Related papers: Great expectations: Unifying Statistical Theory an…
Consider an experiment involving a potentially small number of subjects. Some random variables are observed on each subject: a high-dimensional one called the "observed" random variable, and a one-dimensional one called the "outcome" random…
This paper attributes the sudden emergence of mathematical probability and statistics in the second half of the seventeenth century to Calvin's Reformed theology. Calvin accommodated Epicurean chance with Stoic determinism and synthesised…
We introduce Voevodsky's univalent foundations and univalent mathematics, and explain how to develop them with the computer system Agda, which is based on Martin-L\"of type theory. Agda allows us to write mathematical definitions,…
Initially introduced by Peter Hammer, Logical Analysis of Data is a methodology that aims at computing a logical justification for dividing a group of data in two groups of observations, usually called the positive and negative groups.…
The Curry-Howard correspondence is about a relationship between types and programs on the one hand and propositions and proofs on the other. The implications for programming language design and program verification is an active field of…
We study the statistical properties of random numbers under the Martin-L\"of definition of randomness, proving that random numbers obey analogues of Strong Law of Large Numbers, the Law of the Iterated Logarithm, and that they are normal.…
Data Science and Machine learning have been growing strong for the past decade. We argue that to make the most of this exciting field we should resist the temptation of assuming that forecasting can be reduced to brute-force data analytics.…
This paper introduces a novel type theory and logic for probabilistic reasoning. Its logic is quantitative, with fuzzy predicates. It includes normalisation and conditioning of states. This conditioning uses a key aspect that distinguishes…
Extracting meaning from uncertain, noisy data is a fundamental problem across time series analysis, pattern recognition, and language modeling. This survey presents a unified mathematical framework that connects classical estimation theory,…
So far, following the works of A.M. Turing, the algorithms were considered as the mathematical abstraction from which we could write programs for computers whose principle was based on the theoretical concept of Turing machine. We start…
The notion of programming paradigms, with associated programming languages and methodologies, is a well established tenet of Computer Science pedagogy, enshrined in international curricula. However, this notion sits ill with Kuhn's classic…
Large eliminations provide an expressive mechanism for arity- and type-generic programming. However, as large eliminations are closely tied to a type theory's primitive notion of inductive type, this expressivity is not expected within…
In algorithmic randomness, when one wants to define a randomness notion with respect to some non-computable measure $\lambda $, a choice needs to be made. One approach is to allow randomness tests to access the measure $\lambda $ as an…
The rise of multi-paradigm languages challenges traditional classification methods, leading to practical software engineering issues like interoperability defects. This systematic literature review (SLR) maps the formal foundations of…
In this article, the notion of a mathematical model in science is attempted to be enlightened from several points of view. In particular, it is shown that mathematical models are introduced differently and used differently in different…
Humans can generate reasonable answers to novel queries (Schulz, 2012): if I asked you what kind of food you want to eat for lunch, you would respond with a food, not a time. The thought that one would respond "After 4pm" to "What would you…
In the theory of answer set programming, two groups of rules are called strongly equivalent if, informally speaking, they have the same meaning in any context. The relationship between strong equivalence and the propositional logic of…
Probabilistic programming is the idea of writing models from statistics and machine learning using program notations and reasoning about these models using generic inference engines. Recently its combination with deep learning has been…
Percentage thinking, i.e., assessing quantities as parts per hundred, spread from Roman tax ledgers to modern algorithms. Building on Simon Stevin's La Thiende (1585) and the 19th-century metrication that institutionalized base-10…
We extend the key notion of Martin-L\"of randomness for infinite bit sequences to the quantum setting, where the sequences become states of an infinite dimensional system. We work towards showing an analogy with the Levin-Schnorr theorem to…