## 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…

## Bernoulli’s inequality and probability

Suppose we have independent events , each of which occur with probability . The event that all of the occur is . By using independence we can calculate the probability of , We could also get…

## Solving a system of equations vs inverting a matrix

I used to have trouble understanding why inverting an matrix required the same order of operations as solving an system of linear equations. Specifically, if is an invertible matrix and is a length vector, then computing …

## 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.…

## Concavity of the squared sum of square roots

The -norm of a vector is defined to be: If , then the -norm is convex. When , this function is not convex and actually concave when all the entries of are non-negative. On a recent exam…

## Log-sum-exp trick with negative numbers

Suppose you want to calculate an expression of the form where . 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…

## 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, , is assumed to be binomially distributed with a fixed number of trail but…

## Two sample tests as correlation tests

Suppose we have two samples and 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…

## MCMC for hypothesis testing

A Monte Carlo significance test of the null hypothesis requires creating independent samples . The idea is if and independently are i.i.d. from , then for any test statistic , the rank of among …

## 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…

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