## 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 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.…

## Correlation vs Mutual Information

This post is based on a (very small) part of the (dense and technical) paper Fooled by Correlation by N.N. Taleb, found at (1)

Notes on the main ideas in this post are available from Universidad de Cantabria, found at (2)

The aims of this post are to 1) introduce mutual information as a measure of similarity and 2) to show the nonlinear relationship between correlation and information my means of a relatively simple example

Introduction

A significant part of Statistical analysis is understanding how random variables are related – how much knowledge about the value of one variable tells us about the value of another. This post will consider this issue in the context of Gaussian random variables. More specifically, we will compare- and discuss the relationship between- correlation and mutual information.

Mutual Information

The Mutual Information between 2 random variables is the amount of information that one gains about a random variable by observing the value of the other.…

## The Objective Function

In both Supervised and Unsupervised machine learning, most algorithms are centered around minimising (or, equivalently) maximising some objective function. This function is supposed to somehow represent what the model knows/can get right. Normally, as one would expect, the objective function does not always reflect exactly what we want.

The objective function presents 2 main problems: 1. how do we minimise it (the answer to this is up for debate and there is lots of interesting research about efficient optimisation of non-convex functions and 2) assuming we can minimise it perfectly, is it the correct thing to be minimising?

It is point 2 which is the focus of this post.

Let’s take the example of square-loss-linear-regression. To do so we train a linear regression model with a square loss $\mathcal{L}(\mathbf{w})=\sum_i (y_i - \mathbf{w}^Tx_i)^2$. (Where we are taking the inner product of learned weights with a vector of features for each observation to predict the outcome).…

Introduction

A key consideration when analysing stratified data is how the behaviour of each category differs and how these differences might influence the overall observations about the data. For example, a data set might be split into one large category that dictates the overall behaviour or there may be a category with statistics that are significantly different from the other categories that skews the overall numbers. These features of the data are important to be aware of and go find to prevent drawing erroneous conclusions from your analysis. Context, the source of the data and a careful analysis of the data can prevent this. Simpson’s paradox is an interesting result of some of these effects.

Simpson’s paradox is observed in statistics when a trend is observed in a number of different groups but it is not observed in the overall data or the opposite trend is observed.

Observing the overall data might therefore lead us to draw a conclusion, but when the data is grouped we might conclude something different.…

## The (Central) Cauchy distribution

The core of this post comes from Mathematical Statistics and Data Analysis by John A. Rice which is a useful resource for subjects such as UCT’s STA2004F.

Introduction

The Cauchy distribution has a number of interesting properties and is considered a pathological (badly behaved) distribution. What is interesting about it is that it is a distribution that we can think about in a number of different ways*, and we can formulate the probability density function these ways. This post will handle the derivation of the Cauchy distribution as a ratio of independent standard normals and as a special case of the Student’s t distribution.

Like the normal- and t-distributions, the standard form is centred on, and symmetric about 0. But unlike these distributions, it is known for its very heavy (fat) tails. Whereas you are unlikely to see values that are significantly larger or smaller than 0 coming from a normal distribution, this is just not the case when it comes to the Cauchy distribution.…