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dg.differential geometry – Projections in boundless dimensional statistical manifolds
Are you attempting to assemble an infinite-dimensional (Hilbert) manifold of likelihood measures on a set measurable area? Or you merely need to discover a Bergman divergence analog in organize to generalize a divergence-like metric in an infinite-dimensional manifold? It just isn’t completely limpid within the OP what kindly of analog you might be searching for.
In a finite-dimensional statistical manifold, for instance an exponential household with unaffected parameterizations, the manifold does include a canonical connection as you talked about. In the finite-dimensional status, since we are able to take a parameterization, the geometry of the statistical manifold is embedded within the parameter area. The picture of such a projection to a $nabla$-flat submanifold would coincide to a sub-family of the exponential household, too a restricted assortment of parameters that outline this sub-family.
In an infinite-dimensional statistical manifold, assuming that you just need to examine two factors; you’ll be able to merely compute the norms of their dissimilarity and different kindly of similarity measures are workable [Harandi et.al.]. If you might be speaking about infinite-dimensional statistical manifold that consists of likelihood measures that can’t breathe The inside product construction love we’ve for low-dimensional parameter area, nevertheless, doesn’t at all times live. In a particular illustration identified by [Newton2], we are able to employ an $alpha$-divergence (which is a generalization to Bergman divergence, behold this submit from stat.SE) to partially narrate an identical inside product-like construction for an infinite-dimensional statistical manifold. The asymptotic outcomes from geometric perspective [Kass&Vos] could generalize into infinite-dimensional status if you’ll find a “good finite-dimensional basis approximation”. However, a strict inside product construction just isn’t at all times workable in infinite-dimensional status, due to this fact projections could not generalize. If it does generalize, its acceptation would doubtless breathe a sub-family of likelihood measures.
In one other course, if you happen to want to assemble a realization (or haul samples) from an inifinite-dimensional statistical manifold, the place to begin would breathe Dirichlet processes wiki. And [Newton] proposed to employ transformation (Fenchel–Legendre rework) strategy to assemble fresh infinite-dimensional statistical manifolds in a extra summary route.
Reference[Newton2] Infinite-dimensional statistical manifolds based mostly on a balanced chart, 2016. [Harandi et.al.] Bregman Divergences for Infinite Dimensional Covariance Matrices,2014. [Newton] An infinite-dimensional statistical manifold modelled on Hilbert area,2012. [Kass&Vos] Geometrical Foundations of Asymptotic Inference, 1997.
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