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. BIOGRAPHICAL SKETCH Raaz went from one world to another at an impressionable age. He experientially realized that peoples co-believe what ever they wanted to and was deeply troubled by the relativity of locally co-fabricated cosmologies. He looked for epistemological answers in the fields of mathematics, philosophy and religion. He had to drop out and wander the western lands until the platforms for the troubles vanished by themselves. After a network of intense interactions with other earthlings he studied by a seven mile creek that flowed on the blue earth along a river valley near the ecotone of prairies and deciduous woods in the the land of 10,000 lakes. Then he studied more by the southern tip of a long and narrow lake fingered by a recent glacier. He is currently documenting some statistical ideas that have been haphazardly integrated over some recurring thoughts that bounce back and forth through a wavy transect of the twining vines of minds he tries to sustain. iii To the twining vines of minds within the geodermodynamically coupled, DNA-based, heliotrophic, physicochemical continuum in space-time iv ACKNOWLEDGEMENTS