Wishart exponential families on cones related to An graphs

TL;DR

This paper investigates Wishart exponential families on cones associated with An graphs, providing explicit formulas for their key statistical functions and extending previous models to non-decomposable graphs.

Contribution

It introduces a new approach to study Wishart NEF on cones related to An graphs, including non-decomposable cases, with explicit formulas for moments and covariance functions.

Findings

Derived Riesz measures for Wishart NEF on cones QG and PG.

Provided explicit formulas for mean, covariance, and higher moments.

Extended Wishart models to non-decomposable An graphs.

Abstract

Let G = An be the graph corresponding to the graphical model of nearest neighbour interaction in a Gaussian character. We study Natural Exponential Families( NEF) ofWishart distributions on convex cones QG and PG, where PG is the cone of positive definite real symmetric matrices with obligatory zeros prescribed by G, and QG is the dual cone of PG. The Wishart NEF that we construct include Wishart distributions considered earlier by Lauritzen (1996) and Letac and Massam (2007) for models based on decomposable graphs. Our approach is however different and allows us to study the basic objects ofWishart NEF on the cones QG and PG.We determine Riesz measures generating Wishart exponential families on QG and PG, and we give the quadratic construction of these Riesz measures and exponential families. The mean, inverse-mean, covariance and variance functions, as well as moments of higher order are studied and their explicit formulas are given.

Equations

$$Q_G=\{\eta \in I_G:\forall y\in \overline{P_G}\backslash \{0\}⟨y,\eta ⟩>0\}.$$

$${\Delta }_{\underline{s}}^{\prec }(y)$$

$${\delta }_{\underline{s}}^{\prec }(\eta )$$

$${\Delta }_{\underline{s}}^{(M)}(y)=\prod _{i=1}^{M-1}|y_{\{1:i\}}{|}^{s_i-s_{i+1}}|y{|}^{s_M}\prod _{i=M+1}^n|y_{\{i:n\}}{|}^{s_i-s_{i-1}},$$

$${\delta }_{\underline{s}}^{(M)}(\eta )=\frac{{\prod }_{i=1}^{M-1}|{\eta }_{\{i:i+1\}}{|}^{s_i}{\prod }_{i=M+1}^n|{\eta }_{\{i-1:i\}}{|}^{s_i}}{{\prod }_{i=2}^{M-1}{\eta }_{ii}^{s_{i-1}}\cdot {\eta }_{MM}^{s_{M-1}-s_M+s_{M+1}}\cdot {\prod }_{i=M+1}^{n-1}{\eta }_{ii}^{s_{i+1}}}.$$

$${\delta }_{\underline{s}}^{\prec }(\pi (y^{-1}))={\Delta }_{-\underline{s}}^{\prec }(y)={\Delta }_{-\underline{s}}^{(M)}(y).$$

$$\left|\begin{array}{c}{\eta }_{\{i,i+1\}}\end{array}\right|=|y{|}^{-1}|y_{V\backslash \{i,i+1\}}|.$$

$$i^{+}=\{\begin{array}{ll}\{i+1\}& \text{mbox}ifi\le M-1,\\ \varnothing & \text{mbox}ifi=M,\\ \{i-1\}& \text{mbox}ifi\ge M+1.\end{array}$$

$$1{\prec }_{M}2{\prec }_M\dots {\prec }_{M}M-1{\prec }_{M}n{\prec }_{M}n-1{\prec }_M\dots {\prec }_{M}M+1{\prec }_{M}M.$$

$$\frac{|y_{\{i\}\cup i^{-}}|}{|y_{i^{-}|}}=\frac{|y_{\{1:i\}}||y_{\{n-j:n\}}|}{|y_{\{1:i-1\}}||y_{\{n-j:n\}}|}=\frac{|y_{\{1:i\}}|}{|y_{\{1:i-1\}}|}.$$

$$y=A(b)\left(\begin{array}{cccc}a& 0& \cdots & 0\\ 0& & & \\ ⋮& & z& \\ 0& & & \end{array}\right){}^{t}A(b),A(b)=\left(\begin{array}{cccc}1& & & \\ b& 1& & \\ ⋮& & \ddots & \\ 0& \dots & 0& 1\end{array}\right),$$

$$\eta =\pi \left({}^{t}A(\beta )\left(\begin{array}{cccc}\alpha & 0& \cdots & 0\\ 0& & & \\ ⋮& & x& \\ 0& & & \end{array}\right)A(\beta )\right).$$

$$\tilde{y}={}^{t}B(b)\left(\begin{array}{cccc}& & & 0\\ & z& & ⋮\\ & & & 0\\ 0& \cdots & 0& a\end{array}\right)B(b),B(b)=\left(\begin{array}{cccc}1& & & \\ 0& 1& & \\ ⋮& & \ddots & \\ 0& \dots & b& 1\end{array}\right),$$

$$\tilde{\eta }=\pi \left(B(\beta )\left(\begin{array}{cccc}& & & 0\\ & x& & ⋮\\ & & & 0\\ 0& \cdots & 0& \alpha \end{array}\right){}^{t}B(\beta )\right).$$

$$J_{{\Phi }_n}(a,b,z)=a,J_{{\tilde{\Phi }}_n}(a,b,z)=a.$$

$$J_{{\Psi }_n}(\alpha ,\beta ,x)=x_{22},J_{{\tilde{\Psi }}_n}(\alpha ,\beta ,x)=x_{n-1,n-1}.$$

$$y_{ij}=\{\begin{array}{l}ab\text{if}(i,j)=(1,2)\text{or}(i,j)=(2,1),\\ a{b}^2+z_{22}\text{if}i=j=2,\\ y_{ij}^{\prime }\text{otherwise}.\end{array}$$

$${\eta }_{ij}=\{\begin{array}{l}\alpha +{\beta }^2{x}_{22}\text{if}i=j=1,\\ \beta x_{22}\text{if}(i,j)=(1,2)\text{or}(i,j)=(2,1),\\ {\eta }_{ij}^{\prime }\text{otherwise}.\end{array}$$

$${\tilde{y}}_{ij}=\{\begin{array}{l}ab\text{if}(i,j)=(n-1,n)\text{or}(i,j)=(n,n-1),\\ a{b}^2+z_{n-1,n-1}\text{if}i=j=n-1,\\ {\tilde{y}}_{ij}^{\prime }\text{otherwise},\end{array}$$

$${\tilde{\eta }}_{ij}=\{\begin{array}{l}\alpha +{\beta }^2{x}_{n-1,n-1}\text{if}i=j=n,\\ \beta x_{n-1,n-1}\text{if}(i,j)=(n-1,n)\text{or}(i,j)=(n,n-1),\\ {\tilde{\eta }}_{ij}^{\prime }\text{otherwise}.\end{array}$$

$${\Delta }_{\underline{s}}^{(M)}(y)=a^{s_1}{\Delta }_{(s_2,\dots ,s_n)}^{(M)}(z),$$

$${\delta }_{\underline{s}}^{(M)}(\eta )={\alpha }^{s_1}{\delta }_{(s_2,\dots ,s_n)}^{(M)}(x).$$

$${\Delta }_{\underline{s}}^{(M)}(y)=a^{s_n}{\Delta }_{(s_1,\dots ,s_{n-1})}^{(M)}(z),$$

$${\delta }_{\underline{s}}^{(M)}(\eta )={\alpha }^{s_n}{\delta }_{(s_1,\dots ,s_{n-1})}^{(M)}(x).$$

$${\phi }_{A_n}(\eta )=\stackrel{n-1}{\prod _{i=1}}|{\eta }_{\{i,i+1\}}{|}^{-3/2}\prod _{i\ne 1,n}{\eta }_{ii}.$$

$${\phi }_{A_n}(\eta )=x_{22}^{-1/2}{\alpha }^{-3/2}{\phi }_{A_{n-1}}(x)$$

$${\phi }_{A_n}(\tilde{\eta })=x_{n-1,n-1}^{-1/2}{\alpha }^{-3/2}{\phi }_{A_{n-1}}(x).$$

$$\mathrm{tr}(y\eta )=a\alpha +a{x}_{22}(b+\beta {)}^2+\mathrm{tr}(zx).$$

$$\mathrm{tr}(y\eta )=a\alpha +a{x}_{n-1,n-1}(b+\beta {)}^2+\mathrm{tr}(zx).$$

$$\frac{{\Delta }_{\underline{s}}^{(M)}(y)}{{\Delta }_{(s_2,\dots ,s_n)}^{(M)}(z)}=(y_{11}{)}^{s_1-s_2}\left[\prod _{i=2}^{M-1}{\left(\frac{|y_{\{1:i\}}|}{|z_{\{2:i\}}|}\right)}^{s_i-s_{i+1}}\right]{\left(\frac{|y|}{|z|}\right)}^{s_M}.$$

$$\frac{{\Delta }_{\underline{s}}^{(M)}(y)}{{\Delta }_{(s_2,\dots ,s_n)}^{(M)}(z)}=a^{s_1}.$$

Full text

1 Introduction

The classical Wishart distribution was first derived by Wishart (1928) as the distribution of the maximum likelihood estimator of the covariance matrix of the multivariate normal distribution. In the framework of graphical Gaussian models, the distribution of the maximum likelihood estimator of $\pi(\Sigma)$, where $\pi$ denotes the canonical projection onto $Q_{G}$, was derived by Dawid and Lauritzen (1993), who called it the hyper Wishart distribution. Dawid and Lauritzen (1993) also considered the hyper inverse Wishart distribution which is defined on $Q_{G}$ as the Diaconis-Ylvisaker conjugate prior distribution for $\pi(\Sigma)$, and Roverato (2000) derived the so-called $G$-Wishart distribution on $P_{G}$, that is, the distribution of the concentration matrix $K=\Sigma^{-1}$ when $\pi(\Sigma)$ follows the hyper inverse Wishart distribution. Letac and Massam (2007) constructed two classes of multi-parameter Wishart distributions on the cones $Q_{G}$ and $P_{G}$ associated to a decomposable graph $G$ and called them type I and type II Wishart distributions, respectively. They are more flexible because they have multiple shape parameters. In fact, the type I and type II Wishart distributions generalize the hyper Wishart distribution and the G-Wishart distribution respectively.

The Wishart exponential families introduced and studied in this paper include the type I and type II Wishart distributions of Letac-Massam on the cones $Q_{G}$ and $P_{G}$ associated to $A_{n}$ graphs. Our methods, which are new and different from methods of articles cited above, simplify in a significant way the Wishart theory for graphical models.

In Graczyk and Ishi (2014) and in Ishi (2014) the theory of Wishart distributions on general convex cones was developped, with a strong accent on the quadratic constructions and on applications to homogeneous cones . In this article we apply for the first time the ideas and results of Graczyk and Ishi (2014) to study important families of non-homogeneous cones.

