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Monte Carlo is Bayesian

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Cox was right, Kolmogorov was wrong. The attempt to found probability calculus upon set theory seems to be misguided, because it is necessary to exclude sets of measure zero in order to avoid paradox. It is better to start with unit measure of probabilistic belief, to be distributed among relevant hypotheses regardless of any measure they may possess. This improved viewpoint shows that, contrary to the folklore of the subject, Monte Carlo integration is properly probabilistic (as an algorithm for Bayesian computation should be). Nested sampling is an extension to problems of larger scale: it is an algorithm of wider scope that can deal with a variety of multi-modal problems better than conventional annealing.

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