# Blog: thought

#### Imprecise Probabilities — Part I

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

#### 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

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

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 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

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

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 ...