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.

  • $\begingroup$ Check Eq (5) in that post? $\endgroup$ – user137795 Jan 30 at 2:04
  • $\begingroup$ Ok, I got what is it defined. But what is it called? $\endgroup$ – GENIVI-LEARNER Jan 31 at 14:42
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    $\begingroup$ $\Sigma$ 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. $\endgroup$ – user137795 Jan 31 at 14:54
  • $\begingroup$ ok so to clarify it is covariace and not variance? $\endgroup$ – GENIVI-LEARNER Jan 31 at 15:01
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    $\begingroup$ 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). $\endgroup$ – user137795 Jan 31 at 15:12

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