Machine Interval Experiments
A statistical experiment is a mathematical object that provides a framework for statistical inference, including hypothesis testing and parameter estimation, from observations of an empirical phenomenon. When observations in the continuum of real numbers are not empirically measurable to infinite precision and when conventional floating-point computations used in the inference procedure are not exact, the statistical experiment can become epistemologically invalid. The family of measures of the conventional statistical experiment indexed by a compact finite dimensional continuum is extended to the complete metric space of all compact subsets (of a certain form) of the index set. This is accomplished by the natural interval extension of the likelihood function. The extended experiment allows a statistical decision made with the aid of a computer to be equivalent to a numerical proof of its global optimality. Three open problems in computational statistics were solved using the extended experiment: (1) parametric bootstraps of likelihood ratio test statistics for finite mixture models, (2) rigorous maximum likelihood estimates of the branch lengths of a phylogenetic tree with a fixed topology or shape and (3) Monte Carlo sampling from a multi-modal target density with sharp peaks or witches’ hats.