Applications in estimation and other practical aspects of Wishart distributions are intensely studied, cf. Sugiura and Konno (1988); Tsukuma and Konno (2006); Konno (2007, 2009); Kuriki and Numata (2010).

The focus of this work is on non-homogeneous cones $Q_{A_{n}}$ and $P_{A_{n}}$ appearing in the statistical theory of graphical models, corresponding to the practical model of nearest neighbour interactions. In the Gaussian character $(X_{1},X_{2},\ldots,X_{n})$, non-neighbours $X_{i},X_{j}$, $|i-j|>1$ are conditionally independent with respect to other variables. This family of decomposable graphical models presents many advantages: it encompasses the univariate case ($A_{1}$), a complete graph ($A_{2}$), a non-complete homogeneous graph ($A_{3}$) and an infinite number of non-homogeneous graphs ($A_{n}$, $n\geq 4$).

The methods introduced in this article allowed to solve in Graczyk et al (2016b) the Letac-Massam Conjecture on the cones $Q_{A_{n}}$. Together with the results of this article we achieve in this way the complete study of all classical objects of an exponential family for the Wishart NEF on the cones $Q_{A_{n}}$.

Some of the results of our research may be extended to cones related to all decomposable graphs (work in progress). Many of them are however specific for the cones $Q_{A_{n}}$ and $P_{A_{n}}$ (indexation of Riesz and Wishart measures by $M=1,\ldots,n$, Letac-Massam Conjecture, Inverse Mean Map, Variance function).

Plan of the article. Sections 2, 3 and 4 provide the main tools in order to define and to study the Wishart NEF on the cones $Q_{A_{n}}$ and $P_{A_{n}}$. In Section 2, useful notions of eliminating orders $\prec$ on $A_{n}$ and of generalized power functions $\delta^{\prec}_{\underline{s}}$ and $\Delta^{\prec}_{\underline{s}}$, $\underline{s}\in\operatorname{\mathbb{R}}^{n}$ will be introduced on the cones $Q_{A_{n}}$ and $P_{A_{n}}$ respectively. In Theorem 2.1, a classical relation between the power functions $\delta^{\prec}_{\underline{s}}$ and $\Delta^{\prec}_{-\underline{s}}$ is proved as well as the dependence of $\delta^{\prec}_{\underline{s}}$ and $\Delta^{\prec}_{\underline{s}}$ on the maximal element $M$ of $\prec$ only. Thus, in the sequel of the paper, only generalized power functions $\delta^{(M)}_{\underline{s}}$ and $\Delta^{(M)}_{\underline{s}}$ appear. Next important tool of analysis of Wishart exponential families are recurrent construction of the cones $P_{G}$ and $Q_{G}$ and corresponding changes of variables. They are introduced and studied in Section 3, and are immediately applied in Section 4 in order to compute the Laplace transform of generalized power functions $\delta^{(M)}_{\underline{s}}$ and $\Delta^{(M)}_{\underline{s}}$ (Theorems 4.1 and 4.2).

In Section 5, Wishart natural exponential families on the cones $Q_{A_{n}}$ are defined, and all their classical objects are explicitly determined, beginning with the Riesz generating measures, Wishart densities, Laplace transform, mean and covariance. In Theorem 5.1 and Corollary 3, an explicit formula for the inverse mean map is proved. It provides an infinite number of versions of Lauritzen formulas for bijections between the cones $Q_{G}$ and $P_{G}$. In Section 5.3 two explicit formulas are given for the variance function of a Wishart family. The formula of Theorem 5.2 is surprisingly simple and similar to the case of the symmetric cone $S_{n}^{+}$. Sections 5.4 and 5.5 are devoted to the quadratic constructions of Wishart exponential families on $Q_{G}$ and to the computation of their higher moments in Theorem 5.3. An interesting connection to the Missing Data statistics is mentioned and will be developped in a forthcoming paper.

Section 6 is on Wishart natural exponential families on the cones $P_{A_{n}}$ and follows a similar scheme as Section 5, however the inverse mean map and variance function are not available on the cones $P_{A_{n}}$. The analysis on these cones is more difficult.

In the last Section 7 we establish the relations of the Wishart NEF defined and studied in our paper with the type I and type II Wishart distributions from Letac and Massam (2007). Our methods give a simple proof of the formulas for Laplace transforms of type I and type II Wishart distributions from Letac and Massam (2007).

2 Preliminaries on $A_{n}$ graphs and related cones

In this section we study properties of graphs $A_{n}$ that will be important in the theory of Riesz measures and Wishart distributions on the cones related to these graphs. In particular, we characterize all the eliminating orders of vertices and we introduce generalized power functions related to such orders. We show that they only depend on the maximal element $M\in\{1,\ldots,n\}$ of the order.

An undirected graph is a pair $G=(V,\mathcal{E})$, where $V$ is a finite set and $\mathcal{E}$ is a subset of $\mathcal{P}_{2}(V)$, the set of all subsets of $\mathcal{E}$ with cardinality two. The elements of $V$ are called nodes or vertices and the elements of $\mathcal{E}$ are called edges. If ${\{v,v^{\prime}\}}\in\mathcal{E}$, then $v$ and $v^{\prime}$ are said to be adjacent and this is denoted by $v\sim v^{\prime}$. Graphs are visualized by representing each node by a point and each edge $\{v,v^{\prime}\}$ by a line with the nodes $v$ and $v^{\prime}$ as endpoints. For convenience, we introduce a subset $E\subset V\times V$ defined by $E:=\{(v,v^{\prime}):v\sim v^{\prime}\}\cup\{(v,v):v\in V\}$.

The graph with $V=\{v_{1},v_{2},\ldots,v_{n}\}$ and $\mathcal{E}={\{\{v_{j},\,\,v_{j+1}\}:1\leq j\leq n-1\}}$ is denoted by $A_{n}$ and represented as $1-2-3-\ldots-n$. An $n$-dimensional Gaussian model $(X_{v})_{v\in V}$ is said to be Markov with respect to a graph $G$ if for any $(v,\,\,v^{\prime})\notin E$, the random variables $X_{v}$ and $X_{v^{\prime}}$ are conditionally independent given all the other variables. The conditional independence relations encoded in $A_{n}$ graph are of the form: $X_{v_{i}}\perp X_{v_{j}}|(X_{v_{k}})_{k\neq i,j}$, for all $|i-j|>1$. Thus, $A_{n}$ graphs correspond to nearest neighbour interaction models. In what follows, we often denote the vertex $v_{i}$ by $i$.

Let $S_{n}$ be the space of real symmetric matrices of order $n$ and let $S_{n}^{+}\subset S_{n}$ be the cone of positive definite matrices. The notation for a positive definite matrix $y$ is $y>0$. For a graph $G$, let $Z_{G}\subset S_{n}$ be the vector space consisting of $y\in S_{n}$ such that $y_{ij}=0$ if $(i\,,\,j)\notin E$. Let $I_{G}=Z_{G}^{*}$ be the dual vector space with respect to the scalar product ${\langle y,\eta\rangle=\operatorname{tr}(y\eta)=\sum_{(i,j)\in E}y_{ij}\eta_{ij},\ \ y\in Z_{G},\,\eta\in I_{G}}.$ In the statistical literature, the vector space $I_{G}$ is commonly realised as the space of $n\times n$ symmetric matrices $\eta$, in which only the coefficients $\eta_{ij}$, $(i,j)\in E$, are given. We adapt this realisation of $I_{G}$ in this paper.

If $I\subset V$, we denote by $y_{I}$ the submatrix of $y\in Z_{G}$ obtained by extracting from $y$ the lines and the columns indexed by $I$. The same notation is used for $\eta\in I_{G}$. Let $P_{G}$ be the cone defined by $P_{G}=\{y\in Z_{G}:y>0\}$, and $Q_{G}\subset I_{G}$ the dual cone of $P_{G}$, that is,

[TABLE]

A Gaussian vector model $(X_{v})_{v\in V}$ is Markov with respect to $G$ if and only if the concentration matrix $K=\Sigma^{-1}$ belongs to $P_{G}$.

When $G=A_{n}$, the cone $Q_{G}$ is described as $Q_{G}=\{\eta\in I_{G}:\eta_{\{i,i+1\}}>0,\,\,i=1,\dots,n-1\}$. Let $\pi={\pi_{I_{G}}}$ be the projection of $S_{n}$ onto $I_{G}$, $x\mapsto\eta$ such that $\eta_{ij}=x_{ij}$ if $(i,j)\in E$. Then it is known (cf. Letac and Massam (2007); Andersson and Klein (2010)) that the mapping $P_{G}\longrightarrow Q_{G}$, $y\longmapsto\pi(y^{-1})$ is a bijection.

In the sequel, unless otherwise stated, $G=A_{n}$,

2.1 Eliminating Orders

Different orders of vertices $v_{1},v_{2},\ldots,v_{n}$ should be considered in order to have a harmonious theory of Riesz and Wishart distributions on the cones related to $A_{n}$ graphs. The orders that will be important in this work are called eliminating orders of vertices and will be presented now.

Definition 1

Consider a graph $G={(V,\mathcal{E})}$ and an ordering $\prec$ of the vertices of $G$. The set of future neighbours of a vertex $v$ is defined as $v^{+}=\{w\in V:v\prec w\;\textnormal{and}\;v\sim w\}.$ The set of all predecessors of a vertex $v\in V$ with respect to $\prec$ is defined as $v^{-}=\{u\in V:u\prec v\}$.

Definition 2

An ordering $\prec$ of the vertices of a graph $G$ is said to be an eliminating order if $v^{+}$ is complete for all $v\in V$.

In this section, we present a characterization of the eliminating orders in the case of the graph $A_{n}$. An algorithm that generates all eliminating orders for a general graph is given by Chandran et al (2003).

Proposition 1

Consider a graph $A_{n}:1-2-3-\ldots-n$. All eliminating orders are obtained by an intertwining of two sequences $1\prec\ldots\prec M\ \ {\rm and}\ \ n\prec\ldots\prec M$ for an $M\in V$. There are $2^{n-1}$ eliminating orders on the graph $A_{n}$.

Proof

Consider an eliminating order $\prec$ on $G=A_{n}$. Since the minimal element of an eliminating order $\prec$ on a graph $A_{n}$ is one of the exterior vertices $1,n$ of the graph, it starts with $1$ or $n$, say it is $1$. It follows from Definition 2 that an eliminating order without its minimal element forms again an eliminating order on the graph $A_{n-1}$ obtained from $G$ by suppressing $1$ or $n$. The element following $1$ may be 2 or $n$. This recursive argument proves that in an eliminating order the sequences $1\prec 2\ldots\prec M$ and $n\prec n-1\prec\ldots\prec M$ must appear intertwined. We also see that we construct in this way $2^{n-1}$ different orders.

