“Looking for nothing" bayes linear methods for solving equations

Abstract

Here I will describe and implement Bayes linear methods for finding zeros of deterministic functions. We assume that the zero is known to be unique. Initially, the value of the function is modelled simply as the product of two independent factors, the position of the point from the zero and a "slope" which is assumed to vary "smoothly” with position. Additional prior information specifies first and second order properties of the slopes and the position of the zero: in particular, smoothness is specified by modelling the slope process to be stationary with a decreasing correlation function. This research is motivated by problems arising in large scale computer simulation of mathematical models of complex physical phenomena, where a single run of the code can be expensive and the output difficult to assimilate. Scientists are often confident about the structure of their model as a description of a physical process but may be uncertain about the values of certain model "parameters". Such parameters usually refer directly to physical attributes, and so collateral information about their values is usually available. In some applications, the physical process itself has been observed, and several runs of the code are made at different parameter settings in an attempt to match the realisation of the code with the actual realisation. The eventual aim is to aid scientists to search through the "parameter space” efficiently and systematically, using their knowledge of the process. Obviously, there are several respects in which this formulation does not tackle the real problem, as we mainly consider a single-valued function of a real variable. As well as considering this problem I will review the current state of play in the more general field of statistical numerical analysis and its relationship to deterministic computer experiments; and partial belief specification or Bayes linear methods