Physics – Data Analysis – Statistics and Probability
Scientific paper
2004-06-06
Physics
Data Analysis, Statistics and Probability
9 pages, Latex. Normalisation factor for entropy terms corrected
Scientific paper
We consider the problem of estimating the population probability distribution given a finite set of multivariate samples, using the maximum entropy approach. In strict keeping with Jaynes' original definition, our precise formulation of the problem considers contributions only from the smoothness of the estimated distribution (as measured by its entropy) and the loss functional associated with its goodness-of-fit to the sample data, and in particular does not make use of any additional constraints that cannot be justified from the sample data alone. By mapping the general multivariate problem to a tractable univariate one, we are able to write down exact expressions for the goodness-of-fit of an arbitrary multivariate distribution to any given set of samples using both the traditional likelihood-based approach and a rigorous information-theoretic approach, thus solving a long-standing problem. As a corollary we also give an exact solution to the `forward problem' of determining the expected distributions of samples taken from a population with known probability distribution.
Majumdar Mahbub
Rahman Sabbir
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