Conversely, if an order $\prec$ on $G$ is obtained by intertwining of the sequences $1\prec 2\ldots\prec M$ and $n\prec n-1\prec\ldots\prec M$, it follows that the sets $v^{+}$ of future neighbours of $v$ are singletons or empty(for $v=M$). Thus the order $\prec$ is eliminating.

2.2 Generalized power functions

In this section, we define and study generalized power functions on the cones $P_{G}$ and $Q_{G}$.

Here we introduce useful notation. For $1\leq i\leq j\leq n$, let $\{i:j\}\subset V$ be the set of $a\in V$ for which $i\leq a\leq j$. Then, for $y\in Z_{G}$ and $1\leq i\leq n$, the matrix $y_{\{1:i\}}$ is the upper left submatrix of $y$ of size $i$, and $y_{\{i:n\}}$ is the lower right submatrix of size $n-i+1$. Recall that on the cone $S_{n}^{+}$, the generalized power functions are $\Delta_{\underline{s}}(y)=\prod_{i=1}^{n}|y_{\{1:i\}}|^{s_{i}-s_{i+1}}$ and $\delta_{\underline{s}}(y)=\prod_{i=1}^{n}|y_{\{i:n\}}|^{s_{i}-s_{i-1}}$, with $s_{0}=s_{n+1}=0$.

Definition 3

For $\underline{s}\in\mathbb{R}^{V}$, setting $\det y_{\emptyset}=1=\det\eta_{\emptyset}$, we define

[TABLE]

Note that Definition 3 applied to the complete graph with the usual order $1<\ldots<n$ gives $\Delta_{\underline{s}}$ and $\delta_{\underline{s}}$. For any $\underline{s}$ the following formula $\delta_{\underline{s}}(y^{-1})=\Delta_{-\underline{s}}(y)$ is well known. In Theorem 2.1 we find an analogous formula in the case of the cones $P_{G}$ and $Q_{G}$.

We will see in Theorem 2.1 that on the cones related to the graphs $A_{n}$, different order-depending power functions $\Delta_{\underline{s}}^{\prec}$ and $\delta_{\underline{s}}^{\prec}$ defined in Definition 3 may be expressed in terms of explicit "$M$-power functions" ${\Delta}^{(M)}_{\underline{s}}$ and ${\mathcal{\delta}}^{(M)}_{\underline{s}}$ that will be defined below. They depend only on the choice of $M\in V$.

Definition 4

Let $M\in V$, $y\in P_{G}$ and $\eta\in Q_{G}$. We define the $M$-power functions ${\Delta}^{(M)}_{\underline{s}}(y)$ on $P_{G}$ and ${\delta}^{(M)}_{\underline{s}}(x)$ on $Q_{G}$ by the following formulas:

[TABLE]

[TABLE]

Observe that for $M=1,n$  there are $n-1$ factors in the denominator of (4), and for $M=2,\ldots n-1$ there are $n-2$ factors (powers of $\eta_{22}\ldots\eta_{n-1,n-1}$).

The main result of this section is the following theorem.

Theorem 2.1

Consider a graph $G=A_{n}$ with an eliminating order $\prec$. Let $M$ be the maximal element with respect to $\prec$. Then for all $y\in P_{G}$, we have

[TABLE]

The proof of Theorem 2.1 is preceded by a series of elementary lemmas.

Lemma 1

Let $y\in P_{G}$ and $i<j<j+1<k<m$. The determinant of the submatrix $y_{\{i\,:\,j\}\cup\{k\,:\,m\}}$ can be factorized as $|y_{\{i\,:\,j\}\cup\{k\,:\,m\}}|=|y_{\{i\,:\,j\}}||y_{\{k\,:\,m\}}|$.

Lemma 2

Let $y\in P_{G}$ and $\eta=\pi(y^{-1})$. Then for all $i,i+1\in V$, we have

[TABLE]

Proof

We repeatedly use the cofactor formula for an inverse matrix. We use $\eta_{ii}=|y|^{-1}|y_{V\backslash\{i\}}|$ and show that $\eta_{i,i+1}=-y_{i,i+1}|y|^{-1}|y_{V\backslash\{i,i+1\}}|$.  It follows that

$\begin{vmatrix}\eta_{\{i,i+1\}}\end{vmatrix}=|y|^{-2}|y_{V\backslash\{i,i+1\}}|\left[|y_{\{i+1\,:\,n\}}||y_{\{1\,:\,i\}}|-y_{i,i+1}^{2}|y_{\{1\,:\,i-1\}}||y_{\{i+2\,:\,n\}}|\right]$. The last factor in brackets equals $|y|$.

Proof

(of Theorem 2.1)

Part 1: $\delta_{\underline{s}}^{\prec}(\pi(y^{-1}))={{\Delta}^{(M)}_{-\underline{s}}(y).}$ From Proposition 1, we have

[TABLE]

Using $\eta_{ii}=|y|^{-1}|y_{V\backslash\{i\}}|$ with $\eta=\pi(y^{-1})$ and Lemmas 1 and 2, we get $\delta_{\underline{s}}^{\prec}(\pi(y^{-1}))=\Delta^{(M)}_{-\underline{s}}(y).$

Part 2: ${\Delta_{\underline{s}}^{\prec}(y)={\Delta}^{(M)}_{\underline{s}}(y)}$. Let us first consider the eliminating order $\prec_{M}$ given by

[TABLE]

Using $\eta_{ii}=|y|^{-1}|y_{V\backslash\{i\}}|$, Lemmas 1 and 2 again, we get $\Delta_{\underline{s}}^{\prec_{M}}(y)={\Delta}^{(M)}_{\underline{s}}(y).$

It is easy to see using Proposition 1 and the factorization from Lemma 1 that for any other eliminating order $\prec$, the factors of $\Delta_{\underline{s}}^{\prec}(y)$ under the powers $s_{i}$ are exactly the same as for $\prec_{M}$. Indeed, if $i\leq M-1$, let $n-j$ be the largest vertex greater than M such that $n-j\prec i$. Then, the factor under the power $s_{i}$ is

[TABLE]

A similar argument shows that this is also true for $i=M$ and for $i>M$.

Corollary 1

Let $\prec_{1}$ and $\prec_{2}$ be two eliminating orders on $G$ such that $\max_{\prec_{1}}V=\max_{\prec_{2}}V$. Then  $\delta_{\underline{s}}^{\prec_{1}}(\eta)=\delta_{\underline{s}}^{\prec_{2}}(\eta)$  for all $\eta\in Q_{G}$. If $\max_{\prec}V=M$ then we have $\delta_{\underline{s}}^{\prec}(\eta)=\delta_{\underline{s}}^{(M)}(\eta).$

3 Recurrent construction of the cones $P_{G}$ and $Q_{G}$ and changes of variables

In this section we introduce very useful recurrent constructions of the cones $P_{A_{n}}$ and $Q_{A_{n}}$ from the cones $P_{A_{n-1}}$ and $Q_{A_{n-1}}$. There are two variants of them for $A_{n-1}:2-\ldots-n$ and $A_{n-1}:1-\ldots-{(n-1)}$. Corresponding changes of variables for integration on $P_{A_{n}}$ and $Q_{A_{n}}$ are introduced.

Proposition 2

*For $n\geq 2$, let $\Phi_{n}:\mathbb{R}^{+}\times\mathbb{R}\times P_{A_{n-1}}\longrightarrow P_{A_{n}}$, $(a,b,z)\longmapsto y$ with *

[TABLE]

and let $\Psi_{n}:\mathbb{R}^{+}\times\mathbb{R}\times Q_{A_{n-1}}\longrightarrow Q_{A_{n}}$, $(\alpha,\beta,x)\longmapsto\eta$ with

[TABLE]

Then the maps $\Phi_{n}$ and $\Psi_{n}$ are bijections. 2. 2.

Let $\tilde{\Phi}_{n}:\mathbb{R}^{+}\times\mathbb{R}\times P_{A_{n-1}}\longrightarrow P_{A_{n}}$, $(a,b,z)\longmapsto\tilde{y}$ with

[TABLE]

and let $\tilde{\Psi}_{n}:\mathbb{R}^{+}\times\mathbb{R}\times Q_{A_{n-1}}\longrightarrow Q_{A_{n}}$, $(\alpha,\beta,x)\longmapsto\tilde{\eta}$ with

[TABLE]

Then the maps $\tilde{\Phi}_{n}$ and $\tilde{\Psi}_{n}$ are bijections. 3. 3.

The Jacobians of the changes of variables $y=\Phi_{n}(a,b,z)$ and $y=\tilde{\Phi}_{n}(a,b,z)$ are given by

[TABLE]

The Jacobians of the changes of variables $\eta=\Psi_{n}(\alpha,\beta,x)$ and $\eta=\tilde{\Psi}_{n}(\alpha,\beta,x)$ are given by

[TABLE]

Proof

1. Let $y^{\prime}=\begin{pmatrix}a&0&\ldots&0\\ 0\\ \vdots&&z\\ 0\end{pmatrix}$ and $\eta^{\prime}=\begin{pmatrix}\alpha&0&\ldots&0\\ 0\\ \vdots&&x\\ 0\end{pmatrix}$. Then

[TABLE]

Thus, on the one hand, if $(a,b,z)\in\mathbb{R}^{+}\times\mathbb{R}\times P_{A_{n-1}}$, then $y\in Z_{A_{n}}$. And $z>0$ implies $y^{\prime}>0$ as every principal minor of $y^{\prime}$ equals $a$ times a principal minor of $z$. From $y=Ty^{\prime}{}^{t}T$ with $T=A(b)$, we get $y\in P_{A_{n}}$. On the other hand, if $y\in P_{A_{n}}$, we have $a=y_{11}>0$, $b=\frac{y_{12}}{y_{11}}$, $z_{22}=y_{22}-\frac{y_{12}^{2}}{y_{11}}$ and $z_{ij}=y_{ij}$ for all $i\neq 2$ and $j\neq 2$. We use the notation $z=(z_{ij})_{2\leq i,j\leq n}$. Now, let us show that $z\in P_{A_{n-1}}$. We have $y^{\prime}=T^{-1}y\,{}^{t}T^{-1}>0$. Hence, we have also $z>0$ since each principal minor of $z$ equals $1/a$ times a principal minor of $y^{\prime}$. Therefore, the map $\Phi_{n}$ is indeed a bijection from $\mathbb{R}^{+}\times\mathbb{R}\times P_{A_{n-1}}$ onto $P_{A_{n}}$.

