Randomness is inherently imprecise
Autor: | Jasper De Bock, Gert de Cooman |
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Rok vydání: | 2021 |
Předmět: |
Technology and Engineering
Computer science Imprecise probabilities MODELS Interval (mathematics) 01 natural sciences Theoretical Computer Science MARKOV-CHAINS Set (abstract data type) 010104 statistics & probability Algorithmic random Artificial Intelligence SELF-CALIBRATING PRIORS FOS: Mathematics Limit (mathematics) Interval forecast 0101 mathematics Filter (mathematics) Randomness Sequence Computability Game-theoretic probability Applied Mathematics 010102 general mathematics Probability (math.PR) Supermartingale PROBABILITY Mathematics and Statistics Mathematical structure Constant (mathematics) Algorithm Software Mathematics - Probability |
Zdroj: | INTERNATIONAL JOURNAL OF APPROXIMATE REASONING |
ISSN: | 0888-613X 1873-4731 |
DOI: | 10.48550/arxiv.2103.00071 |
Popis: | We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define several notions of randomness associated with interval, rather than precise, forecasting systems, and study their properties. The richer mathematical structure that thus arises lets us, amongst other things, better understand and place existing results for the precise limit. When we focus on constant interval forecasts, we find that every sequence of binary outcomes has an associated filter of intervals it is random for. It may happen that none of these intervals is precise -- a single real number -- which justifies the title of this paper. We illustrate this by showing that randomness associated with non-stationary precise forecasting systems can be captured by a constant interval forecast, which must then be less precise: a gain in model simplicity is thus paid for by a loss in precision. But imprecise randomness can't always be explained away as a result of oversimplification: we show that there are sequences that are random for a constant interval forecast, but never random for any {\comp} (more) precise forecasting system. We also show that the set of sequences that are random for a non-vacuous interval forecasting system is meagre, as it is for precise forecasting systems. Comment: 49 pages, 8 figures. arXiv admin note: text overlap with arXiv:1703.00931 |
Databáze: | OpenAIRE |
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