#### The Universal Approximation Theorem

*The Capability of Neural Networks as General Function Approximator*s

**Introduction**Artificial Intelligence has become very present in the media in ...

Get me more...

#### 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 ...

Get me more...

#### Function (Vector) Spaces

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

Get me more...

#### Outer Measures

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

Get me more...

Get me more...

#### Aleatory and Epistemic Probabilities

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

Get me more...

#### Sequences in Metric Spaces

Convergence ca be defined in many different ways. In this post, we study the most popular way to define convergence ...

Get me more...

Get me more...

#### 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 ...

Get me more...

Get me more...

#### 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 ...

Get me more...

#### 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 ...

Get me more...

Get me more...

#### 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 ...**

Get me more...

#### 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 ...

Get me more...

Get me more...

#### Conditional Probability & Bayes Rule

Conditional Probabilities Let us consider a probability measure $P: \mathcal{A} \rightarrow \mathbb{R}$ of a measurable space $(\Omega, \mathcal{A})$. Further, let ...

Get me more...

Get me more...

#### 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 ...

Get me more...

Get me more...

#### 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 ...

Get me more...

Get me more...

#### 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 ...

Get me more...

Get me more...

#### Poisson Distribution

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

Get me more...

Get me more...

#### 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 ...

Get me more...

Get me more...

#### 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 ...

Get me more...

Get me more...