Let us turn to $\Psi_{n}$. The relation between $\eta$ and $\eta^{\prime}$ is given by

[TABLE]

First we show that if $(\alpha,\beta,x)\in\mathbb{R}^{+}\times\mathbb{R}\times Q_{A_{n-1}}$, then $\eta\in I_{A_{n}}$. Actually, since $x_{\{2,3\}}>0$, we have $\alpha+\beta^{2}x_{22}>0$ and $\eta_{\{1,2\}}=\begin{pmatrix}\alpha+\beta^{2}x_{22}&\beta x_{22}\\ \beta x_{22}&x_{22}\end{pmatrix}>0$. On the other hand, if $\eta\in Q_{A_{n}}$, we have $x_{ij}=\eta_{ij}$ for all $i,j=2,\ldots,n$. Thus, $\eta\in Q_{A_{n}}$ implies $x\in Q_{A_{n-1}}$.

2. Let $\tilde{y}^{\prime}=\begin{pmatrix}&&&0\\ &z&&\vdots\\ &&&0\\ 0&\ldots&0&a\end{pmatrix}$ and $\tilde{\eta}^{\prime}=\begin{pmatrix}&&&0\\ &x&&\vdots\\ &&&0\\ 0&\ldots&0&\alpha\end{pmatrix}$. Then we have

[TABLE]

and

[TABLE]

Similar reasoning as above shows that $\tilde{\Phi}$ and $\tilde{\Psi}$ are indeed bijections.

The proof is by direct computation.

Lemma 3

Let $y=\Phi_{n}(a,b,z)$ and $\eta=\Psi_{n}(\alpha,\beta,x)$. Then, for all $M=2,\ldots,n$,

[TABLE]

[TABLE]

Let $y=\tilde{\Phi}_{n}(a,b,z)$ and $\eta=\tilde{\Psi}_{n}(\alpha,\beta,x)$. Then, for all $M=1,\ldots,n-1$,

[TABLE]

[TABLE] 2. 2.

Let us define $\varphi_{A_{n}}:Q_{A_{n}}\to\mathbb{R}_{+}$ by $\varphi_{A_{1}}(\eta)=\eta^{-1}$, and for $n\geq 2$

[TABLE]

Let $\eta=\Psi_{n}(\alpha,\beta,x)$ and $\tilde{\eta}=\tilde{\Psi}_{n}(\alpha,\beta,x)$. Then,

[TABLE]

and

[TABLE] 3. 3.

If $y=\Phi_{n}(a,b,z)$ and $\eta=\Psi_{n}(\alpha,\beta,x)$, then

[TABLE]

If $y=\tilde{\Phi}_{n}(a,b,z)$ and $\eta=\tilde{\Psi}_{n}(\alpha,\beta,x)$, then

[TABLE]

Proof

1. For $M\geq 2$, we have

[TABLE]

Using Lemma 7, we have $|y_{\{1:i\}}|=a|z_{\{2:i\}}|$. Thus,

[TABLE]

Noting that $a=y_{nn}$, we have for $M=1,\dots,n-1$,

[TABLE]

Similarly, we show that $\delta_{\underline{s}}^{(M)}(\eta)=\alpha^{s_{1}}\delta_{(s_{2},\ldots,s_{n})}^{(M)}(x)$ for $M\geq 2$ and that $\delta_{\underline{s}}^{(M)}(\eta)=\alpha^{s_{n}}\delta_{\underline{s}}^{(M)}(x)$ for all $M\leq n-1$.

Let $\eta=\Psi(\alpha,\beta,x)$ and $\tilde{\eta}=\tilde{\Psi}(\alpha,\beta,x)$. For $n=2$, we have

[TABLE]

For $n>2$, using (10), we have

[TABLE]

The proof of the second part is analogous.

The proof is by direct computation.

4 Laplace transform of generalized power functions on $Q_{G}$ and $P_{G}$

Theorem 4.1

*For all $n\geq 1$, for all  $1\leq M\leq n$  and for all $y\in P_{A_{n}}$, the integral

$\int_{Q_{A_{n}}}e^{-\operatorname{tr}(y\eta)}\delta_{\underline{s}}^{(M)}(\eta)\varphi_{A_{n}}(\eta)d\eta$ converges if and only if $s_{i}>\frac{1}{2}$ for all $i\neq M$, and $s_{M}>0$. In this case, we have*

[TABLE]

Proof

We will proceed by induction on the number $n$ of vertices. For $n=1$, we have the gamma integral that converges if and only if $s>0$, so that

[TABLE]

Now assume that the assertion holds for a graph with $n-1$ vertices.

Case $M>1$. Let $y=\Phi_{n}(a,b,z)$ and let us make the change of variable $\eta=\Psi_{n}(\alpha,\beta,x)$. The induction hypothesis gives

[TABLE]

if and only if $s_{i}>\frac{1}{2}$ for all $i\neq M$, and $s_{M}>0$. By Lemma 3, the change of variable $\eta=\Psi_{n}(\alpha,\beta,x)$ gives $d\eta=x_{22}d\alpha d\beta dx$. Thus, we have

[TABLE]

where we used parts 3 and 1 of Lemma 3. Now, using the Gaussian integral

[TABLE]

and the gamma integral

[TABLE]

that is finite if and only if $s_{1}>\frac{1}{2}$, we get

[TABLE]

Finally, using Formulas (23) and (13) completes the proof in the case $M>1$.

Case $M=1$. Let $y=\tilde{\Phi}_{n}(a,b,z)$ and let us make the change of variable $\eta=\tilde{\Psi}_{n}(\alpha,\beta,x)$. The proof is similar.

Theorem 4.2

*For all $n\geq 1$, for all  $1\leq M\leq n$  and  for all $\eta\in Q_{A_{n}}$, the integral

$\int_{P_{A_{n}}}e^{-\operatorname{tr}(y\eta)}\Delta_{\underline{s}}^{(M)}(y)dy$ converges if and only if $s_{i}>-\frac{3}{2}$ for all $i\neq M$, and $s_{M}>-1$. In this case, we have*

[TABLE]

Proof

Similar to the proof of Theorem 4.1 using Proposition 2 and Lemma 3.

On a convex cone $\Omega$ we define the characteristic function $\varphi_{\Omega}$ of the cone as the Laplace transform of the Lebesgue measure of the dual cone. The measure $\varphi_{\Omega}(x)dx$ is called the canonical measure of $\Omega$. It is invariant by the linear automorphisms of $\Omega$ Faraut and Korányi (1994).

Corollary 2

$\varphi_{Q_{A_{n}}}=\mathrm{const}\,.\,\varphi_{{A_{n}}}$.

Proof

The result,  $\left(\frac{4}{\pi^{2}}\right)^{\frac{n-1}{2}}\int_{P_{A_{n}}}e^{-\operatorname{tr}(y\eta)}dy\,=\,\varphi_{A_{n}}(\eta)$, is obtained by substituting $\underline{s}=(0,\ldots,0)$ into Theorem 4.2.

Remark 1

Formulas (22) and (25) may seem similar but in (25) the integrand does not contain the characteristic function of the cone $P_{A_{n}}$. This function is unknown except for $A_{4}$ when it is not a power function (Letac and Massam, 2007, Prop.3.2).

5 Wishart exponential families on $Q_{G}$

Let us define the Riesz measure $R_{\underline{s}}^{(M)}$ on $Q_{G}$ by

[TABLE]

where $C_{\underline{s}}^{-1}=\pi^{(n-1)/2}\left(\underset{i\neq M}{\prod}\Gamma(s_{i}-\frac{1}{2})\right)\Gamma(s_{M})$. Therefore, from Theorem 4.1, the Laplace transform of the measure $dR_{\underline{s}}^{(M)}$ is given for all $s_{i}>\frac{1}{2}$, $i\neq M$ and $s_{M}>0$ by

[TABLE]

Wishart natural exponential family $\gamma_{\underline{s},y}^{(M)}$ on $Q_{G}$ is, by definition, generated by the Riesz measure $dR_{\underline{s}}^{(M)}$. The density function of the Wishart distribution on $Q_{G}$ is given by

[TABLE]

The Laplace transform of $\gamma_{\underline{s},y}^{(M)}(dx)$ is

[TABLE]

The family $\gamma_{\underline{s},y}^{(M)}$ does not depend on the normalization of the Riesz measure.

5.1 Mean and covariance of the Wishart distributions on $Q_{G}$

In this subsection we derive a formula for the mean of the Wishart exponential family on the cones $Q_{G}$. It is known from the general theory of exponential families of distributions, that the mean of $\gamma^{(M)}_{\underline{s},y}$ is obtained by differentiation with respect to $y$ of the Laplace transform of the Riesz measure:

[TABLE]

For all matrix $A$ in $Z_{G}$ and a subset $B\subset V$ of the set of vertices $V$ of $G$ we note $(A_{B})^{0}$ the matrix in $Z_{G}$ such that $(A_{B})^{0}_{ij}=\begin{cases}A_{ij}&\quad\textnormal{if}\quad{i,j\in V},\\ 0&\textnormal{otherwise}.\end{cases}$

Proposition 3

The mean function of the Wishart family $\gamma_{\underline{s},y}^{(M)}$ on $Q_{G}$ is equal to

[TABLE]

Proof

Use formulas (3), (29) and  $\operatorname{grad}\log|y_{A}|=\left((y_{A})^{-1}\right)^{0}$.

Proposition 4

For all $y\in P_{G}$, we have

[TABLE]

where the constant $\kappa(\underline{s})$ is $\sum_{i=1}^{n}s_{i}-(n-M)s_{M}$.

Proof

Observe that by (3), for any $c>0$,  $\Delta_{-\underline{s}}^{(M)}(cy)=c^{-\kappa(\underline{s})}\Delta_{-\underline{s}}^{(M)}(y).$ By (29), $\langle m^{(M)}_{\underline{s}}(y),y\rangle=-\langle{\rm grad}_{y}\log\Delta^{(M)}_{-\underline{s}}(y),y\rangle$. Set $F(y)=\log\Delta^{(M)}_{-\underline{s}}(y)$. By the chain rule, \langle{\rm grad}_{y}F(y),y\rangle=\frac{d}{dt}F(ty)\big{|}_{t=1}. The map $t\rightarrow F(ty)=\log\varphi(t),\operatorname{\mathbb{R}}^{+}\rightarrow\operatorname{\mathbb{R}},$ where $\varphi(t)=\Delta^{(M)}_{-\underline{s}}(ty)$, satisfies $\varphi(ct)=c^{-\kappa(\underline{s})}\varphi(t)$. Hence $\varphi(c)=c^{-\kappa(\underline{s})}\varphi(1)$ and \frac{d}{dt}F(ty)\big{|}_{t=1}=\frac{\varphi^{\prime}(1)}{\varphi(1)}=-\kappa(\underline{s}). Thus $\langle{\rm grad}_{y}F(y),y\rangle=-\kappa(\underline{s})$ and the result follows.

Differentiating the mean function gives the covariance function. For $A\in S_{n}$, let $\operatorname{\mathbb{P}}(A):Z_{G}\rightarrow{I_{G}}$ be the quadratic operator defined by $\operatorname{\mathbb{P}}(A)u={\pi}(AuA),\,\,\,u\in Z_{G}$.

