基于Sabot-Tarrès-Zeng积分的多元倒易逆高斯分布
Multivariate reciprocal inverse Gaussian distributions from the Sabot -Tarrès -Zeng integral
摘要：
在Sabot和Tarrès（2015）中，作者明确计算了积分$$STZ_n=\int \exp( -\langle x,y\rangle)(\det M_x)^{-1/2}dx$$，其中$M_x$是具有固定非正非对角系数和对角$(2x_1,\ldots,2x_n)$的$n$阶对称矩阵。积分域是$M_x$正定的$\mathbb{R}^n$的一部分。我们计算更一般的$ b_1\geq 0,\ldots b_n\geq 0$的积分$$GSTZ_n=\int \exp \left(-\langle x,y\rangle-\frac{1}{2}b^*M_x^{-1}b\right)(\det M_x)^{-1/2}dx,$$，我们表明，它导致一个自然的家庭分布在$\mathbb{R}^n$，称为$GSTZ_n$概率定律。这个家庭是稳定的边缘化和条件，它有一些属性，这是多变量版本的熟悉性质的单变量倒数逆高斯分布。我们还证明了，如果图的顶点集$V=\{1,\ldots,n\}$和边集$E$是树，则积分$$\int \exp( -\langle x,y\rangle)(\det M_x)^{q-1}dx$$，其中$q>0,$是可用MacDonald函数$K_q.$计算的
英文摘要：
In Sabot and Tarrès (2015), the authors have explicitly computed the integral $$STZ_n=\int \exp( -\langle x,y\rangle)(\det M_x)^{-1/2}dx$$ where $M_x$ is a symmetric matrix of order $n$ with fixed non positive off-diagonal coefficients and with diagonal $(2x_1,\ldots,2x_n)$. The domain of integration is the part of $\mathbb{R}^n$ for which $M_x$ is positive definite. We calculate more generally for $ b_1\geq 0,\ldots b_n\geq 0$ the integral $$GSTZ_n=\int \exp \left(-\langle x,y\rangle-\frac{1}{2}b^*M_x^{-1}b\right)(\det M_x)^{-1/2}dx,$$ we show that it leads to a natural family of distributions in $\mathbb{R}^n$, called the $GSTZ_n$ probability laws. This family is stable by marginalization and by conditioning, and it has number of properties which are multivariate versions of familiar properties of univariate reciprocal inverse Gaussian distribution. We also show that if the graph with the set of vertices $V=\{1,\ldots,n\}$ and the set $E$ of edges $\{i,j\}'$ s of non zero entries of $M_x$ is a tree, then the integral $$\int \exp( -\langle x,y\rangle)(\det M_x)^{q-1}dx$$ where $q>0,$ is computable in terms of the MacDonald function $K_q.$
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