions of the Wishart exponential families satisfy VF(R^∗_s)(ρ^∗(T)IN) =ρ^∗(T)VF(R^∗_s)(IN)ρ(T), T ∈H_V, (15)

VF(Rs)(ρ(T)I_N) =ρ(T)VF(Rs)(I_N)ρ^∗(T), T ∈H_V. (16)

Proof. The identity (15) for the variance function follows from the invariance ofF(R^∗_s) underρ(HV) (see for example formula (2.2) in [18]). The invariance property ofRs results in identity (16).

In [18, Theorem 7] it was shown that property (16) actually characterizes measureRs. The same is true for (15) andR^∗_s.

Recall that N = n₁+. . .+n_r = N_r. For z ∈ Sym(N_k,R), we define the matrix z₀ ∈ Sym(N,R) completed with zeros, that is, (z₀)_{1:k}=zand (z₀)_ij = 0 if max{i, j}> N_k. SetJ_k := (I_Nk)₀∈ Z_V and J_k^∗:=I_N −J_k.

Proposition 6.1 If y=T^>T ∈Sym₊(N,R) withT ∈H_V, then T^>J_k^∗ T =y−h

y⁻¹

{1:k}

i−1 0 . Proof. We have to show thatT^>Jk T =h

y⁻¹

{1:k}

i−1 0

. Observe first that T^>Jk T =

T_{1:k}^> T_{1:k}

0.

SetS=T⁻¹∈H_V. Theny⁻¹=SS^>. Since (SS^>)_{1:k}=S_{1:k}S_{1:k}^> , we have (y⁻¹)_{1:k}−1

= (S_{1:k}^> )⁻¹(S_{1:k})⁻¹.

Thanks to (S_{1:k})⁻¹= (S⁻¹)_{1:k}=T_{1:k}, we get the assertion.

Now we are ready to state and prove our main theorem.

Theorem 6.1 Let s∈X∩R^r>0. Then the variance function ofF(R^∗_s)is given by VF(R^∗_s)(m) =π◦

(n₁ s1

ρ( ˆm) +

r

X

i=2

n_i si

−n_i−1 si−1

ρ

ˆ m−h

ˆ m⁻¹

{1:i−1}

i⁻¹

0

)

, m∈ QV. Proof. Similarly as in the proof of Theorem 5.1, using formula (11) from Proposition 5.1, we obtain

D

α, ψ⁰_s(m)βE

=−D²_α,βlogδ−s(m)

=s_rD²_α,βlogφ_r(m) +

r−1

X

i=1

s_i

D²_α,βlog detφ_i(m)−D²_α,βlog det b φ_i(m)

=−s_rφr(α)φr(β) φ_r(m)² −

r−1

X

i=1

s_i

trφ_i(α)φ_i(m)⁻¹φ_i(β)φ_i(m)⁻¹−tr b φ_i(α)

b φ_i(m)⁻¹

b φ_i(β)

b

φ_i(m)⁻¹ . Settingm=I_N, we obtain by Proposition 3.1 (iv)

D

α, ψ⁰_s(IN)βE

=−srαrrβrr−

r−1

X

i=1

si

ni

niαiiβii+ 2

r

X

l=i+1

nl(αli|βli)

! .

Recall that J_k^∗ = 0N_k

In_k+1+...+n_r

∈ Z_V and set Pk := ρ(J_k^∗), k = 1, . . . , r−1, P0 = Id_ZV. Pi is the orthogonal projection ontoL

i<k,l≤rVlk. Then, through direct computation, one can show that for i= 1, . . . , r−1,

niαiiβii+ 2

r

X

l=i+1

nl(αli|βli) =hα,(Pi−1−Pi)βi, and

n_rα_rrβ_rr =hα,Pr−1βi.

These imply that (Pr:= 0ZV)

ψ_s⁰(IN) =−

r

X

i=1

si

ni

(Pi−1−Pi).

Since (Pi−1−Pi)(Pj−1−Pj) =δij(Pi−1−Pi), we have [ψ_s⁰(IN)]⁻¹=−

r

X

i=1

ni

si

(Pi−1−Pi).

(17) Indeed,

ψ_s⁰(I_N)

r

X

j=1

−n_j sj

(Pj−1−Pj)

=

r

X

i=1

(Pi−1−Pi) =P0= Id_ZV. Thus, (17) gives us

VF(R^∗_s)(IN) =−[ψ⁰_s(IN)]⁻¹= n1

s1

Id_ZV+

r

X

i=2

ni

si

−n_i−1 s_i−1

Pi−1. Finally, using (15) we obtain form=ρ^∗(T)IN,

VF(R^∗_s)(m) =ρ^∗(T)VF(R^∗_s)(I_N)ρ(T) = n₁ s1

ρ^∗(T)ρ(T) +

r

X

i=2

n_i si

−n_i−1 si−1

ρ^∗(T)Pi−1ρ(T).

