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We develop the contextual measurement model (CMM) which is used for clarification of the quantum foundations. This model matches with Bohr's views on the role of experimental contexts. CMM is based on contextual probability theory which is…
The hypothesis of randomness is fundamental in statistical machine learning and in many areas of nonparametric statistics; it says that the observations are assumed to be independent and coming from the same unknown probability…
In this paper we develop the theory of {\it polymorphisms} of measure spaces, which is a generalization of the theory of measure-preserving transformations; we describe the main notions and discuss relations to the theory of Markov…
We explore the interplay between random and deterministic phenomena using a representation of uncertainty based on the measure-theoretic concept of outer measure. The meaning of the analogues of different probabilistic concepts is…
The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying…
Tennenbaum's theorem states that the only countable model of Peano arithmetic (PA) with computable arithmetical operations is the standard model of natural numbers. In this paper, we use constructive type theory as a framework to revisit,…
The present work provides an original framework for random matrix analysis based on revisiting the concentration of measure theory from a probabilistic point of view. By providing various notions of vector concentration ($q$-exponential,…
Lecture notes as per the title. In the first part, the concepts of a measurable space, measurable maps between measurable spaces and that of a measure on a measurable space are introduced, after which the fundamentals of the theory of…
It is possible in principle to construct quantum mechanical observables and unitary operators which, if implemented in physical systems as measurements and dynamical evolution, would contradict the Church-Turing thesis, which lies at the…
This article begins with a brief review of random matrix theory, followed by a discussion of how the large-$N$ limit of random matrix models can be realized using operator algebras. I then explain the notion of "Brown measure," which play…
We introduce and discuss the notion of monotonicity for the complexity measures of general probability distributions, patterned after the resource theory of quantum entanglement. Then, we explore whether this property is satisfied by the…
We examine the connections between deterministic, complete, and general global optimisation of continuous functions and a general concept of regression from the perspective of constructive type theory via the concept of 'searchability'. We…
To develop rigorous knowledge about ML models -- and the systems in which they are embedded -- we need reliable measurements. But reliable measurement is fundamentally challenging, and touches on issues of reproducibility, scalability,…
In this note, we develop some of the basic theory of s-finite (measures and) kernels, a little-studied class that Staton has recently argued convincingly to be precisely the semantic counterpart of (first-order) probabilistic programs. We…
Compound random measures (CoRM's) are a flexible and tractable framework for vectors of completely random measure. In this paper, we provide conditions to guarantee the existence of a CoRM. Furthermore, we prove some interesting properties…
We address the issue of coupling variables which are essentially classical to variables that are quantum. Two approaches are discussed. In the first (based on collaborative work with L.Di\'osi), continuous quantum measurement theory is used…
We use machine learning to provide a tractable measure of the amount of predictable variation in the data that a theory captures, which we call its "completeness." We apply this measure to three problems: assigning certain equivalents to…
The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong…
Randomized higher-order computation can be seen as being captured by a lambda calculus endowed with a single algebraic operation, namely a construct for binary probabilistic choice. What matters about such computations is the probability of…
Probability metrics constitute an important tool in probability theory and statistics \cite{DKS91}, \cite{R91}, \cite{Z83} as they are specific metrics on spaces of random variables which, by satisfying an extra condition, concord well with…