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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

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

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

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

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

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

Looking back on the blog

Next week I’ll be starting the second year of my statistics PhD. I’ve learnt a lot from taking the first year classes and studying for the qualifying exams. Some of what I’ve learnt has given me a new perspective on some of my old blog posts. Here are three things that I’ve written about before…

by michaelnhowes

Total Variation and Marathon Running

Total variation is a way of measuring how much a function f:[a,b] \to \mathbb{R} “wiggles”. In this post, I want to motivate the definition of total variation by talking about elevation in marathon running. Comparing marathon courses On July 24th I ran the 2022 San Francisco (SF) marathon. All marathons are the same distance,…

by michaelnhowes

Finding Australia’s youngest electorates with R

My partner recently wrote an article for Changing Times, a grassroots newspaper that focuses on social change. Her article, Who’s not voting? Engaging with First Nations voters and young voters, is about voter turn-out in Australia and an important read. While doing research for the article, she wanted to know which electorates in Australia had the…

by michaelnhowes

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