Proposition 5

The covariance function of the Wishart family $\gamma_{\underline{s},y}^{(M)}$ on $Q_{G}$ is equal

[TABLE]

5.2 Inverse mean map

In the study of the exponential family $(\gamma^{(M)}_{\underline{s},y})_{y\in P_{G}}$ it is important to determine explicitly the inverse of the mean map $\psi^{(M)}_{\underline{s}}:\ m=m^{(M)}_{\underline{s}}(y)\mapsto y,$ which we refer to as the inverse mean map in the sequel. The following theorem is known for Wishart exponential families on homogeneous cones Ishi (2014). Surprisingly, it is also true on $Q_{G}$.

Theorem 5.1

The inverse mean map $\psi^{(M)}_{\underline{s}}$ is given by the formula

[TABLE]

The proof consists in following steps:

1. One shows that there exists a constant $c_{\underline{s}}$ depending only on $\underline{s}$ such that for any $y\in P_{G}$

[TABLE]

This is done in Proposition 6 below.

2. One uses a differential calculus argument, based on the Legendre transform methods.

Proposition 6

The following formula holds for any $y\in P_{G}$ and $\underline{s}\in\mathbb{R}^{n}$:

[TABLE]

The proof of Proposition 6 will need a generalization of Lemma 2, where coefficients of inverse matrices of principal submatrices $y_{\{1:k\}}$ (or of $y_{\{k:n\}}$) are simultanously considered. Define for $y\in P_{G}$,  $\eta^{(k)}=(y_{\{1:k\}})^{-1},\ \ \ \eta^{[k]}=(y_{\{k:n\}})^{-1}.$ The rows and the columns of the matrix $\eta^{(k)}$ are numbered by $i=1,\ldots,k$ and the rows and the columns of the matrix $\eta^{[k]}$ are numbered by $i=k,\ldots,n$.

Lemma 4

Let $y\in P_{G}$.

For all $i\in V$ and $k,m\geq i+1$ we have

[TABLE] 2. 2.

For all $i\in V$ and $k,m\leq i<n$ we have

[TABLE]

Proof

Similar to the proof of Lemma 2; instead of $y$ use $y_{\{1:k\}}$ or $y_{\{k:n\}}$.

Proof

(of Proposition 6) We will deal with $\delta^{(M)}_{\underline{s}}(m^{(M)}_{\underline{s}}(y))=\delta^{\prec_{M}}_{\underline{s}}(m^{(M)}_{\underline{s}}(y))$ where the order $\prec_{M}$ was defined in (6). By formula (30) and by the definition of $\delta^{\prec_{M}}_{\underline{s}}$ we obtain that $\delta^{\prec_{M}}_{\underline{s}}(m_{\underline{s}}(y))$ equals

[TABLE]

where $x_{i}=(s_{i}-s_{i+1})\eta^{(i)}_{ii},\,\,a_{i}=\sum_{k=i+1}^{M-1}(s_{k}-s_{k+1})\eta_{ii}^{(k)}+s_{M}\eta_{ii}^{(n)}$,

[TABLE]

and $x^{\prime}_{i}=(s_{i}-s_{i-1})\eta^{[i]}_{ii}$. Let us first compute the factors $\begin{vmatrix}x_{i}+a_{i}&b_{i}\\ b_{i}&c_{i}\end{vmatrix}/c_{i}$ for $i=1,\ldots,M-1$. We will show that

[TABLE]

We have ${\displaystyle\frac{1}{c_{i}}\begin{vmatrix}x_{i}+a_{i}&b_{i}\\ b_{i}&c_{i}\end{vmatrix}=x_{i}+\frac{1}{c_{i}}\begin{vmatrix}a_{i}&b_{i}\\ b_{i}&c_{i}\end{vmatrix},}$ so in order to prove (35), it is sufficient to prove that

[TABLE]

In order to prove (36), we first use the multilinearity of the determinant with respect to its columns and we write, using the notation $D_{i}^{k,m}$ from Lemma 4,

[TABLE]

By Part 1 of Lemma 4 we have $D_{i}^{k,m}=|y_{\{1:m\}}|^{-1}|y_{\{1:m\}\backslash\{i,i+1\}}|$, which is independent of the left index $k$. The last fact allows to write

[TABLE]

We factorize the determinants $|y_{\{1:m\}\backslash\{i,i+1\}}|$ and $|y_{\{1:n\}\backslash\{i,i+1\}}|$ in the last sum according to Lemma 1 and we write this sum as

[TABLE]

We have $|y_{\{1:m\}}|^{-1}|y_{\{1:i\}}||y_{\{i+2:m\}}|=\eta^{(m)}_{i+1,i+1}$. By definition of $c_{i}$ we finally obtain

[TABLE]

and formulas (36) and (35) are proved.

A "mirror" proof based on Part 2 of Lemma 4 shows that

[TABLE]

and that  $\delta^{(M)}_{\underline{s}}(m^{(M)}_{\underline{s}}(y))=\prod_{i=1}^{n}s_{i}^{s_{i}}\prod_{i=1}^{M-1}(\eta^{(i)}_{ii})^{s_{i}}(\eta^{(n)}_{MM})^{s_{M}}\prod_{i=M+1}^{n}(\eta^{[i]}_{ii})^{s_{i}}.$

Recall that

[TABLE]

so we deduce, using formula (3) that

[TABLE]

Applying Theorem 2.1, we see that $\delta^{(M)}_{\underline{s}}(m^{(M)}_{\underline{s}}(y))=\prod_{i=1}^{n}s_{i}^{s_{i}}\delta_{\underline{s}}^{(M)}(\pi(y^{-1})).$

Proof

(of Theorem 5.1). By Proposition 6 and formula (29) we have for $y\in P_{G}$

[TABLE]

where

[TABLE]

We know that $m^{(M)}_{\underline{s}}:y\mapsto m$ is a diffeomorphism. Our goal is to investigate the inverse map

[TABLE]

Using Legendre-Fenchel transform methods, we get

[TABLE]

where the function $g$ is defined by

[TABLE]

Thus $g(m)=\langle m,y\rangle+\log\delta^{(M)}_{\underline{s}}(m)$, so that

[TABLE]

We use now Part 2 of Proposition 4. It follows that

[TABLE]

Corollary 3

The inverse mean map $\psi_{\underline{s}}^{(M)}:Q_{G}\rightarrow P_{G}$ is given by

[TABLE]

Proof

The result is obtained by computing the gradient of $\log\delta_{\underline{s}}^{(M)}(m)$, as indicated in (32). We use the formula (4).

The Lauritzen formula Lauritzen (1996) is an explicit formula for a bijection between $Q_{G}$ and $P_{G}$. It states that for all $x\in Q_{G}$, the unique $y\in P_{G}$ such that $\pi(y^{-1})=x$ is given by

[TABLE]

Setting $s_{1}=\ldots=s_{n}=1$ in formula (30) for the mean function, we get

[TABLE]

Thus,

[TABLE]

is the Lauritzen formula. Indeed, for $s_{1}=\ldots=s_{n}=1$, formula (3) gives

[TABLE]

Thus we found a new proof of the Lauritzen formula, based on the observation that the Lauritzen map is the inverse mean map for $\underline{s}={\bf 1}=(1,1,\ldots,1)$. At the same time we find an infinite number of explicit isomorphisms from $Q_{G}$ onto $P_{G}$, given by

the inverse mean maps $\psi_{\underline{s}}^{(M)}$. It is an essential generalization of the Lauritzen formula. Each map $\psi_{\underline{s}}^{(M)}$ is a generalized Lauritzen map.

5.3 Variance function

5.3.1 Properties of lower-upper $M$-triangular matrices

Here, we define and prove basic properties of lower-upper $M$-triangular matrices, that we will denote by LU(M). They are very important in proofs of this section.

Definition 5

A matrix $T$ is said to be an LU(M) triangular matrix if for all $i<M$, $T_{ij}=0$ if $j>i$ and for all $i>M$, $T_{ij}=0$ if $i>j$.

In particular, $T$ is an LU(n) triangular matrix if and only if it is lower triangular, and $T$ is an LU(1) triangular matrix if and only if it is upper triangular. An LU(M) triangular matrix $T$ is a succession of an $M\times M$ lower triangular matrix $L=T_{\{1:M\}}$ and an $(N-M)\times(N-M)$ upper triangular matrix $U=T_{\{M:n\}}$ with diagonal term $T_{MM}$ in common. We write $T=s(L,U)$.

T=$$T_{MM}$$L$$U

Proposition 7

$s(L,U)s(L^{\prime},U^{\prime})=s(LL^{\prime},UU^{\prime})$. 2. 2.

If $s(L,U)$ is invertible, then $\left(s(L,U)\right)^{-1}$ is also an LU(M) triangular matrix and

$\left(s(L,U)\right)^{-1}=s(L^{-1},U^{-1})$. 3. 3.

The set of $LU(M)$ triangular matrices is a group.

Proof

1. Part 1 is proved by block matrix multiplication. Part 2 is straightforward using Part 1. Part 3 follows from Parts 1 and 2.

Lemma 5

Let $S$ and $T$ be LU(M) triangular $n\times n$ matrices.

2. (a)

Let $A=K^{0}$ with $K=A_{\{1:k\}}$. If $k\leq M-1$, then ${}^{t}SAT=\left({}^{t}S_{\{1:k\}}KT_{\{1:k\}}\right)^{0}$. 2. (b)

Let $B=K^{0}$ with $K=B_{\{k:n\}}$. If $k\geq M+1$, then ${}^{t}SBT=\left({}^{t}S_{\{k:n\}}KT_{\{k:n\}}\right)^{{0}}.$ 2. 2.

Let $A$ be an $n\times n$ matrix. Then $(TA{}^{t}S)_{\{1:i\}}=T_{\{1:i\}}A_{\{1:i\}}{}^{t}S_{\{1:i\}}$ for $i\leq M-1$, and $(TA{}^{t}S)_{\{i:n\}}=T_{\{i:n\}}A_{\{i:n\}}{}^{t}S_{\{i:n\}}$ for $i\geq M+1$. 3. 3.

If $T$ is invertible, then

1. (a)

$(T_{\{1:k\}})^{-1}=(T^{-1})_{\{1:k\}}$* for all $k\leq M-1$;* 2. (b)

$(T_{\{k:n\}})^{-1}=(T^{-1})_{\{k:n\}}$* for all $k\geq M+1$.*

Proof

Part 1 is straightforward using block matrix multiplication and Part 1 of Lemma 7 in Appendix; for Part 2 just imagine which lines and columns intervene in the computation); Part 3 follows from Part 2 of Proposition 7 and Part 3 of Lemma 7.

