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 ...