Randomness and Imprecision: A Discussion of Recent Results

We discuss our recent work on incorporating imprecision in the field of algorithmic randomness, based on the martingale-theoretic approach of game-theoretic probability. We consider several notions of randomness associated with interval, rather than precise, forecasting systems. We study their properties and argue that there are quite a number of reasons for wanting to do so. First, the richer mathematical structure in this generalisation provides a useful backdrop for a better understanding of precise randomness. Second, randomness associated with non-stationary precise forecasting systems can be captured by a constant but less precise interval forecast: greater model simplicity requires more imprecision. Third, imprecise randomness can’t always be explained away as a result of (over)simplification: there are sequences that are random for a constant interval forecast, but never random for any computable (more) precise forecasting system. Incorporating imprecision into randomness therefore allows us to do more than was hitherto possible. Finally, the random sequences for a non-vacuous interval forecast constitute a meagre set, as they do for precise forecasts: imprecise and precise random sequences are equally rare from a topological point of view, and are, in that sense, equally interesting.