Proposition 8

For all $y\in P_{A_{n}}$, for all $1\leq M\leq n$, there exists an LU(M) triangular matrix $T$ satisfying $T_{ij}=0$ if $i\not\sim j$ and such that $y={T\,{}^{t}T}$.

Proof

We will proceed by induction on $n$. The statement is obviously true for $n=1$. Let us assume that the statement is true for $n-1$. Let $y\in P_{A_{n}}$ and $M\neq 1$. Let us write $y=\Phi_{n}(a,b,z)$ with $z\in P_{A_{n-1}}$. The induction assumption implies that there exists $V$ an $(n-1)\times(n-1)$ LU(M) triangular matrix such that $V_{ij}=0$ if $i\not\sim j$ and such that $z=V\,{}^{t}V$. Let us write

[TABLE]

$T$ is LU(M) triangular satisfying $T_{ij}=0$ if $i\not\sim j$ and $y=T\,{}^{t}T$.

For $M=1$, we use $y=\tilde{\Phi}_{n}(a,b,z)$ with $z\in P_{A_{n-1}}$.

5.3.2 Two formulas for the variance function

Let $m\in Q_{G}$. We note $\hat{m}\in S^{+}_{n}$ the unique symmetric positive definite matrix verifying ${\pi}(\hat{m})=m,\ \ \hat{m}^{-1}\in P_{G}.$ Define $y=\psi_{\underline{s}}^{(M)}(m)\in P_{G}$. Decompose $y={T\,{}^{t}T}$, with $T$ an LU(M) triangular matrix such that $T_{ij}=0$ when $i\not\sim j$.

Lemma 6

We have

[TABLE]

Proof

Note that $y=\psi_{\underline{s}}^{(M)}(m)$ is equivalent to $m=m_{\underline{s}}^{(M)}(y)$. The formula of the mean function (30) gives $m=\pi(Z)$, where

[TABLE]

Using Part 2 of Lemma 5, we have $y_{\{1:i\}}=T_{\{1:i\}}I_{\{1:i\}}\,^{t}T_{\{1:i\}}$ for $i\leq M-1$. By Part 3 of Lemma 5, we get $(y_{\{1:i\}})^{-1}=\,^{t}(T^{-1})_{\{1:i\}}I_{\{1:i\}}(T^{-1})_{\{1:i\}}$. Finally, using Part 1 of Lemma 5, we obtain

[TABLE]

Similarly, we have

[TABLE]

Thus,

[TABLE]

Therefore, $Z$ is positive definite and $Z^{-1}=T\begin{pmatrix}s_{1}^{-1}&\ldots&0\\ &\ddots&\\ 0&\ldots&s_{n}^{-1}\end{pmatrix}{}^{t}T\in P_{A_{n}}.$ Indeed, for all $i<i+1<j$, we have $(Z^{-1})_{ij}=\sum_{k=1}^{n}T_{ik}T_{jk}s_{k}^{-1}.$ Since $T_{ik}=0$ for $|k-i|>1$, $(Z^{-1})_{ij}=T_{i,i-1}T_{j,i-1}s_{i-1}^{-1}+T_{ii}T_{ji}s_{i}^{-1}+T_{i,i+1}T_{j,i+1}s_{i+1}^{-1}.$ But since $|j-i|>1$, we have $T_{j,i-1}=0=T_{ji}$ and $(Z^{-1})_{ij}=T_{i,i+1}T_{j,i+1}s_{i+1}^{-1}$. Now since $T$ is LU(M), we have $T_{i,i+1}T_{j,i+1}=0$. In fact, $T_{i,i+1}=0$ for $i\leq M-1$ and $T_{j,i+1}=0$ for $i\geq M$. In conclusion, we have shown that $m=\pi(Z)$ with $Z^{-1}\in P_{A_{n}}$, which implies $Z=\hat{m}$.

The following Proposition derives the formula for the variance function $V(m)$ which, for each fixed $m\in Q_{G}$ is a continuous operator $V(m):Z_{G}\rightarrow{I_{G}}$ Casalis and Letac (1996). Recall that $\operatorname{\mathbb{P}}(A):Z_{G}\rightarrow{I_{G}}$ is the quadratic operator defined by $\operatorname{\mathbb{P}}(A)u={\pi}(AuA)$. For $A,B\in S_{n}$, let $\operatorname{\mathbb{P}}(A,B)u=\frac{1}{2}{\pi}(AuB+BuA)$. For all $m\in Q_{G}$ and $I\subset V$, we note

[TABLE]

Proposition 9

The variance function $V(m)$ of a Wishart NEF on $Q_{G}$ is equal to

[TABLE]

Proof

The variance function is given for all $m\in Q_{A_{n}}$ by $V(m)=v(\psi_{\underline{s}}^{(M)}(m))$, where $v(y)$ is given by (31). Let $y=\psi_{\underline{s}}^{(M)}(m)=T\,^{t}T$, where $T$ is LU(M). From Lemma 6, we have

[TABLE]

Using Lemma 5, we get

[TABLE]

and

[TABLE]

Thus, for all  $2\leq i\leq M-1$, we have

[TABLE]

and for all $n-1\geq i\geq M+1$, we have

[TABLE]

where $e_{i}$ is the matrix with $e_{ii}=1$ and $e_{ij}=0$ for all $i\neq j$. Observing that $(I_{\{1:i\}})^{0}=\sum_{k=1}^{i}e_{i}\quad\textnormal{and}\;(I_{\{i:n\}})^{0}=\sum_{k=i}^{n}e_{i},$ and using (45) and (51), we obtain for $i\leq M-1$

[TABLE]

Similarly, using (46) and (52), we obtain for $i\geq M+1$,

[TABLE]

We also observe that

[TABLE]

Thus, by (51), (52) and (53), we get

[TABLE]

Substituting these expressions of $[(y_{\{1:i\}})^{-1}]^{0}$, $y^{-1}$ and $[(y_{\{i:n\}})^{-1}]^{0}$  into $v(y)$ given by (31), we obtain the stated result.

We prove now a much simpler formula for the variance function on $Q_{G}$, surprisingly similar to the variance function on a homogeneous cone, in particular on the symmetric cone $S^{+}_{n}$ (cf. Graczyk et al (2016a)).

Theorem 5.2

The variance function of the Wishart exponential family $\gamma_{\underline{s},y}^{(M)}$ is

[TABLE]

where $M_{\{1:i\}}$ and $M_{\{i:n\}}$ are defined in (47).

Proof

Using $\operatorname{\mathbb{P}}(a-b)=\operatorname{\mathbb{P}}(a)+\operatorname{\mathbb{P}}(b)-2\operatorname{\mathbb{P}}(a,b)$, we see that (55) is equivalent to

[TABLE]

We show that the right hand sides of (9) and (5.3.2) are the same. Below, we expand (9) using $\operatorname{\mathbb{P}}(a+b)=\operatorname{\mathbb{P}}(a)+\operatorname{\mathbb{P}}(b)+2\operatorname{\mathbb{P}}(a,b)$ and compute the coefficients in the expanded formula. Note that for all $Z\in Z_{G}$, $\operatorname{\mathbb{P}}(M_{\{1:i\}},M_{\{k:n\}})Z=0$ for all $i\leq M-1$ and $k\geq M+1$, since ${Z_{\{1:i\},\{k:n\}}}=0$.

For a fixed $r\leq M-1$, the coefficient of $\operatorname{\mathbb{P}}({M_{\{1:r\}}})$ is

[TABLE]

By a mirror argument, for a fixed $r\geq M+1$, the coefficient of $\operatorname{\mathbb{P}}({M_{\{r:n\}}})$ is $\frac{1}{s_{r-1}}-\frac{1}{s_{r}}$. On the other hand, the coefficient of $\operatorname{\mathbb{P}}(\hat{m})$ is $\frac{1}{s_{M}}$.

For a fixed $r$, the coefficient of $\operatorname{\mathbb{P}}(\hat{m},M_{\{1:r\}})$ is $\frac{1}{s_{r}}-\frac{1}{s_{r+1}}$ if $r\leq M-1$, and the coefficient of $\operatorname{\mathbb{P}}(\hat{m},M_{\{r:n\}})$ is $\frac{1}{s_{r}}-\frac{1}{s_{r-1}}$ if $r\geq M+1$. Moreover, if $k<r\leq M-1$, the coefficient of $\operatorname{\mathbb{P}}(M_{\{1:r\}},\,M_{\{1:k\}})$ is

[TABLE]

By a mirror argument, for a fixed $M+1\leq k<r$, the coefficient of $\operatorname{\mathbb{P}}(M_{\{k:n\}},{M_{\{r:n\}}})$ is [math].

Remark 2

$\hat{m}$* is easy to compute, using, for non adjacent $i$ and $j$ the formula

$\hat{m}_{ij}=m_{i,V\setminus\{i,j\}}(\hat{m}^{-1}_{V\setminus\{i,j\},V\setminus\{i,j\}})m_{V\setminus\{i,j\},j}.$ (Letac and Massam, 2007, p.1279).*

In the next Corollary, we consider $\underline{s}=p\mathbf{1}$, $p>1/2$. We note that $\delta_{p\mathbf{1}}^{(M)}$ and $\gamma_{p\mathbf{1},\,y}^{(M)}:=\gamma_{p,y}$ do not depend on $M$.

Corollary 4

The variance function of the Wishart exponential family $\gamma_{p,y}$ is

[TABLE]

5.3.3 A relation between the inverse mean map and $m_{\frac{1}{\underline{s}}}$

Recall that for the classical Wishart exponential families $W_{s{\bf 1},y}$ on the symmetric cone $Sym^{+}_{n}$ the bijection between the cone $Q_{G}$ and $P_{G}$ is given by $L(m)=m^{-1}$. The mean map is $m_{s}(y)=sy^{-1}$ and the inverse mean map $\psi_{s}(m)=sm^{-1}$. It follows that

[TABLE]

that is, the maps $\psi_{s}$ and $m_{\frac{1}{s}}$ are intertwined by the bijection $L$.

An analogous property holds on the cone $Q_{A_{n}}$, with the intertwiner given by the Lauritzen map. The bijection $L:Q_{A_{n}}\rightarrow P_{A_{n}}$ is the Lauritzen map $L(m)=(\hat{m})^{-1}$. Its inverse $L^{-1}:P_{A_{n}}\rightarrow Q_{A_{n}}$ is $L^{-1}(y)=\pi(y^{-1})$.

