In information geometry, a strictly convex and smooth function induces a dually flat Hessian manifold equipped with a pair of dual Bregman divergences, hereby termed a Bregman manifold. Two common types of such Bregman manifolds met in statistics are (1) the exponential family manifolds induced by the cumulant functions of regular exponential families, and (2) the mixture family manifolds induced by the Shannon negentropies of statistical mixture families with prescribed linearly independent mixture components. However, the differential entropy of a mixture of continuous probability densities sharing the same support is hitherto not known in closed form making implementation of mixture family manifolds in practice difficult. In this work, we report an exception: The family of mixtures of two prescribed and distinct Cauchy distributions. We exemplify the explicit construction of a dually flat manifold induced by the differential negentropy for this very particular setting. This construction allows one to use the geometric toolbox of Bregman algorithms, and to obtain closed-form formula (albeit being large) for the Kullback-Leibler divergence and the Jensen-Shannon divergence between two mixtures of two prescribed Cauchy components.
翻译:在信息几何学中,一个严格的混凝土和光滑功能导致一个双平的黑森元件,配有双倍Bregman差异,此处称为Bregman元件。在统计中遇到的两种常见的Bregman元件是:(1) 正常指数家庭累积功能引发的指数式家庭元件,(2) 由具有线性独立混合成分的统计混合家庭的香农内源体引发的混合家庭元件。然而,一个连续概率密度的混合物的差别性酶,共用同样的支持,迄今尚未以封闭的形式为人们所知,使得混合家庭元件难以实际实施。我们在此工作中报告一个例外:两种指定且独特的宽度分布的混合物的组合。我们为这一非常特殊的环境举例说明了由差异性内分泌液引起的双平式组合。这一构造允许一种使用布雷格曼算法的几何工具箱,并获得库尔背利伯利弗尔差异和两个规定制的正向-Shan混合物之间的硬度差异的封闭式公式(尽管是大)。