Numerical application of the fundamental theorem of prevision

摘要

The fundamental theorem of the operational subjective theory of probability, from the viewpoint established by Bruno de Finetti, provides the solutions to probability problems in terms of the results of a pair of specific linear or nonlinear programming problems. We state a general version of this theorem in a computationally feasible form and present numerical examples of its use. The examples display interesting extensions of the Bienaym#xE9;-Chebyshev inequality and a variation on the Kolmogorov inequality in the context of finite discrete quantities. The Bienaym#xE9;-Chebyshev application is extended to exemplify the use of a nonlinear programming algorithm to resolve a common question regarding coherent inference. In concluding discussion, we comment on the sizes of realistic problems and suggest a variety of applications for such computations, among them the safety assessment of complex engi-neering systems, the analysis of agricultural production statistics, and a synthesis of subjective judgments in macroeconomic forecasting.