Related papers: How much randomness is needed for statistics?

In this work we study randomised reduction strategies,a notion already known in the context of abstract reduction systems, for the $\lambda$-calculus. We develop a simple framework that allows us to prove a randomised strategy to be…

This paper is about how we study statistical methods. As an example, it uses the random regressions model, in which the intercept and slope of cluster-specific regression lines are modeled as a bivariate random effect. Maximizing this…

Nies and Scholz defined quantum Martin-L\"of randomness (q-MLR) for states (infinite qubitstrings). We define a notion of quantum Solovay randomness and show it to be equivalent to q-MLR using purely linear algebraic methods. Quantum…

We make precise sense of the idea of "molecular chaos" through algorithmic randomness of microscopic trajectories, and ground macroscopic irreversibility in the lack of symmetry under time reversal of this property. This concept of…

Following Fisher, it is widely believed that randomization "relieves the experimenter from the anxiety of considering innumerable causes by which the data may be disturbed." In particular, it is said to control for known and unknown…

A common assumption in causal inference is that random treatment assignment ensures that potential outcomes are independent of treatment, or in one word, unconfoundedness. This paper highlights that randomization and unconfoundedness are…

The mission of statistics is to provide adequate statistical hypotheses (models) for observed data. But what is an "adequate" model? To answer this question, one needs to use the notions of algorithmic information theory. It turns out that…

Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter $p$. By the law of large numbers, the frequency of zeros in the sequence tends to~$p$, and thus we can get better and better…

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,…

Prediction of events is the challenge in many different disciplines, from meteorology to finance; the more this task is difficult, the more a system is {\it complex}. Nevertheless, even according to this restricted definition, a general…

In 1970, Donald Ornstein proved a landmark result in dynamical systems, viz., two Bernoulli systems with the same entropy are isomorphic except for a measure 0 set. Keane and Smorodinsky gave a finitary proof of this result. They also…

Do completely unpredictable events exist in nature? Classical theory, being fully deterministic, completely excludes fundamental randomness. On the contrary, quantum theory allows for randomness within its axiomatic structure. Yet, the fact…

Statistical analysis is an important tool to distinguish systematic from chance findings. Current statistical analyses rely on distributional assumptions reflecting the structure of some underlying model, which if not met lead to problems…

Neural network models have become the leading solution for a large variety of tasks, such as classification, language processing, protein folding, and others. However, their reliability is heavily plagued by adversarial inputs: small input…

Stability is a central property in learning and statistics promising the output of an algorithm $A$ does not change substantially when applied to similar datasets $S$ and $S'$. It is an elementary fact that any sufficiently stable algorithm…

Contrary to traditional deterministic notions of algorithmic fairness, this paper argues that fairly allocating scarce resources using machine learning often requires randomness. We address why, when, and how to randomize by proposing…

Randomness in the sense of Martin-L\"of can be defined in terms of lower semicomputable supermartingales. We show that such a supermartingale cannot be replaced by a pair of supermartingales that bet only on the even bits (the first one)…

We extend the notion of randomness (in the version introduced by Schnorr) to computable Probability Spaces and compare it to a dynamical notion of randomness: typicality. Roughly, a point is typical for some dynamic, if it follows the…

Consider a probability measure supported by a regular geodesic ball in a manifold. For any p larger than or equal to 1 we define a stochastic algorithm which converges almost surely to the p-mean of the measure. Assuming furthermore that…

When an experimentalist measures a time series of qubits, the outcomes generate a classical stochastic process. We show that measurement induces high complexity in these processes in two specific senses: they are inherently unpredictable…