Maths
Theoretical articles about machinelearning, statistics and other mathematical topics

Introduction to PAC Learning
What is “learning” and do we have a formal model for it? I’ve decided to dive into the theoretical underpinnings of machinelearning, so here’s a quick introduction to...

Understanding pvalues
Hypothesis testing and pvalues are often misused and misunderstood. In this article, I explain what a pvalue is, and how to use it.

Introduction to hypothesis testing
We introduce the basic vocabulary required to understand hypothesis testing and define the pvalue.

MLE: an information theory viewpoint
We show that the MLE is obtained by minimizing the KLdivergence from an empirical distribution and interpret what it means.

The maximum likelihood estimator (MLE)
The maximum likelihood estimator is one of the most used estimators in statistics. In this article, we introduce this estimator and study its properties.

Introduction to statistical estimators
In this article we define what an estimator is. We focus on the theory to compare and assess estimators, rather than how to find one.

The effect of L2regularization
When fitting a model to some training dataset, we want to avoid overfitting. A common method to do so is to use regularization. In this article, we discuss the impact of L2regularization on the estimated parameters of a linear model.

Underfitting and overfitting illustrated
In this article, we define underfitting and overfitting and show some nice ways to vizualize them on polynomial regressions.

How to assess an OLS regression?
We’ve just fitted OLS to our trainset. How to assess whether it was a good model to use? We will answer this question from the point of view...

OLS regressions from the probabilistic viewpoint
We will show that the loss function used by ordinary leastsquares (OLS) stems from the statistical theory of maximum likelihood estimation applied to the normal distribution.

Vector notation for linear regressions
A linear regression attempts to estimate an output value using a linear function. Those functions can be expressed concisely using the vector notations. In this article, we define...

Why there is more to classification than dicrete regression
We can use regressions methods to do classification, but this is suboptimal. Here's why.

Primer on stochastic convergence
Types of convergence: in distribution, in probability and the fundamental convergence theorems.

What is a statistic and why do we care?
In this article, we explain that a statistic is a way of compressing information contained in the data, and we show how it can be used for inference....

Derivative, Gradient and Jacobian unified
A summary about scalar and vector derivatives.

What is a generalized linear model?
To understand what a generalized linear model does, let’s look back at linear models.

Conditional expectations and regression with squared error loss
In this article we review the solution to a regression with squared error loss. We start with the theoretical formulation before tackling the problem in practice.

The MoorePenrose (pseudoinverse) matrix
The MoorePenrose inverse of a matrix is used to approximatively solve a degenerate system of linear equations.

Understanding and solving the normal equations
The normal equations arise in several branches of mathematics, from statistics to geometry. In this article, we discuss how they emerge and how to solve them.

The geometry of the normal equations
In this article, I show that the normal equations define the orthogonal projection of a vector onto a linear subspace.

Why do we care about convexity?
In machine learning, the best parameters for a model are chosen so as to minimize the training objective. Strictly convex functions are paticularly interesting because they have a...

A probability distribution to quantify measurement errors
In this article we will derive the normal distribution as the probability distribution that models measurement errors. We start with a dart game and follow Herschel’s derivation.

The geometry of (normal) parameter estimation
This article shows geometrically where the best estimates for the mean and variance of a normally distributed random vector can be found. We start with a simple question...

Why bayesian inference is more powerful than logic
In a previous article I showed that the inference rules of propositional logic can be obtained from probability calculus. But actually, we can obtain much more, and even...

Propositional logic derived as a special case of probability calculus
In this article, I will apply the rules of probability calculus to derive the rules of propositional logic (also called propositional calculus).

Key ideas in probability and statistics illustrated on a simple problem

Extending logic to deal with uncertainty
This article sketches a construction of probability calculus as an extension of classical logic to account for uncertainty so that by construction, it can be used to automate...

A Bayesian Perspective
Probability is not a property of an event or state; there is no such thing as the probability that the coin lands showing head. Probability expresses a strength...

An information theory perspective on probability
In 1948, Claude Shannon invented information theory based on probability theory. The basic definition is entropy. Given of a set of messages mi, each one occurring with probability...

The change of basis matrix
This article explains the intuition behind the change of basis matrix.

A nontechnical introduction to statistics
This article explains in simple terms the purpose of statistical theory and gives an overview of how it is used.