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Neighbourhoods of independence and associated geometry in manifolds of bivariate Gaussian and Freund distributions. (English) Zbl 1126.53010

Let \( \Omega \) be a measure space, \( \Theta \) an open subset of \( \mathbb R^{n} \), and assume \( (p_{\theta}) \) to be a family of positive functions \( p_{\theta}: \Omega \to \mathbb R \), one for each \( \theta \in \Theta \), such that \( \theta \mapsto p_{\theta}(x) \) is smooth for any \( x \in \Omega \) and \( \int_{\Omega} p_{\theta} = 1 \) for all \( \theta \in \Theta \). More regularity assumptions are to be made if necessary, in particular for the following basic construction. Forming the functions \( \ell := \ln\; p_{\theta} \), one can introduce a Riemannian metric on \( \Theta \) by \( g_{ij} := \int_{\Omega} p_{\theta} \cdot (\partial \ell/\partial \theta^{i}) (\partial \ell/\partial \theta^{j}) = -\int_{\Omega} p_{\theta} \cdot (\partial^{2} \ell/\partial \theta^{i}\partial\theta^{j}) \), the so-called Fisher metric.
The authors study the case where \( \Omega \) is one-dimensional, i.e. \( \Omega \subseteq \mathbb R \), and the \( p_{\theta} \) have the exponential form \( p_{\theta}(x) = \exp(C(x)-\varphi(\theta)+\sum_{i}\theta^{i}F_{i}(x)) \), where \( \varphi \) is called the potential function. In this case, the \( g_{ij} \) turn out to be the coefficients \( \varphi_{ij} \) of the Hessian of \( \varphi \). So a bridge is built to the Riemannian geometry of the Hessian of a smooth convex function \( \varphi: \Theta \to \mathbb R \). In addition to the Levi-Cività-connection \( \nabla \) one may consider its constant multiples \( \nabla^{(\alpha)} := (1-\alpha)/2 \nabla\). In this way, an \( \alpha \)-geometry becomes possible, just arising from these measure theoretic assumptions.
In fact, this setting has a statistical interpretation, \( \Omega \) taking the role of an event space and the \( p_{\theta} \) forming an \(n\)-dimensional family of probability density functions. This is a central piece of a whole theory of differential-geometrical statistics, as comprehensively developed in the book of [S. Amari, Differential-geometrical methods in statistics. Lecture Notes in Statistics 28. Berlin etc.: Springer-Verlag (1985; Zbl 0559.62001)].
In the present paper two statistical distributions are pursued in detail, namely, bivariate Gaussian and Freund distributions. Both fit in the concept of exponential families of density functions with dimensions \( 5 \), resp., \( 4 \). In either case, several geometric quantities, like the \( \alpha \)-curvature tensor, the corresponding Ricci tensor, etc., are calculated explicitly. This is also done for certain submanifolds of these distribution families, in particular for such which correspond to independent random variables. Implications are discussed for the geometry of the graph of the potential function \( \varphi \). As is well known, the Hessian of \( \varphi \) is related to the Euclidean curvature of the graph and is conformally equivalent to the first fundamental form of affine hypersurface theory. With these means, the authors establish neighbourhoods of independence limiting cases and emphasize their applicability in models and stochastic processes.

MSC:

53B20 Local Riemannian geometry
60G05 Foundations of stochastic processes
53B05 Linear and affine connections
53B25 Local submanifolds

Citations:

Zbl 0559.62001
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References:

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