## Sticky Post – Read this first. Categories and Links in Mathemafrica

The navigability of Mathemafrica isn’t ideal, so I have created this post which might guide you to what you are looking for. Here are a number of different categories of post which you might like to take a look at:

Always write in a comment if there is anything you would like to see us write about, or you would like to write about.…

## CRL Task 3: Counterfactual Decision Making

In the previous blog post we discussed some theory of how to select optimal and possibly optimal interventions in a causal framework. For those interested in the decision science, this blog post may be more inspiring. This next task involves applying counterfactual quantities to boost learning performance. This is clearly very important for an RL agent where its entire learning mechanism is based on interventions in a system. What if intervention isn’t possible? Let’s begin!

## This Series

1. Causal Reinforcement Learning
2. Preliminaries for CRL
3. CRL Task 1: Generalised Policy Learning
4. CRL Task 2: Interventions – When and Where?
5. CRL Task 3: Counterfactual Decision Making
6. CRL Task 4: Generalisability and Robustness
7. (Coming soon) Task 5: Learning Causal Models
8. (Coming soon) Task 6: Causal Imitation Learning
9. (Coming soon) Wrapping Up: Where To From Here?

## Counterfactual Decision Making

A key feature of causal inference is its ability to deal with counterfactual queries. Reinforcement learning, by its nature, deals with interventional quantities in a trial-and-error style of learning.…

## CRL Task 2: Interventions – When and Where?

In the previous blog post we discussed the gorey details of generalised policy learning – the first task of CRL. We went into some very detailed mathematical description of dynamic treatment regimes and generalised modes of learning for data processing agents. The next task is a bit more conceptual and focuses on the question on how to identfy optimal areas of intervention in a system. This is clearly very important for an RL agent where its entire learning mechanism is based on these very interventions in some system with a feedback mechanism. Let’s begin!

## This Series

1. Causal Reinforcement Learning
2. Preliminaries for CRL
3. CRL Task 1: Generalised Policy Learning
4. CRL Task 2: Interventions – When and Where?
5. CRL Task 3: Counterfactual Decision Making
6. CRL Task 4: Generalisability and Robustness
7. (Coming soon) Task 5: Learning Causal Models
8. (Coming soon) Task 6: Causal Imitation Learning
9. (Coming soon) Wrapping Up: Where To From Here?

## CRL Task 1: Generalised Policy Learning

In the previous blog post we developed some ideas and theory needed to discuss a causal approach to reinforcement learning. We formalised notions of multi-armed bandits (MABs), Markov Decision Processes (MDPs), and some causal notions. In this blog post we’ll finally get to developing some causal reinforcement learning ideas. The first of which is dubbed Task 1, for CRL can help solve. This is Generalised Policy Learning. Let’s begin.

## This Series

1. Causal Reinforcement Learning
2. Preliminaries for CRL
3. CRL Task 1: Generalised Policy Learning
4. CRL Task 2: Interventions – When and Where?
5. CRL Task 3: Counterfactual Decision Making
6. CRL Task 4: Generalisability and Robustness
7. (Coming soon) Task 5: Learning Causal Models
8. (Coming soon) Task 6: Causal Imitation Learning
9. (Coming soon) Wrapping Up: Where To From Here?

## Generalised Policy Learning

Reinforcement learning typically involves learning and optimising some policy about how to interact in an environment to maximise some reward signal.…

## Preliminaries for CRL

In the previous blog post we discussed and motivated the need for a causal approach to reinforcement learning. We argued that reinforcement learning naturally falls on the interventional rung of the ladder of causation. In this blog post we’ll develop some ideas necessary for understanding the material covered in this series. This might get quite technical, but don’t worry. There is still always something to take away. Let’s begin.

## This Series

1. Causal Reinforcement Learning
2. Preliminaries for CRL
3. CRL Task 1: Generalised Policy Learning
4. CRL Task 2: Interventions – When and Where?
5. CRL Task 3: Counterfactual Decision Making
6. CRL Task 4: Generalisability and Robustness
7. (Coming soon) Task 5: Learning Causal Models
8. (Coming soon) Task 6: Causal Imitation Learning
9. (Coming soon) Wrapping Up: Where To From Here?

## Preliminaries

As you probably recall from high school, probability and statistics are almost entirely formulated on the idea of drawing random samples from an experiment. One imagines observing realisations of outcomes from some set of possibilities when drawing from an assortment of independent and identically distributed (i.i.d.) events.…

## Causal Reinforcement Learning: A Primer

As part of any honours degree at the University of Cape Town, one is obliged to write a thesis ‘droning’ on about some topic. Luckily for me, applied mathematics can pertain to pretty much anything of interest. Lo and behold, my thesis on merging causality and reinforcement learning. This was entitled Climbing the Ladder: A Survey of Counterfactual Methods in Decision Making Processes and was supervised by Dr Jonathan Shock.

