gaussian process
Tags: #math #gaussian processEquation
$$\log p(y|X) \propto -[y^{T}(K + \sigma^{2}I)^{-1}y+\log|K + \sigma^{2}I|] \\ f(X)=\[f(x_{1}),f(x_{2}),...,f(x_{N}))\]^{T} \sim \mathcal{N}(\mu, K_{X,X}) \\ f_{*}|X_{*},X,y \sim \mathcal{N}(\mathbb{E}(f_{*}),\text{cov}(f_{*})) \\ \text{cov}(f_{*})=K_{X_{*},X_{*}}-K_{X_{*},X}[K_{X,X}+\sigma^{2}I]^{-1}K_{X,X_{*}}$$Latex Code
\log p(y|X) \propto -[y^{T}(K + \sigma^{2}I)^{-1}y+\log|K + \sigma^{2}I|] \\
f(X)=\[f(x_{1}),f(x_{2}),...,f(x_{N}))\]^{T} \sim \mathcal{N}(\mu, K_{X,X}) \\
f_{*}|X_{*},X,y \sim \mathcal{N}(\mathbb{E}(f_{*}),\text{cov}(f_{*})) \\
\text{cov}(f_{*})=K_{X_{*},X_{*}}-K_{X_{*},X}[K_{X,X}+\sigma^{2}I]^{-1}K_{X,X_{*}}
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Introduction
Equation
Joint Gaussian Distribution assumption
Probabilistic framework for GP
Prediction on new unseen data
Latex Code
// Joint Gaussian Distribution assumption
f(X)=\[f(x_{1}),f(x_{2}),...,f(x_{N}))\]^{T} \sim \mathcal{N}(\mu, K_{X,X})
// Probabilistic framework for GP
\log p(y|X) \propto -[y^{T}(K + \sigma^{2}I)^{-1}y+\log|K + \sigma^{2}I|]
// Prediction on new unseen data
f_{*}|X_{*},X,y \sim \mathcal{N}(\mathbb{E}(f_{*}),\text{cov}(f_{*})) \\
\mathbb{E}(f_{*}) = \mu_{X_{*}}+K_{X_{*},X}[K_{X,X}+\sigma^{2}I]^{-1}(y-\mu_{x}) \\
\text{cov}(f_{*})=K_{X_{*},X_{*}}-K_{X_{*},X}[K_{X,X}+\sigma^{2}I]^{-1}K_{X,X_{*}}
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.
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