کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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4591058 | 1335004 | 2012 | 21 صفحه PDF | دانلود رایگان |
We construct an infinite-dimensional Hilbert manifold of probability measures on an abstract measurable space. The manifold, M, retains the first- and second-order features of finite-dimensional information geometry: the α-divergences admit first derivatives and mixed second derivatives, enabling the definition of the Fisher metric as a pseudo-Riemannian metric. This is enough for many applications; for example, it justifies certain projections of Markov processes onto finite-dimensional submanifolds in recursive estimation problems. M was constructed with the Fenchel–Legendre transform between Kullback–Leibler divergences, and its role in Bayesian estimation, in mind. This transform retains, on M, the symmetry of the finite-dimensional case. Many of the manifolds of finite-dimensional information geometry are shown to be C∞-embedded submanifolds of M. In establishing this, we provide a framework in which many of the formal results of the finite-dimensional subject can be proved with full rigour.
Journal: Journal of Functional Analysis - Volume 263, Issue 6, 15 September 2012, Pages 1661-1681