Compass Blog – Page 11

Video: The Data Science behind COVID Modelling

Posted on 20th May 202121st May 2021 by crina.radu

We are excited to share Dr Daniel Lawson’s (Compass CDT Co-Director) latest video where he will tell you about the Data Science behind Bristol’s COVID Modelling.

Mathematics has had a hidden role in predicting how we can best fight COVID-19. How is mathematics used with data science and machine learning? Why is modelling epidemics such a hard problem? How can we do it better next time? What will data science be able to do in the future, and how do you become a part of it?

Find out more at https://www.bristolmathsresearch.org/data-science/ .

DataScience@work seminar: CheckRisk

Posted on 18th May 202124th September 2021 by Liz Wainwright

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Student Perspectives: Embedding probability distributions in RKHSs

Posted on 13th May 20216th October 2021 by jake.spiteri

A post by Jake Spiteri, Compass PhD student.

Recent advancements in kernel methods have introduced a framework for nonparametric statistics by embedding and manipulating probability distributions in Hilbert spaces. In this blog post we will look at how to embed marginal and conditional distributions, and how to perform probability operations such as the sum, product, and Bayes’ rule with embeddings.

Embedding marginal distributions

Throughout this blog post we will make use of reproducing kernel Hilbert spaces (RKHS). A reproducing kernel Hilbert space is simply a Hilbert space with some additional structure, and a Hilbert space is just a topological vector space equipped with an inner product, which is also complete.

We will frequently refer to a random variable $X$ which has domain $\mathcal{X}$ endowed with the $\sigma$ -algebra $\mathcal{B}_\mathcal{X}$ .

Definition. A reproducing kernel Hilbert space $\mathcal{H}$ on $\mathcal{X}$ with associated positive semi-definite kernel function $k: \mathcal{X} \times \mathcal{X} \rightarrow \mathbb{R}$ is a Hilbert space of functions $f: \mathcal{X} \rightarrow \mathbb{R}$ . The inner product $\langle\cdot,\cdot\rangle_\mathcal{H}$ satisfies the reproducing property: $\langle f, k(x, \cdot) \rangle_\mathcal{H}, \forall f \in \mathcal{H}, x \in \mathcal{X}$ . We also have $k(x, \cdot) \in \mathcal{H}, \forall x \in \mathcal{X}$ . (more…)