Proposition 10

The inverse mean map $\psi_{\underline{s}}^{(M)}:Q_{G}\rightarrow P_{G}$ satisfies

[TABLE]

Equivalently, for any $m\in Q_{G}$,   $\pi(\psi_{\underline{s}}^{(M)}(m)^{-1})=m_{\frac{1}{\underline{s}}}^{(M)}(\hat{m}^{-1}).$

[TABLE]

Proof

Using formula (30) of the mean function and definition (47) of $M_{\{1:i\}}$ and $M_{\{i:n\}}$, we see that $m_{\frac{1}{\underline{s}}}^{(M)}(\hat{m}^{-1})$ equals

[TABLE]

Confronting this result with (5.3.2), we obtain   $m_{\frac{1}{\underline{s}}}^{(M)}(\hat{m}^{-1})={\pi}\left(\psi_{\underline{s}}^{(M)}(m)^{-1}\right).$

5.4 Quadratic construction of Riesz measures and Wishart distributions on $Q_{G}$

Let $I\subset\{1,\ldots,n\}$. We define $|I|$-dimensional subspaces $W_{I}$ of $\operatorname{\mathbb{R}}^{n}$ by

[TABLE]

Denote by $q^{I}$ the quadratic map $q^{I}(x)=x{}^{t}x$ from $W_{I}$ into Sym$(n,\operatorname{\mathbb{R}})$ and by $q^{I}_{\ast}$ its projection onto ${I_{G}}$, i.e. $q^{I}_{\ast}={\pi}\circ q^{I}$. The maps $q^{I}_{\ast}$ are clearly $Q_{G}$-positive (submatrices $y_{I}$ of a positive definite matrix $y$ are positive definite for any ${I}\subset\{1,\ldots,n\}).$ In Graczyk and Ishi (2014), p.322, Riesz measures $\mu_{q}$ associated to a quadratic map $q$ were defined and their Laplace transform computed. Recall that the measure $\mu_{q^{I}_{\ast}}$ is the image of the Lebesgue measure on $W_{I}$ by $q^{I}_{\ast}$ and that its Laplace transform equals

[TABLE]

When $I=\{1,\ldots,k\}$, we write $q^{I}_{\ast}=q^{k}_{\ast}$. When $I=\{k,\ldots,n\}$, we write $q^{I}_{\ast}=\tilde{q}^{k}_{\ast}$.

Fix $M\in\{1,\ldots,n\}$. We define the set $B_{M}$ of basic quadratic maps for the Riesz $R_{\underline{s}}^{(M)}$ and Wishart $\gamma^{(M)}_{\underline{s},y}$ families on $Q_{G}$ by $B_{M}=\{q^{1}_{\ast},\ldots,q^{M-1}_{\ast},q^{n}_{\ast},\tilde{q}^{M+1}_{\ast},\ldots,\tilde{q}^{n}_{\ast}\}$. Note that the basic quadratic maps with values in $Q_{G}$ are different for each fixed $M=1,\ldots,n$.

Proposition 11

Let $\sigma_{i}\in\operatorname{\mathbb{R}},i=1,\ldots,m$. A virtual quadratic map

$\hskip 56.9055ptq^{\underline{\sigma}}_{\ast}=\sum_{i<M}^{\oplus}(q^{i}_{\ast})^{\oplus\sigma_{i}}\oplus(q^{n}_{\ast})^{\oplus\sigma_{M}}\oplus\sum_{i>M}^{\oplus}(\tilde{q}^{i}_{\ast})^{\oplus\sigma_{i}}.$

exists if there exists $\underline{s}$ satisfying $s_{i}>\frac{1}{2},i\neq M$, $s_{M}>0$ and

[TABLE]

Proof

We compare the Laplace transform of $\mu_{q^{\underline{\sigma}}_{\ast}}$, computed thanks to (57), with (27). As a result, we see that there exists a constant $c>0$ such that $R_{\underline{s}}^{(M)}=c\mu_{q^{\underline{\sigma}}_{\ast}}$.

Thus all the Riesz $R_{\underline{s}}^{(M)}$ measures on $Q_{G}$ defined in this paper are obtained as virtual or true (i.e. for $\sigma_{i}\in\operatorname{\mathbb{N}}$) quadratic Riesz families, with basic maps from $B_{M}$.

Observe that by the quadratic construction, we can obtain absolutely continuous Riesz measures on $Q_{G}$ not belonging to $\cup_{M}\{R_{\underline{s}}^{(M)}\}$, e.g. when $n=3$, consider $\mu_{q}$ associated to the quadratic map $q=q_{\ast}^{2}\oplus(q_{\ast}^{3})^{\oplus 2}\oplus\tilde{q}_{\ast}^{2}$.

5.4.1 Relation to missing data

Let us mention a nice application of the (true) quadratic Wishart distributions constructed from the basic set $B_{M}$, as laws of the Maximum Likelihood Estimators for the covariance matrix $\Sigma$ restricted to the terms $(\Sigma_{ij})_{i\sim j}$, i.e. to the three neighbour diagonals, in a two-sided monotonous missing data problem, when the sample contains: $\sigma_{i}$ observations of $(X_{1},\ldots,X_{i})$, $i<M$,  $\sigma_{k}$ observations of $(X_{k},\ldots,X_{n})$, $k>M$ and $\sigma_{M}$ observations of the complete $n$-dimensional Gaussian character $(X_{1},\ldots,X_{n})$.

5.5 Higher order moments of Wishart families on $Q_{A_{n}}$

Thanks to the identification of Wishart families $\gamma^{(M)}_{\underline{s},y}$ with quadratically constructed Wishart distributions $\gamma_{q^{\underline{\sigma}}_{\ast}}$ in Section 5.4, we can compute moments of any order $N$ of a Wishart random variable $X$ on $Q_{A_{n}}$.

Theorem 5.3

Let $X$ be a $Q_{A_{n}}$-valued random variable with the Wishart law $\gamma^{(M)}_{\underline{s},y}$. Let $z^{(1)},z^{(2)},\ldots z^{(N)}\in Z_{G}$. Then, denoting by $C(\pi)$ the set of cycles of a permutation $\pi\in S_{N}$, the $N$-th moment ${\bf E}(\langle X,z^{(1)}\rangle\ldots\langle X,z^{(N)}\rangle)$ equals

[TABLE]

Proof

We apply Theorem 2.13 from Graczyk and Ishi (2014) and formula (58).

Corollary 5

If $\underline{s}=s{\bf 1}$, $s>\frac{1}{2}$, then $\gamma^{(M)}_{\underline{s},y}=\gamma_{s,y}$ does not depend on $M$. Moreover, for $X$ with law $\gamma_{s,y}$, we have

[TABLE]

Example. For any graph $A_{n}$ and $N=3$ we get for $X$ with law $\gamma_{s,y}$:

${\bf E}(\langle X,z^{(1)}\rangle\langle X,z^{(2)}\rangle\langle X,z^{(3)}\rangle)=s^{3}\prod_{j=1}^{3}\operatorname{tr}(y^{-1}z^{(j)})+\\ s^{2}[\operatorname{tr}y^{-1}z^{(1)}y^{-1}z^{(2)}\operatorname{tr}y^{-1}z^{(3)}+\operatorname{tr}y^{-1}z^{(1)}y^{-1}z^{(3)}\operatorname{tr}y^{-1}z^{(2)}+\\ \operatorname{tr}y^{-1}z^{(2)}y^{-1}z^{(3)}\operatorname{tr}y^{-1}z^{(1)}]+s[\operatorname{tr}\prod_{j=1}^{3}y^{-1}z^{(j)}+\operatorname{tr}y^{-1}z^{(1)}y^{-1}z^{(3)}y^{-1}z^{(2)}].$

6 Wishart exponential families on the cone $P_{G}$.

A measure $\tilde{R}$ on $P_{G}$ is said to be a Riesz measure if, for some $1\leq M\leq n$, $s_{M}>-1$ and $s_{i}>-3/2$, $i\neq M$,   its Laplace transform is given by

[TABLE]

From formula (22), the measure $\tilde{R}_{\underline{s}}^{(M)}(dy)=C_{\underline{s}}\Delta_{\underline{s}}^{(M)}(y)dy$, where

[TABLE]

is a Riesz measure. The exponential family of generated by $\tilde{R}_{\underline{s}}^{(M)}$ will be called the exponential family of Wishart distributions on $P_{G}$. Its density function is

[TABLE]

Its Laplace transform is

[TABLE]

6.1 Mean and covariance

Theorem 6.1

The mean function of the Wishart exponential family on $P_{G}$ is for all $s_{i}>-\frac{3}{2}$ and $x\in Q_{G}$,

[TABLE]

The covariance function $\tilde{v}(x):I_{G}\to Z_{G}$ of the Wishart exponential family on $P_{G}$ equals

[TABLE]

where we identify $I_{G}$ with $Z_{G}$ by the trace inner product.

Proof

We have $\tilde{m}_{\underline{s}}^{(M)}(x)=-\operatorname{grad}\log L_{\mu_{\underline{s}}^{(M)}}(x)=-\operatorname{grad}\log\delta_{-\underline{s}}^{(M)}(x)\varphi_{Q_{A_{n}}}(x).$ The covariance operator is obtained by differentiation of (62).

6.2 Quadratic construction of Riesz measures and Wishart distributions on $P_{G}$

Let $M\in\{1,\ldots n\}$. Suppose $s_{i}>-\frac{3}{2}$, for all $i\neq M$ and $s_{M}>-1$. Let $\theta\in Q_{G}$. In order to establish a relation between quadratically constructed Riesz measures $\tilde{\mu}_{q}$ on $P_{G}$ and the measures $\tilde{R}_{\underline{s}}^{(M)}$ we consider the sets $J_{k}=\{k,k+1\}$ and $J_{k}^{\prime}=\{k\}$. As basic quadratic maps we choose the quadratic $P_{G}$-positive maps $q^{J_{k}}$ and $q^{J_{k}^{\prime}}$. For $\alpha=(\alpha_{1},\ldots,\alpha_{n-1})$ and $\beta=(\beta_{1},\ldots,\beta_{n})$ with $\alpha_{i},\beta_{j}\in\operatorname{\mathbb{N}}$ define $q^{\alpha,\beta}=\sum^{\oplus}_{k<n}(q^{J_{k}})^{\oplus\alpha_{k}}\oplus\sum^{\oplus}_{k\leq n}(q^{J_{k}^{\prime}})^{\oplus\beta_{k}}$. The following proposition is easy to prove by comparing $\delta^{(M)}_{-\underline{s}}(\eta)\varphi_{Q_{A_{n}}}(\eta)$ with the Laplace transform

[TABLE]

Proposition 12

Let $M\in\{2,\ldots,n-1\}$. Then there exists a constant $c>0$ such that $c\tilde{R}_{\underline{s}}^{(M)}=\tilde{\mu}_{q^{\alpha,\beta}}$ if and only if $\alpha_{i}/2=s_{i}+3/2$, $i\leq M-1$, $\alpha_{i}/2=s_{i+1}+3/2$, $i\geq M$, $\beta_{1}=0$, $\beta_{i}/2=-s_{i-1}-1,2\leq i\leq M-1,\beta_{M}/2=-s_{M-1}+s_{M}-s_{M+1}-1,\beta_{i}/2=-s_{i+1}-1,M+1\leq i<n,\beta_{n}=0$. For $M=1,n$ the condition $\beta_{M}=0$ is supressed.

