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Developing new ways to estimate probabilities can be valuable for science,statistics, and engineering. By considering the information content ofdifferent output patterns, recent work invoking algorithmic information theoryhas shown that a priori probability predictions based on pattern complexitiescan be made in a broad class of input-output maps. These algorithmicprobability predictions do not depend on a detailed knowledge of how outputpatterns were produced, or historical statistical data. Although quantitativelyfairly accurate, a main weakness of these predictions is that they are given asan upper bound on the probability of a pattern, but many low complexity, lowprobability patterns occur, for which the upper bound has little predictivevalue. Here we study this low complexity, low probability phenomenon by lookingat example maps, namely a finite state transducer, natural time series data,RNA molecule structures, and polynomial curves. Some mechanisms causing lowcomplexity, low probability behaviour are identified, and we argue thisbehaviour should be assumed as a default in the real world algorithmicprobability studies. Additionally, we examine some applications of algorithmicprobability and discuss some implications of low complexity, low probabilitypatterns for several research areas including simplicity in physics andbiology, a priori probability predictions, Solomonoff induction and Occam'srazor, machine learning, and password guessing.

Low complexity, low probability patterns and consequences for algorithmic probability applications