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Entropy 2015, 17(7), 4986-4999; doi:10.3390/e17074986

Maximum Entropy Estimation of Probability Distribution of Variables in Higher Dimensions from Lower Dimensional Data

1
Battelle Center for Mathematical Medicine, Research Institute at the Nationwide Children's Hospital, 700 Children's Drive, OH 43205, USA
2
Department of Pediatrics, The Ohio State University, Columbus, OH 43205, USA
3
Department of Physics, The Ohio State University, Columbus, OH 43210, USA
4
Department of Biophysics Program, The Ohio State University, Columbus, OH 43210, USA
*
Author to whom correspondence should be addressed.
Academic Editor: Kevin H. Knuth
Received: 5 May 2015 / Revised: 1 July 2015 / Accepted: 3 July 2015 / Published: 15 July 2015
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Abstract

A common statistical situation concerns inferring an unknown distribution Q(x) from a known distribution P(y), where X (dimension n), and Y (dimension m) have a known functional relationship. Most commonly, n ≤ m, and the task is relatively straightforward for well-defined functional relationships. For example, if Y1 and Y2 are independent random variables, each uniform on [0, 1], one can determine the distribution of X = Y1 + Y2; here m = 2 and n = 1. However, biological and physical situations can arise where n > m and the functional relation Y→X is non-unique. In general, in the absence of additional information, there is no unique solution to Q in those cases. Nevertheless, one may still want to draw some inferences about Q. To this end, we propose a novel maximum entropy (MaxEnt) approach that estimates Q(x) based only on the available data, namely, P(y). The method has the additional advantage that one does not need to explicitly calculate the Lagrange multipliers. In this paper we develop the approach, for both discrete and continuous probability distributions, and demonstrate its validity. We give an intuitive justification as well, and we illustrate with examples. View Full-Text
Keywords: maximum entropy; joint probability distribution; microbial ecology maximum entropy; joint probability distribution; microbial ecology
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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, J.D.; Mukherjee, S.; Hodge, S.E. Maximum Entropy Estimation of Probability Distribution of Variables in Higher Dimensions from Lower Dimensional Data. Entropy 2015, 17, 4986-4999.

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