# Randomness is inherently imprecise

@article{deCooman2021RandomnessII, title={Randomness is inherently imprecise}, author={Gert de Cooman and Jasper De Bock}, journal={International Journal of Approximate Reasoning}, year={2021} }

Abstract 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… Expand

#### 3 Citations

A Remarkable Equivalence between Non-Stationary Precise and Stationary Imprecise Uncertainty Models in Computable Randomness

- Computer Science
- ISIPTA
- 2021

It is shown that there are stationary imprecise models and non-stationary precise models that have the exact same set of computably random sequences. Expand

Randomness and Imprecision: A Discussion of Recent Results

- Computer Science
- ISIPTA
- 2021

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… Expand

The Smallest Probability Interval a Sequence Is Random for: A Study for Six Types of Randomness

- Mathematics, Computer Science
- ECSQARU
- 2021

It is shown that for many randomness notions, every sequence has a smallest interval it is (almost) random for. Expand

#### References

SHOWING 1-10 OF 86 REFERENCES

A Remarkable Equivalence between Non-Stationary Precise and Stationary Imprecise Uncertainty Models in Computable Randomness

- Computer Science
- ISIPTA
- 2021

It is shown that there are stationary imprecise models and non-stationary precise models that have the exact same set of computably random sequences. Expand

A particular upper expectation as global belief model for discrete-time finite-state uncertain processes

- Computer Science, Mathematics
- Int. J. Approx. Reason.
- 2021

It is shown that the most conservative upper expectation satisfying the proposed axioms, that is, the model of choice, coincides with a particular version of the game-theoretic upper expectation introduced by Shafer and Vovk. Expand

Global Upper Expectations for Discrete-Time Stochastic Processes: In Practice, They Are All The Same!

- Mathematics, Computer Science
- ISIPTA
- 2021

It is argued that these domains cover most practical inferences, and that therefore, in practice, it does not matter which type of global uncertainty models are considered. Expand

Sum-Product Laws and Efficient Algorithms for Imprecise Markov Chains

- 2021

We propose two sum-product laws for imprecise Markov chains, and use these laws to derive two algorithms to efficiently compute lower and upper expectations for imprecise Markov chains under complete… Expand

Computable Randomness Is About More Than Probabilities

- Mathematics, Computer Science
- SUM
- 2020

Interestingly, it is found that every sequence is computable random with respect to at least one lower expectation, and that lower expectations that are more informative have fewer computably random sequences. Expand

- 2019

A Recursive Algorithm for Computing Inferences in Imprecise Markov Chains

- Mathematics, Computer Science
- ECSQARU
- 2019

We present an algorithm that can efficiently compute a broad class of inferences for discrete-time imprecise Markov chains, a generalised type of Markov chains that allows one to take into account… Expand

Continuity Properties of Game-theoretic Upper Expectations.

- Mathematics
- 2019

We consider discrete-time uncertain processes with finite state space and study the properties of game-theoretic upper expectations developed by Shafer and Vovk. We start by proving some basic… Expand

Continuity properties of gametheoretic upper expectations

- 2019

Game‐Theoretic Foundations for Probability and Finance

- Computer Science
- Wiley Series in Probability and Statistics
- 2019