# $\Sigma_*$ in Gaussian Process Prediction Formula

So posterior predictive distribution of the Gaussian process is given by the following equation.

$$\mathbf{f}_*$$ given new input $$\mathbf{X}_*:$$

$$p(\mathbf{f}_* \lvert \mathbf{X}_*,\mathbf{X},\mathbf{y}) = \int{p(\mathbf{f}_* \lvert \mathbf{X}_*,\mathbf{f})p(\mathbf{f} \lvert \mathbf{X},\mathbf{y})}\ d\mathbf{f} \\ = \mathcal{N}(\mathbf{f}_* \lvert \boldsymbol{\mu}_*, \boldsymbol{\Sigma}_*)$$

What I would like to ask is what is $$\Sigma_*$$ here? it was not defined in this post I read.

• Check Eq (5) in that post? – user137795 Jan 30 at 2:04
• Ok, I got what is it defined. But what is it called? – GENIVI-LEARNER Jan 31 at 14:42
• $\Sigma$ is convention notation for covariance matrix in multivariate normal distribution, see: en.wikipedia.org/wiki/Multivariate_normal_distribution. $\Sigma_*$ is used to denote "new covariance matrix" or "derived covariance matrix" I guess. – user137795 Jan 31 at 14:54
• ok so to clarify it is covariace and not variance? – GENIVI-LEARNER Jan 31 at 15:01
• Because in most common multivariate normal distribution parameterization, $N(\mu,\Sigma)$, $\mu$ is called expectation and $\Sigma$ is called covariance matrix. You can read "Conditional distributions" section in wikipedia to figure out how that post derived Eq (4) (5). – user137795 Jan 31 at 15:12