Related papers: The Infinity of Randomness
We identify a fundamental incompatibility between the goals of accuracy, trust, and human-level reasoning in artificial intelligence (AI) systems, for strict mathematical definitions of these notions. We define accuracy of a system as the…
In the following pages we discuss infinite sequences defined on a finite alphabet, and more specially those which are generated by finite automata. We have divided our paper into seven parts which are more or less self-contained. Needless…
Recent tremendous development of quantum information theory led to a number of quantum technological projects, e.g., quantum random generators. This development stimulates a new wave of interest in quantum foundations. One of the most…
The field of algorithmic randomness studies what it means for infinite binary sequences to be random for some given uncertainty model. Classically, martingale-theoretic notions of such randomness involve precise uncertainty models, and it…
This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a…
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…
In this paper, we consider robust control using randomized algorithms. We extend the existing order statistics distribution theory to the general case in which the distribution of population is not assumed to be continuous and the order…
We introduce a concept for random tilings which, comprising the conventional one, is also applicable to tiling ensembles without height representation. In particular, we focus on the random tiling entropy as a function of the tile…
The notions of potential infinity (understood as expressing a direction) and actual infinity (expressing a quantity) are investigated. It is shown that the notion of actual infinity is inconsistent, because the set of all (finite) natural…
Randomness is a valuable resource in science, cryptography, engineering, and information technology. Quantum-mechanical sources of randomness are attractive because of the indeterminism of individual quantum processes. Here we consider the…
The classical concept of bounded completeness and its relation to sufficiency and ancillarity play a fundamental role in unbiased estimation, unbiased testing, and the validity of inference in the presence of nuisance parameters. In this…
The extremal characteristics of random structures, including trees, graphs, and networks, are discussed. A statistical physics approach is employed in which extremal properties are obtained through suitably defined rate equations. A variety…
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…
This text provides a practical introduction to randomness and data analysis, in particular in the context of computer simulations. At the beginning, the most basics concepts of probability are given, in particular discrete and continuous…
Some asymptotic notions for random variables are discussed. In particular, different versions of O and o for sequences of random variables are studied. The results are elementary and more or less well-known, but collected here for future…
The paper considers quantitative versions of different randomness notions: algorithmic test measures the amount of non-randomness (and is infinite for non-random sequences). We start with computable measures on Cantor space (and Martin-Lof…
Asymptotic properties of random graph sequences, like occurrence of a giant component or full connectivity in Erd\H{o}s-R\'enyi graphs, are usually derived with very specific choices for defining parameters. The question arises to which…
We introduce the notion of a reproducible algorithm in the context of learning. A reproducible learning algorithm is resilient to variations in its samples -- with high probability, it returns the exact same output when run on two samples…
Algorithms for continuous optimization problems have a rich history of design and innovation over the past several decades, in which mathematical analysis of their convergence and complexity properties plays a central role. Besides their…
I will propose the notion that the universe is digital, not as a claim about what the universe is made of but rather about the way it unfolds. Central to the argument will be the concepts of symmetry breaking and algorithmic probability,…