Blog: thought

CoverPage Imprecise Probabilities -- Part I

Imprecise Probabilities — Part I

Introduction The world is an uncertain place. How to measure and deal with uncertain quantities is an important problem for ...
Universal Approximation Theorem

The Universal Approximation Theorem

The Capability of Neural Networks as General Function Approximators Introduction Artificial Intelligence has become very present in the media in ...
Measures of Risk

Measures of Risk

Introduction The quantification and even the definition of 'risk' is a hard problem. Questions like the following are therefore--in general--hard ...
Function Vector Spaces

Function (Vector) Spaces

Introduction Vector spaces are one of the most fundamental and important algebraic structures that are used far beyond math and ...

Outer Measures

In this post, we will introduce the concept of an outer measure, and we will also illustrate the connection to ...
Aleatory & Epistemic Probability

Aleatory and Epistemic Probabilities

Introduction In order to develop a useful theory of events with 'uncertain' outcome, it seems to be reasonable to give ...
Sequences in Metric Spaces

Sequences in Metric Spaces

Convergence ca be defined in many different ways. In this post, we study the most popular way to define convergence ...
Fundamentals of Set-Theoretic Topology

Fundamentals of Set-Theoretic Topology

Why would one want to generalize notions such as convergence and continuity to a setting even more abstract than metric ...

Nature of Continuity for Measure & Probability Theory – Part I

Introduction There are many textbooks, posts, videos and papers about continuity. However, it is hard to find a cohesive introduction ...

Fundamentals of Topologies & Metric Spaces

In this brief post, we investigate the topological foundations of analysis. We limit the scope to the topology of a ...

Probability Integral Transform & Quantile Function Theorem

Introduction We present simple illustrations, explanations and proofs for two very important theorems,
  • the probability integral transformation, and,
  • the quantile ...

Inner Products, Norms and Metrics

Most people do have an intuitive understanding of the real number system: it is the number system that should be ...

Conditional Probability & Bayes Rule

This article is about conditional probabilities and Bayes Rule / Theorem. In a second part, we are going to delve ...

Uncertainty and Capacities in Finance

Monotone Set Functions and the Choquet Integral The quote of the famous statistician George E. P. Box that "all models ...

Decision Problems, Risk and Uncertainty

In this post, an introduction to decision-making under risk and uncertainty is provided. To this end, basic concepts and components ...

Binomial- and Poisson-Mixture Model

Introduction The assumption of independent and identically distributed random variables, short i.i.d., might be quite handy since it simplifies several ...

Poisson Distribution

Poisson distributions are very important not only for counting events during a fixed period of time but also for different ...

GPU TensorFlow Installation Guide for Windows

Before we actually start the installation process of the GPU-accelerated Python API of TensorFlow on a Windows platform, we shortly ...

What is an Empirical Copula?

Introduction Copulas are an important concept in statistics and beyond to describe dependency structures of continuous distributions. However, what can ...