(18)

Sinceρ^∗(T) =π◦ρ(T^>) and ˆm=T^>T, we have

ρ^∗(T)ρ(T) =π◦ρ(T^>)ρ(T) =π◦ρ(T^>T) =π◦ρ( ˆm), and, by Proposition 6.1, fori= 2, . . . , r,

ρ^∗(T)Pi−1ρ(T) =π◦ρ(T^>J_i−1^∗ T) =π◦ρ

ˆ m−h

ˆ m⁻¹

{1:i−1}

i−1 0

.

Remark 6.1 Note that the formula for the inverse of the mean map given in Theorem 5.1 is not necessary for the proof of Theorem 6.1. Formula (11)from Proposition 5.1 is sufficient.

Example 6.1 Let us apply Theorem 6.1 to the Wishart exponential families on the Vinberg cone. Let

Z_V:=











x11 0 x31

0 x22 x32

x31 x32 x33



: x11, x22, x33, x31, x32∈R





 . Conditions(V1)–(V3)are satisfied and we have n1=n2=n3= 1,N =r= 3. Then,

PV=ZV∩Sym₊(3,R) ={x∈ ZV: x₁₁>0, x₂₂>0,detx >0∈R} and its dual cone is given by

Q_V=

ξ∈ Z_V:ξ₃₃>0, ξ₁₁ξ₃₃> ξ₃₁² , ξ₂₂ξ₃₃> ξ₃₂² .

The coneQV is called the Vinberg cone, whilePV is called the dual Vinberg cone. The cones QV andPV

are the lowest dimensional non-symmetric homogeneous cones.

Forξ∈ ZV, we have φ1(ξ) =

ξ11 ξ31

ξ31 ξ33

=:ξ_{1,3}, φ2(ξ) =

ξ22 ξ32

ξ32 ξ33

=:ξ_{2,3}, φ₃(ξ) =

b φ₁(ξ) =

b

φ₂(ξ) =ξ₃₃,

so that

φ^∗₁ a b

b c

=





a 0 b 0 0 0 b 0 c



, φ^∗₂ a b

b c

=





0 0 0 0 a b 0 b c



,

φ^∗₃(a) = b φ^∗₁(a) =

b φ^∗₂(a) =





0 0 0 0 0 0 0 0 a





fora, b, c∈R. Therefore, form∈ Q_V ands= (s1, s2, s3)we obtain by (12) ψ_s(m) =s₁







m33

|m_{1,3}| 0 −_|m^m31

{1,3}|

0 0 0

−_|m^m31

{1,3}| 0 _|m^m11

{1,3}|





+s₂







0 0 0

0 _|m^m33

{2,3}| −_|m^m32

{2,3}|

0 −_|m^m32

{2,3}|

m₂₂

|m{2,3}|





+ (s₃−s₁−s₂)





0 0 0

0 0 0

0 0 _m¹

33



. In particular,(14)tells us that

(19) mˆ⁻¹=







m₃₃

|m{1,3}| 0 −_|m^m31

{1,3}|

0 0 0

−_|m^m31

{1,3}| 0 _|m^m33

{1,3}|





+







0 0 0

0 _|m^m33

{2,3}| −_|m^m32

{2,3}|

0 −_|m^m32

{2,3}|

m₂₂

|m{2,3}|





−





0 0 0

0 0 0

0 0 _m¹

33



,

where |m_{1,3}|=m11m33−m²₃₁ and |m_{2,3}| =m22m33−m²₃₂. This is exactly the Lauritzen formula.

Moreover, we have

ˆ m=





m₁₁ ^m31_m^m32

33 m₃₁

m₃₁m₃₂

m33 m22 m32

m31 m32 m33





and it is easy to see that the projection π: Sym(3,R)→ ZV sets 0 in(1,2) and (2,1)entries remaining all other entries unchanged.

We shall use Theorem 6.1 in order to give explicitlyVF(R^∗_s). We denoteM_i= ^|m_m^{i,3}|

33 E_ii fori= 1,2, whereE_ii is the diagonal3×3 matrix with1 in(i, i)entry and all else 0. Then we have by (19)

h ˆ m⁻¹

{1:1}

i−1

0 =M1, h ˆ m⁻¹

{1:2}

i−1

0 =M1+M2. Theorem 6.1 gives for s∈X∩R³>0,

(20) VF(R^∗_s)(m) =π◦ 1

s1

ρ( ˆm) + 1

s2

− 1 s1

ρ( ˆm−M₁) + 1

s3

− 1 s2

ρ( ˆm−M₁−M₂)

.

