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The role of the loss function in the probabilistic function approximation

Dodd, T.J., Harrison, R.F. and Harris, C.J. (2002) The role of the loss function in the probabilistic function approximation. Research Report. ACSE Research Reports no. 820 . Automatic Control and Systems Engineering, University of Sheffield

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Abstract

Generalising results on the time series estimation it is natural to consider function approximation with finite data observations in a probabilistic setting. The function is treated as a stochastic process where for each point in the functions domain the function is a random variable. Equivalently the function can be considered as a single random variable whose range is a space of functions. In this paper two results well known within the context of time series estimation and stochastic control are generalised to probabilistic function approximation problems. Under mild conditions on the space of functions it is shown that the optimal function estimate corresponds for all reasonable symmetrical loss functions to the pointwise conditioned expectation given the observed data. Further in the case where the space of functions belongs to the class of Gaussian process the optimal estimate is the conditional expectation even for asymmetric loss functions.

Item Type: Monograph (Research Report)
Copyright, Publisher and Additional Information: The Department of Automatic Control and Systems Engineering research reports offer a forum for the research output of the academic staff and research students of the Department at the University of Sheffield. Papers are reviewed for quality and presentation by a departmental editor. However, the contents and opinions expressed remain the responsibility of the authors. Some papers in the series may have been subsequently published elsewhere and you are advised to cite the later published version in these instances.
Institution: The University of Sheffield
Academic Units: The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Automatic Control and Systems Engineering (Sheffield) > ACSE Research Reports
Depositing User: Miss Anthea Tucker
Date Deposited: 06 Sep 2012 14:49
Last Modified: 10 Jun 2014 05:30
Status: Published
Publisher: Automatic Control and Systems Engineering, University of Sheffield
URI: http://eprints.whiterose.ac.uk/id/eprint/74478

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