By Javier Jiménez

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The MSE as a function of the model parameter α (two minima are clearly visible). 5. The solution function which resulted from the empirical risk minimisation principle (dashed) together with the observed data points and the “true” underlying mapping (solid line). 26 Learning and regularisation 27 It is clear that this solution is not a favourable one in terms of generalisation capabilities. Apart from a couple of observables where both curves cross each other, the model produces estimations irrelevant to the underlying mapping.

Latter on, in almost his first published work (1794), Laplace rediscovered Bayes’ principle in greater clarity and generality, and then for the next 40 years proceeded to apply it to various problems of astronomy, geodesy, meteorology and statistics. The Bayesian learning paradigm is founded upon the premise that all forms of uncertainty can be expressed and measured by probabilities (Bernardo and Smith, 1994). Although the paradigm can be expressed in a formal framework, based on mathematical abstraction and rigorous analysis, it relies upon subjective experience.

The bound on the risk is the sum of the empirical risk and the VC confidence. The smallest bound on the risk is achieved by taking particular trained machine on an appropriate subset of the structure whose sum of the empirical risk and the VC confidence is minimal (adopted from Vapnik, 1998). 3. The structure can be imposed on the input representation to the ANN with fixed architecture. g. the width of the smoothing kernel). A structure can be introduced in a set of functions S={f(K(x,β),θ), } through β≥cp and c1>c2>…> cn.

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