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This package uses the term Bayesian Integration to mean approximating a function with a kriging metamodel (aka a gaussian process model) and then integrating under it. A kriging metamodel has the nice feature that uncertainty about the nature of the function is explicitly modelled (unlike for instance a approximation with Chebyshev polynomials) and the Bayesian Integral uses this feature to give a Gaussian distribution representing the probabilities of various integral values. The output of the bayesian_integral_gaussian_exponential function is the expectation and variance of this distribution.