# Notes on Gaussian Process

** Published:**

My study notes on Gaussian Process and some useful resources.

## Useful Resources

A good GP textbook: Gaussian Processes for Machine Learning.

## Notes on Gaussian Process

# I. Introduction

Gaussian process (GP) is a non-parametric supervised machine learning method, which has been widely used to model nonlinear system dynamics as well. GP works to infer an unknown function \(y = f(x)\) based on the training set \(\mathcal{D}:= \{(x_i, y_i): i=1,\cdots,n\}\) with \(n\) noisy observations. Comparing with other machine learning techniques, GP has the following main merits:

GP provides an estimate of uncertainty or confidence in the predictions through the predictive variance, in addition to using the predictive mean as the prediction.

GP can work well with small datasets.

In the nature of Bayesian learning, GP incorporates prior domain knowledge of the unknwon system by defining kernel covariance function or setting hyperparameters.

Formally, a GP is defined as a collection of random variables, any Gaussian process finite number of which have a joint Gaussian distribution. A GP is fully specified by a mean function \(m(x)\) and a (kernel) covariance function \(k(x,x')\), which is denoted as \begin{align} f(x)\sim\mathcal{GP}(m(x),k(x,x’)) \end{align}

It aims to infer the function value \(f(x_*)\) on a new point \(x_{*}\) based on the observations \(\mathcal{D}\). According to the formal definition, the collection \((\boldsymbol f_{\mathcal{D}}, f(x_*))\) follows a joint Gaussian distribution with

\[[\boldsymbol f_{\mathcal{D}}; f(x_*)] \sim \mathcal{N} \Big( [ \boldsymbol m_{\mathcal{D}}; m(x_*) ], [ K_{\mathcal{D},\mathcal{D}}, \boldsymbol k_{ *,\mathcal{D}}; \boldsymbol k_{ *,\mathcal{D}}^\top, k(x_*,x_*) ] \Big)\]where vector \(\boldsymbol k_{*, \mathcal{D}}:= [ k(x_*,x_1); \cdots; k(x_*, x_n)]\), and matrix \(K_{\mathcal{D},\mathcal{D}}\) is the covariance matrix, whose \(ij\)-component is \(k(x_i,x_j)\). Then conditioning on the given observations \(\mathcal{D}\), it is known that the posterior distribution \(f(x_*)|(\boldsymbol f_{\mathcal{D}} =\boldsymbol y_{\mathcal{D}})\) is also a Gaussian distribution \(\mathcal{N}(\mu_{*|\mathcal{D}}, \sigma^2_{*|\mathcal{D}} )\) with the closed form

\begin{align*} \mu_{*|\mathcal{D}} & = m(x_*) + \sigma^2_{*|\mathcal{D}} & = \end{align*}