Maths to Share

Doing Calvin’s homework

Growing up, my siblings and I would read a lot of Bill Watterson’s Calvin and Hobbes. I have fond memories of spending hours reading and re-reading our grandparents collection during the school holidays. The comic strip is witty, heartwarming and beautifully drawn. Recently, I came across this strip online Seeing Calvin take this test took…

by michaelnhowes 6th May 2023

Bernoulli’s inequality and probability

Suppose we have independent events $E_1,E_2,\ldots,E_m$ , each of which occur with probability $1-\varepsilon$ . The event that all of the $E_i$ occur is $E = \bigcap_{i=1}^m E_i$ . By using independence we can calculate the probability of $E$ , $P(E) = P\left(\bigcap_{i=1}^m E_i\right) = \prod_{i=1}^m P(E_i) = (1-\varepsilon)^m.$ We could also get…

by michaelnhowes 27th Apr 202327th Apr 2023

Solving a system of equations vs inverting a matrix

I used to have trouble understanding why inverting an $n \times n$ matrix required the same order of operations as solving an $n \times n$ system of linear equations. Specifically, if $A$ is an $n\times n$ invertible matrix and $b$ is a length $n$ vector, then computing $A^{-1}$ …

by michaelnhowes 16th Apr 2023

Braids and the Yang-Baxter equation

I recently gave a talk on the Yang-Baxter equation. The focus of the talk was to state the connection between the Yang-Baxter equation and the braid relation. This connection comes from a system of interacting particles. In this post, I’ll go over part of my talk. You can access the full set of notes here.…

by michaelnhowes 12th Mar 2023

Concavity of the squared sum of square roots

The $p$ -norm of a vector $x\in \mathbb{R}^n$ is defined to be: $\displaystyle{\Vert x \Vert_p = \left(\sum_{i=1}^n |x_i|^p\right)^{1/p}}.$ If $p \ge 1$ , then the $p$ -norm is convex. When $0<p<1$ , this function is not convex and actually concave when all the entries of $x$ are non-negative. On a recent exam…

by michaelnhowes 13th Feb 202325th Feb 2023

Log-sum-exp trick with negative numbers

Suppose you want to calculate an expression of the form $\displaystyle{\log\left(\sum_{i=1}^n \exp(a_i) – \sum_{j=1}^m \exp(b_j)\right)},$ where $\sum_{i=1}^n \exp(a_i) > \sum_{j=1}^m \exp(b_j)$ . Such expressions can be difficult to evaluate directly since the exponentials can easily cause overflow errors. In this post, I’ll talk about a clever way to avoid such errors. If there were…

by michaelnhowes 21st Jan 2023

The beta-binomial distribution

The beta-binomial model is a Bayesian model used to analyze rates. For a great derivation and explanation of this model, I highly recommend watching the second lecture from Richard McElreath’s course Statistical Rethinking. In this model, the data, $X$ , is assumed to be binomially distributed with a fixed number of trail $N$ but…

by michaelnhowes 15th Jan 202315th Jan 2023

Two sample tests as correlation tests

Suppose we have two samples $Y_1^{(0)}, Y_2^{(0)},\ldots, Y_{n_0}^{(0)}$ and $Y_1^{(1)},Y_2^{(1)},\ldots, Y_{n_1}^{(1)}$ and we want to test if they are from the same distribution. Many popular tests can be reinterpreted as correlation tests by pooling the two samples and introducing a dummy variable that encodes which sample each data point comes from. In this…

by michaelnhowes 30th Nov 2022

MCMC for hypothesis testing

A Monte Carlo significance test of the null hypothesis $X_0 \sim H$ requires creating independent samples $X_1,\ldots,X_B \sim H$ . The idea is if $X_0 \sim H$ and independently $X_1,\ldots,X_B$ are i.i.d. from $H$ , then for any test statistic $T$ , the rank of $T(X_0)$ among $T(X_0), T(X_1),\ldots, T(X_B)$ …

by michaelnhowes 13th Nov 2022

How to Bake Pi, Sherman-Morrison and log-sum-exp

A few months ago, I had the pleasure of reading Eugenia Cheng’s book How to Bake Pi. Each chapter starts with a recipe which Cheng links to the mathematical concepts contained in the chapter. The book is full of interesting connections between mathematics and the rest of the world. One of my favourite ideas in…

by michaelnhowes 31st Oct 20221st Nov 2022

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