Proposition 12 implies easily two following facts.

Corollary 6

All Riesz measures $\tilde{R}_{\underline{s}}^{(M)}$ are equal (up to a factor) to a virtual quadratic Riesz measure $\tilde{\mu}_{q^{\alpha,\beta}}$. 2. 2.

For $n\geq 4$, no true quadratic Riesz measure $\tilde{\mu}_{q^{\alpha,\beta}}$, $\alpha_{i},\beta_{j}\in\operatorname{\mathbb{N}}$, belongs (up to a factor) to the set of Riesz measures $\tilde{R}_{\underline{s}}^{(M)}$.

Proof

To prove Part 2, we have conditions $s_{i}+\frac{3}{2}=\alpha_{i^{\prime}}/2$ and $s_{j}+1=-\beta_{j^{\prime}}/2$, so all (except at most one) $s_{i}\geq-1$, and all (except at most one) $s_{j}\leq-3/2$ simultanously.

6.3 Higher order moments of Wishart families on $P_{A_{n}}$

Thanks to Part 1 of Corollary 6, all the moments of the Wishart Exponential Families $\tilde{\gamma}_{\underline{s},\theta}^{(M)}$ can be computed, using Theorem 2.13 from Graczyk and Ishi (2014) and Proposition 12.

Theorem 6.2

Let $Y$ be a $P_{A_{n}}$-valued random variable with the Wishart law $\tilde{\gamma}^{(M)}_{\underline{s},\theta}$. Let $x^{(1)},x^{(2)},\ldots x^{(N)}\in I_{G}$. Then, denoting by $C(\pi)$ the set of cycles of a permutation $\pi\in S_{N}$, the $N$-th moment ${\bf E}(\langle Y,x^{(1)}\rangle\ldots\langle Y,x^{(N)}\rangle)$ equals

[TABLE]

7 Relations with the type I and type II Wishart distributions of Letac and Massam (2007)

In this section we will explain the relation between our work and type 1 and type 2 Wishart distributions constructed by Letac and Massam (2007).

Letac and Massam (2007) introduced, studied and used the function $H(\alpha,\beta,x)$ on $Q_{G}$ as a generalized power function for constructing type I and type II Wishart distributions. The reader is referred to the cited paper for the general definition of the function $H(\alpha,\beta,x)$ as well as for graphical theoretic notions such as cliques, separators and perfect order of cliques (see also Lauritzen (1996)). For our purpose, it is sufficient to recall that for $\alpha\in\operatorname{\mathbb{R}}^{n-1}$ and $\beta\in\operatorname{\mathbb{R}}^{n-2}$

[TABLE]

that the cliques(i.e. the sets of vertices of maximal complete subgraphs) are $\{1,2\},\ldots,\\ \{n-1,n\}$ and the separators $\{2\},\ldots,\{n-1\}$. The definition of the function $H(\alpha,\beta;x)$ does not include any restrictions on the values of the parameter $(\alpha,\beta)$ of dimension $2n-3$.

However, the existence of type I Wishart distributions on $Q_{G}$ is only showed for $(\alpha,\beta)$ belonging to some set $A_{P}$ dependent on a perfect order of cliques $P$, i.e. for $(\alpha,\beta)\in{\mathcal{A}}_{0}=\cup_{P}A_{P}$, where the union is on all perfect order of cliques. Proposition 14 describes this set for $A_{n}$ graphs. It also makes clear a phenomenon observed by Letac and Massam (2007) for the graph $A_{4}$, where there are only two different sets $A_{P}$ although there are $4$ perfect orders of cliques. To prove Proposition 14 we use the following explicit relation between two concepts: perfect orders of cliques used by Letac and Massam (2007) and eliminating orders of vertices used in this work.

Proposition 13

Let $G=A_{n}:1-2-3-\ldots-n$. A clique ordering $C^{\prime}_{1}<\ldots<C^{\prime}_{n-1}$ is perfect if and only if $C^{\prime}_{n-1}\prec\ldots\prec C^{\prime}_{1}$ is an eliminating order on the $A_{n-1}$ graph $G^{\prime}:C_{1}-C_{2}\ldots-C_{n-1}$. There are $2^{n-2}$ perfect orders of cliques on $A_{n}$.

Proof

The proof is in two parts, for the two inclusions of the claimed equality. Both parts are straightforward and based on the definitions of a perfect order of cliques and of an eliminating order on a graph. We omit the details.

Proposition 14

*Let $P^{\prime}:C_{1}^{\prime}<C^{\prime}_{2}<\ldots<C^{\prime}_{n-1}$ and $P^{\prime\prime}:C_{1}^{\prime\prime}<C^{\prime\prime}_{2}<\ldots<C^{\prime\prime}_{n-1}$ be two perfect orders of cliques on $G=A_{n}$. Let $S_{2}^{\prime}$ and $S_{2}^{\prime\prime}$ be the first separators of $P^{\prime}$ and $P^{\prime\prime}$. If $S_{2}^{\prime}=S_{2}^{\prime\prime}$ then $A_{P^{\prime}}=A_{P^{\prime\prime}}$, i.e. the parameter set $A_{P}$ depends only on the first separator $S_{2}$ with respect to the clique order $P$. If $S_{2}=\{M\}$ then the set $A_{P}$ is described by the conditions:

[TABLE]

and

[TABLE]

Thus $\mathcal{A}_{0}=\cup_{P}A_{P}$ is the set of $(\alpha,\beta)$ such that there exists $2\leq M\leq n-1$ for which (64) and (65) are satisfied.

Proof

We use Propositions 1 and 13.

The reference measure $\mu_{G}$ used by Letac and Massam (2007) is, on the cone $Q_{A_{n}}$,

[TABLE]

By (17), we observe that $\mu_{A_{n}}(x)(dx)=\varphi_{Q_{A_{n}}}(x)1_{Q_{A_{n}}}(x)dx$. Namely, the reference measure $\mu_{G}$ is the characteristic measure of the cone $G=Q_{A_{n}}$.

Theorem 7.1

(Letac and Massam (2007) Theorem 3.3) If $(\alpha,\beta)\in\mathcal{A}_{0}$, then, for a constant ${\Gamma_{1}}_{(\alpha,\beta)}$, and for all $y\in P_{A_{n}}$

[TABLE]

The methods of our article give a new simple proof of Theorem 7.1, see the proof of Corollary 7 below.

Let us compare now the functions $H(\alpha,\beta;x)$ and $H(\alpha,\beta;{\pi(y^{-1})})$ with the generalized power functions $\delta_{\underline{s}}^{(M)}$ and $\Delta_{\underline{s}}^{(M)}$.

Proposition 15

Let $\alpha\in\operatorname{\mathbb{R}}^{n-1}$ and $\beta\in\operatorname{\mathbb{R}}^{n-2}$. There exists $\underline{s}\in\operatorname{\mathbb{R}}^{n}$ such that $H(\alpha,\beta;x)=\delta_{\underline{s}}^{(M)}(x)$ if and only if (64) holds for some $2\leq M\leq n-1$.

Then $s_{j}=\alpha_{j}\ {\it if}\ 1\leq j\leq M-1,\ s_{M}=\alpha_{M-1}+\alpha_{M}-\beta_{M}$ and $s_{j}=\alpha_{j-1}\ {\it if}\ M+1\leq j\leq n.$ 2. 2.

Moreover, under the hypothesis of Part 1, we have $H(\alpha,\beta;\pi(y^{-1}))=\Delta_{-\underline{s}}^{(M)}(y)$.

Proof

The equality of $H(\alpha,\beta;x)$ and $\delta_{\underline{s}}^{(M)}(x)$ is easily verified by confronting their definitions (63) and (4). Part 2 follows from Theorem 2.1.

Corollary 7

The type I Wishart distributions indexed by the set ${\mathcal{A}}_{0}$ are equal to the subset $\bigcup_{M=2}^{n-1}(\gamma_{\underline{s},y}^{(M)})_{y\in P_{G}}$ of Wishart NEF families defined in Section 5. Thus they are strictly contained in the set of all Wishart NEF families on $Q_{G}$, equal to $\bigcup_{M=1}^{n}(\gamma_{\underline{s},y}^{(M)})_{y\in P_{G}}$.

Proof

It is a direct application of Proposition 15 and Theorem 4.1. Note that Theorem 4.1 implies Theorem 7.1 of Letac and Massam (2007).

The family of functions $H(\alpha,\beta,x)$ does not contain the power functions $\delta_{\underline{s}}^{(1)}$ or $\delta_{\underline{s}}^{(n)}$. In fact, the last functions contain powers of $n-1$ diagonal elements $x_{ii}$, whereas the function $H(\alpha,\beta,x)$ contains powers of $n-2$ such elements.

Similar comparisons can be done on the cones $P_{G}$. In this case, Letac and Massam (2007) define type II Wishart distributions on $P_{G}$ indexed by a set ${\mathcal{B}}_{0}$, analogous to the set ${\mathcal{A}}_{0}$ for $Q_{G}$. Similar arguments as on the cone $Q_{G}$ lead to

Corollary 8

The type II Wishart distributions on $P_{G}$ indexed by the set ${\mathcal{B}}_{0}$ are equal to the subset $\bigcup_{M=2}^{n-1}(\tilde{\gamma}_{\underline{s},x}^{(M)})_{x\in Q_{G}}$ of Wishart NEF families defined in Section 6. Thus they are strictly contained in the set of all Wishart NEF families on $P_{G}$, equal to $\bigcup_{M=1}^{n}(\tilde{\gamma}_{\underline{s},x}^{(M)})_{x\in Q_{G}}$.

8 Appendix

We list here some properties of triangular matrices, used in proofs.

Lemma 7

Let $A=K^{0}$, where $K=A_{\{1:k\}}$ and let $L$ be lower triangular and $U$ upper triangular $n\times n$ matrices. Then $UAL=\left(U_{\{1:k\}}KL_{\{1:k\}}\right)^{0}.$ 2. 2.

Let $M,L,U$ be matrices $n\times n$, with $L$ lower triangular and $U$ upper triangular. Then, for all $i=1,\ldots,n$, $(LMU)_{\{1:i\}}=L_{\{1:i\}}M_{\{1:i\}}U_{\{1:i\}}$ and $(UML)_{\{i:n\}}=U_{\{i:n\}}M_{\{i:n\}}L_{\{i:n\}}$. 3. 3.

If $T$ is an invertible triangular matrix then $(T_{\{1:k\}})^{-1}=(T^{-1})_{\{1:k\}}$ for all $k=1,\ldots,n$.

All these properties are elementary and easy to prove, by block multiplication of matrices (1,2) or by inverse matrix formula with cofactors (3).