Computable Randomness is Inherently Imprecise

We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define a notion of computable randomness associated with interval, rather than precise, forecasting systems, and study its properties. The richer mathematical structure that thus arises lets us better understand and place existing results for the precise limit. When we focus on constant interval forecasts, we find that every infinite sequence of zeroes and ones has an associated filter of intervals with respect to which it is computably random. It may happen that none of these intervals is precise, which justifies the title of this paper. We illustrate this by showing that computable randomness associated with non-stationary precise forecasting systems can be captured by a stationary interval forecast, which must then be less precise: a gain in model simplicity is thus paid for by a loss in precision.Comment: 29 pages, 12 of which constitute the main text, and 17 of which constitute an appendix with proofs and additional material. 3 figures. Conference paper (ISIPTA 2017

Prequential Randomness

Die TU Dresden verfügt über viele spannende Sammlungen: 150 Jahre alte Daguerreotypien in der Hermann-Krone-Sammlung, hunderte von historischen Farbstoffen aus Lapislazuli und Indischgelb aus Urin von Mango essenden Kühen, Objekten zur Farbenlehre ab Goethe, mathematische Modelle, Visualisierungen von fernen und heimischen Landschaften in der  kartographischen Reliefsammlung usw. Der Direktor der Kustodie, Herr Mauersberger, beschreibt die Sammlungen als identitätsstiftend für die TU. Folglic..

Universal Coding and Prediction on Martin-L\"of Random Points

We perform an effectivization of classical results concerning universal coding and prediction for stationary ergodic processes over an arbitrary finite alphabet. That is, we lift the well-known almost sure statements to statements about Martin-L\"of random sequences. Most of this work is quite mechanical but, by the way, we complete a result of Ryabko from 2008 by showing that each universal probability measure in the sense of universal coding induces a universal predictor in the prequential sense. Surprisingly, the effectivization of this implication holds true provided the universal measure does not ascribe too low conditional probabilities to individual symbols. As an example, we show that the Prediction by Partial Matching (PPM) measure satisfies this requirement. In the almost sure setting, the requirement is superfluous.Comment: 12 page

Merging of opinions in game-theoretic probability

This paper gives game-theoretic versions of several results on "merging of opinions" obtained in measure-theoretic probability and algorithmic randomness theory. An advantage of the game-theoretic versions over the measure-theoretic results is that they are pointwise, their advantage over the algorithmic randomness results is that they are non-asymptotic, but the most important advantage over both is that they are very constructive, giving explicit and efficient strategies for players in a game of prediction.Comment: 26 page

Randomness is inherently imprecise

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 for which it is random. 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 cannot 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 computable (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, 7 figures. arXiv admin note: text overlap with arXiv:1703.0093

On impossibility of sequential algorithmic forecasting

The problem of prediction future event given an individual sequence of past events is considered. Predictions are given in form of real numbers $p_n$ which are computed by some algorithm $varphi$ using initial fragments $omega_1,dots, omega_{n-1}$ of an individual binary sequence $omega=omega_1,omega_2,dots$ and can be interpreted as probabilities of the event $omega_n=1$ given this fragment. According to Dawid\u27s {it prequential framework} %we do not consider %numbers $p_n$ as conditional probabilities generating by some %overall probability distribution on the set of all possible events. we consider partial forecasting algorithms $varphi$ which are defined on all initial fragments of $omega$ and can be undefined outside the given sequence of outcomes. We show that even for this large class of forecasting algorithms combining outcomes of coin-tossing and transducer algorithm it is possible to efficiently generate with probability close to one sequences for which any partial forecasting algorithm is failed by the method of verifying called {it calibration}