Is there a category whose objects are probability spaces and morphisms are measurable functions? The answer is yes.

In this post, Terance Tao writes ”probability theory is only ‘allowed’ to study concepts and perform operations which are preserved with respect to extension of the underlying sample space.”. An extension of a probability space to means that there is a function from to which is measurable, probability preserving and surjective, where and are samples spaces, and are -algebras and and are probability measures. But is there a category whose morphisms are extensions (his post indicates so in passing)? We show here that the answer is yes.

Example of a category whose objects are measurable spaces and morphisms are measurable functions

In this post we consider a category whose objects are measurable spaces and morphisms are measurable functions. We limit ourselves to two objects, which let us enumerate all possible morphism. By doing so we get at better feeling of what is going on when working with categories and measurable spaces.

Concrete example of almost sure convergence using the definition

I had difficulties finding a concrete and simple example of almost sure convergence using the standard measure theoretic definition (Gut, 2009, p. 147), i.e. random variables converges almost surely to random variable iff , where is a member of a sample space . So here is one example

Measuring international relations using artifical intelligence (AI)

Abstract
We look at methodology for measuring national influence in international organizations using artificial intelligence (AI). More specifically, we analyze to what extent the United Nations (UN) and NATO uses keywords in their Twitter communication, that are more common in Swedish tweets relative to tweets by the United States or Norway. The idea is that similarities in terms of keywords works as a crude indicator of national influence.

Using a Poisson process to model the price of a security

Abstract
A popular way of modeling the price of a security is to use a Poisson process, where changes in price are due to random shocks. In this post we derive such a model and show how its parameters can be fitted using the method of moments.

Development aid statistics: Neural network vs. Bayes classifier

Introduction
In a previous post we used a naive Bayes classifier to map words in project descriptions to policy objectives. What we wanted to do was to predict whether a policy objective were present for an aid contribution, e.g. gender equality, based on its description. In this post we try to do the same thing but use a neural network instead of the Bayes classifier. Our assumption is that the neural network is better equipped to handle interdependencies between words, which the naive Bayes classifier is know to ignore, and therefore will produce better results.

We find that the neural network indeed performs (much) better. On average it correctly predicts the policy objective for 91% of the contributions, compared with 76% for the Bayes classifier. These results are promising considering civil servants in donor countries put much effort into filling in mandatory fields, such as policy objectives, when reporting official aid statistics to the OECD DAC (about about 123 000 bilateral contributions per year). Automatization may also assist additional qualitative checking.

Using artificial intelligence to classify development aid contributions (400k dataset)

Introduction
Official development assistance (ODA) is defined by the OECD Development Assistance Committee (DAC) as government aid that promotes and specifically targets the economic development and welfare of developing countries. In 2017, ODA disbursements by the DAC countries amounted to 119 billion USD and were reported to the DAC. Part of the reporting included filling in a set of statistical parameters, including, for each aid contribution, whether gender equality, environment, trade or participatory development/good governance had been a principal or significant policy objective.

In this post we use artificial intelligence, in the form of a Bayes classifier, to guess the policy objectives of ODA contributions based on their project descriptions. In total we analyze 396 000 different contributions by 29 countries from 2015-2017. Motivation for doing so is to explore the possibility of mechanically filling in (preliminary) statistical parameters and also to facilitate quality assurance by flagging contributions which seem missclassified based on their descriptions.

Principal component analysis of 374 large, mid and small cap stocks on NasdaqOMX Stockholm

Introduction
In this post we use principal component analysis to explore how large, mid and small cap stocks differ when eight key performance figures are decomposed into just two principal components. All 374 stocks on NasdaqOMX Stockholm were analyzed. Some possible applications are mentioned.

A concise explanation of stationary ARMA models

Motivation
Stationary autoregressive moving average (ARMA) models play an important role in financial forecasting (we referred to it here and here). My experience is that online explanations of stationarity often are either very long or implicit about certain steps. Therefore, this post presents a fairly short but hopefully complete explanation of when an ARMA model is stationary.

We use a neural network to identify unknown time series

Motivation
In an earlier post we simulated time series data from an ARIMA model in Python. In this post we use simulated data to train a neural network. Our goal is to make the neural network correctly predict ARIMA models for unknown time series.