# Total Internal Reflection

Technology and Art

### Contact

Avishek Sen Gupta on 17 April 2021

This article aims to start the road towards a theoretical intuition behind Gaussian Processes, another Machine Learning technique based on Bayes’ Rule. However, there is a raft of material that I needed to understand and relearn before fully appreciating some of the underpinnings of this technique.

I’d like to do some high level dives into some of the topics I believe will help practictioners go a little deeper than “it’s just a Gaussian Process of many variables”.

## Theory Track for Gaussian Processes

The map below shows the rough order in which the preliminary material will be presented.

graph TD; quad[Quadratic Form of Matrix]-->chol[Cholesky Factorisation]; tri[Triangular Matrices]-->chol[Cholesky Factorisation]; det[Determinants]-->chol[Cholesky Factorisation]; jac[Jacobian]-->jaclin[Jacobian of Linear Transformations] cov[Covariance Matrix]-->mvn[Multivariate Gaussian] chol[Cholesky Factorisation]-->mvn[Multivariate Gaussian] mvn[Multivariate Gaussian]-->mvnlin[MVN as Linearly Transformed Sums of Uncorrelated Random Variables] crv[Change of Random Variable]-->mvnlin[MVN as Linearly Transformed Sums of Uncorrelated Random Variables] jaclin[Jacobian of Linear Transformations]-->mvnlin[MVN as Linearly Transformed Sums of Uncorrelated Random Variables] diffeq[Difference Equations]-->diffmat[Difference Matrix]-->gp[Gaussian Processes] mvnlin[MVN as Linearly Transformed Sums of Uncorrelated Random Variables]-->Conditioning mvnlin[MVN as Linearly Transformed Sums of Uncorrelated Random Variables]-->Marginalisation Conditioning-->gp[Gaussian Processes] Marginalisation-->gp[Gaussian Processes] style chol fill:#006f00,stroke:#000,stroke-width:2px,color:#fff style mvn fill:#006fff,stroke:#000,stroke-width:2px,color:#fff style gp fill:#8f0f00,stroke:#000,stroke-width:2px,color:#fff