gaussian process

Tags: #math #gaussian process

Equation

$$\begin{gathered} \log p(y|X)\propto -[y^T(K+{\sigma }^{2}I{)}^{-1}y+\log |K+{\sigma }^{2}I|] \\ f(X)=\text{[}f(x_1),f(x_2),...,f(x_N)){\text{]}}^T\sim \mathcal{N}(\mu ,K_{X,X}) \\ {f}_{\ast }|X_{\ast },X,y\sim \mathcal{N}(\mathbb{E}(f_{\ast }),\text{cov}(f_{\ast })) \\ \text{cov}(f_{\ast })=K_{X_{\ast },X_{\ast }}-K_{X_{\ast },X}[K_{X,X}+{\sigma }^{2}I{]}^{-1}{K}_{X,X_{\ast }} \end{gathered}$$

Latex Code

Have Fun

Let's Vote for the Most Difficult Equation!

Introduction

Equation

Joint Gaussian Distribution assumption

Probabilistic framework for GP

Prediction on new unseen data

Latex Code

Explanation

Gaussian process assumes that the output of N function are not independent but correlated. It assumes the collection of N function values, represented by N-dimensional vector f, has a joint Gaussian distribution with mean vector and covariance matrix(kernel matrix). The predicted value of n^{*} test values are given by mean and variance as \mathbb{E}(f_{*}) and \text{cov}(f_{*}) respectively. See below link Deep Kernel Learning for more details.