Elementary properties of the quadratic operatorρand of its bilinear extensionρ(a, b)x= ¹₂(axb^>+bxa^>), and the fact thatρ(M₁, M₂) = 0onZV, imply the following formula, proven by different methods in [10]:

(21) VF(R^∗_s)(m) =π◦ 1

s₁ + 1 s₂ − 1

s₃

ρ( ˆm) + 1

s₃ − 1 s₁

ρ( ˆm−M₁) + 1

s₃ − 1 s₂

ρ( ˆm−M₂)

. Observe that formulas (20)and (21)imply analogous formulas for the homogeneous coneQ, dual to the coneP in the vector space

Z:=











x11 x21 0 x21 x22 x32

0 x32 x33



:x11, x21, x22, x32, x33∈R





 ,

i.e. P =Z∩Sym₊(3,R). Note thatZ∩Sym₊(3,R)is not a matrix realization of the coneP, so Theorem 6.1 does not apply directly to Wishart families on Q. Instead, we use the permutation(1,3,2)7→(1,2,3)

andmˆ with mˆ13=^m21_m^m32

22 . For example, formula (21)gives the following formula proven in [10]

VQ(m) = 1

s1

+ 1 s3

− 1 s2

ρ( ˆm) +

1 s2

− 1 s1

ρ( ˆm−M₁⁰) + 1

s2

− 1 s3

ρ( ˆm−M₃⁰), whereM₁⁰ := ^x11x_x^22−x221

22 E₁₁andM₃⁰ := ^x22x_x^33−x232

22 E₃₃. Note that fors= (p, p, p)withp > ¹₂, the variance function ofF(R^∗_s) on the Vinberg cone is

(22) Vp(m) = 1

pρ( ˆm).

Remark 6.2 A formula for the variance function of a Wishart family on a homogeneous cone is an- nounced in the unpublished article [4], Theorem 4.2. That formula does not coincide with the formulas proven in the present article. The conflict is caused by incorrect treatment of inverse elements in T- algebras in [4].

7. Variance function of Wishart exponential families on PV

We are going to find the variance function of the NEF generated byRs on the cone PV. Using the similar approach (see (18)) as in the proof of Theorem 6.1, one can show the following Proposition.

Proposition 7.1 Let s∈Ξ∩R^r>0. For anyT ∈H_V one has VF(Rs)(ρ(T)IN) = nr

s_rρ(T)ρ^∗(T) +

r−1

X

k=1

nk

s_k −nk+1

s_k+1

ρ(T)ρ(Jk)ρ^∗(T).

Proof. We have Θ(Rs) =Q_V and M_F(Rs)=P_V. By definition,m_Rs(θ) = −(logδ_−s)⁰(θ),θ ∈ Q_V. Use Lemma 2.1 to show thatψR_s(m) =−(log ∆−s)⁰(m),m∈ PV and proceed analogously as in the proof of

Theorem 6.1.

Now we will use also another approach to this problem. We will use the duality of the cones P_V and Q_V and a matrix realization ofQ_V. The objective is to boil down to the results of the preceding Section and apply the formula for the variance function from Theorem 6.1.

Dual coneQ_V to a homogeneous cone is also homogeneous. Thus, due to [13, Theorem D] it admits (under suitable linear isomorphism) a matrix realization. There exists a familyVe ={eVlk}_1≤k<l≤˜r sat- isfying (V1)-(V3) and a linear isomorphism l: Z

eV → Z_V such that Q_V =l(P

eV). It can be shown that

˜ r=r.

Sincel(P

Ve) =QV andl is an isomorphism, for anyT ∈HV there exists a uniqueTe∈H

Ve such that l(ρ(T)I_N^Ve_˜) =ρ^∗(Te)I_N^V.

The linear isomorphism l can be taken in such a way that if T has diagonal (t11, . . . , trr) then Te has diagonal (trr, . . . , t11) (see the choice of the permutation w in Proposition 7.4). In such case we have χ^V_s(Te) =χ^V_s^e∗(T), where s^∗= (s_r, . . . , s₁). This implies the following Proposition

Proposition 7.2 There exists a familyVe={eVlk}1≤k<l≤rsatisfying(V1)-(V3)and a linear isomorphism l: Z

Ve → Z_V such that Q_V=l(P

Ve)and

∆^V_s^e∗(x) =δ_s^V(l(x)), x∈ P

Ve. (23)

In this case, s∈Ξ_V if and only ifs^∗∈X

Ve.