Preprint Article Version 1 NOT YET PEER-REVIEWED

Tensor Properties of Joint Probability Densities on Riemannian Parametric Manifold

  1. Health Organization, Tehran, Iran
Version 1 : Received: 28 October 2016 / Approved: 31 October 2016 / Online: 31 October 2016 (02:39:45 CET)

How to cite: Amiri, M. Tensor Properties of Joint Probability Densities on Riemannian Parametric Manifold. Preprints 2016, 2016100131 (doi: 10.20944/preprints201610.0131.v1). Amiri, M. Tensor Properties of Joint Probability Densities on Riemannian Parametric Manifold. Preprints 2016, 2016100131 (doi: 10.20944/preprints201610.0131.v1).

Abstract

We show the tensor properties of joint probabilities on a Riemannian parametric manifold. Initially we develop a binary data matrix for parameter measurements of a large number of particles confining in a closed system in order to retrieve the joint probability densities of related parameters. By introducing a new generalized inner product as a multilinear operation on dual basis of parametric space, we extract the set of joint probabilities and prove them to meet contravariant tensor properties in a general Riemannian parametric space. We show these contravariant tensors reduce to classical ordinary partial derivatives definition in ordinary Euclidean parametric space. Finally we prove by a theorem that symmetrized iterative contravariant derivative of cumulative probability function on Riemannian manifold gives the set of joint probabilities in those manifold. We bring some examples for compatibility with physical tensors.

Subject Areas

joint probability density; tensor properties; probability on Riemannian manifold; covariant derivative; contravariant derivative

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