In this series of posts I will break down my thesis into digestible blog chucks and go into quite some detail of the emerging field of Causal Reinforcement Learning (CRL) – which is being spearheaded by Elias Bareinboim and Judea Pearl, among others. I will try to present this in such a way as to satisfy those craving some mathematical detail whilst also trying to paint a broader picture as to why this is generally useful and important. Each of these blog posts will be self contained in some way.…

## Covid-19 tests: probabilities

Bayes’ Theorem is applied to medical tests, to calculate the probability of being infected with a virus, given a positive or negative test result. What drives the uncertainty is false negative results, or false positive results. In this article, I give a practical outline as to how one can interpret one’s test result, after calculating the relevant probability using Bayes’ Theorem.

To start off with, we need two estimates. For a negative covid-19 test, we need the rate of false negative results, and the current actual prevalence of the disease in the community. On the other hand, for a positive covid-19 test, we need the rate of false positives, and the current prevalence of the disease. False outcomes in tests vary according to the laboratory doing the test, and probably also the skill with which each individual test is carried out, but, for the sake of a rational understanding of the usefulness of these tests, we can use common statistics to calculate feasible probabilities.…

By | January 1st, 2021|News|0 Comments

## A challenging limit

This post comes mostly from the youtube video by BlackPenRedPen found here: https://www.youtube.com/watch?v=89d5f8WUf1Y&t=3s

This in turn comes from Brilliant.com – details and links can be found in the original video

In this post we will have a look at a complicated-looking limit that has an interesting solution. Here it is:

$\lim_{n \rightarrow \infty} ( \frac{n!}{n^n})^{\frac{1}{n}}$

This looks pretty daunting – but we will break the solution down into sections:

• taking the logarithms and rearranging
• recognising something familiar
• finding the numerical value

Step 1: Taking the Logarithm

The first step here is to take the logarithm, a generally useful trick when applying limits. First we assign the variable L to the limit (so that we can solve for it in the end). Now lets do some algebra:

$L = \lim_{n \rightarrow \infty} ( \frac{n!}{n^n})^{\frac{1}{n}}$

$\ln(L) = \ln(\lim_{n \rightarrow \infty} ( \frac{n!}{n^n})^{\frac{1}{n}})$

Noting that the natural logarithm $\ln$ is a continuous function and therefore we can take the limit outside of the function:

$\ln(L) = \lim_{n \rightarrow \infty} \ln( (\frac{n!}{n^n})^{\frac{1}{n}})$

Next we can use the logarithm laws to bring down the exponent:

$\ln(L) = \lim_{n \rightarrow \infty} \frac{1}{n} \ln(\frac{n!}{n^n})$

Alright, now we have taken the logarithm, step 1 is complete.…

Introduction

In this post we will have a look at Parrondos paradox. In a paper* entitled “Information Entropy and Parrondo’s Discrete-Time Ratchet”** the authors demonstrate a situation where, by switching between 2 losing strategies, we can create a winning strategy.

Setup

The setup to this paradox is as follows:

We have 2 games that we can play – if we win we get 1 unit of wealth, if we lose, it costs 1 unit of wealth. Game A gives us a payout of 1 with a probability of slightly less than 0.5. Clearly if we play this game for long enough we will end up losing.

Game B is a little more complicated in that it is defined with reference to our existing winnings. If our current level of wealth is a multiple of M we play a game where the probability of winning is slightly less than 0.1. If it is not a multiple of M, the probability of winning is slightly less than 0.75.…

## Basic Reverse Image Search Using an Autoencoder

Introduction

In this post we are going to create a simple reverse image search on the MNIST handwritten image dataset. That is to say, given any image, we want to return images that look most similar to it. To do this, we will use an autoencoder, trained using Tensorflow 2.

The dataset

The MNIST dataset is a commonly-used dataset in machine learning comprised of 28-by-28 images of handwritten digits between 0 and 9. For our purposes we would be interested in our image searcher returning images of the same number as the query images, i.e. if we input a 3 we want the images returned to all be 3s. However, if we had, say, four 3s and one 2 that mightn’t be too bad, considering how 2 and 3 look a bit similar. However, if we had three 3s, one 1 and a 7 we might say that the performance is not up to standard.…

## A simple introduction to causal inference

Introduction

Causal inference is a branch of Statistics that is increasing in popularity. This is because it allows us to answer questions in a more direct way than do other methods. Usually, we can make inference about association or correlation between a variable and an outcome of interest, but these are often subject to outside influences and may not help us answer the questions in which we are most interested.

Causal inference seeks to remedy this by measuring the effect on the outcome (or response variable) that we see when we change another variable (the ‘treatment’). In a sense, we are looking to reproduce the situation that we have when we do an designed experiment (with a ‘treated’ and a ‘control’ group). The goal here is to have groups that are otherwise the same (with regard to factors that might influence the outcome) but where one is ‘treated’ and the other is not.…