As far as I undestand Gaussian process (GP) is generalization of the Normal Distribution to infinite dimensional case.
- $1d$ case : $x \sim e^{-(x-\mu)^2 / 2 \sigma^2}$
- $Nd$ case: $\mathbf{x} \sim e^{-(\mathbf{x}-\mu^T) \Sigma^{-1} (\mathbf{x}- \mu)/2}$
- (GP) : $x(t) \sim e^{-\int dt^{'} (x(t) - \mu(t)) \Sigma^{-1}(t,t^{'}) (x(t^{'}) - \mu(t^{'})) /2}$
It is alsl claimed in Wikipedia https://en.wikipedia.org/wiki/Gaussian_process, that restriction to the finite set of points is the multivariate Gaussian random variable.
My question is how to imagine visually the GP. I would think about a plot, where there is some function, representing the mean $\mu(t)$, and the shading around this mean represents the probability density.
And the width of the band at each point would correspond to the standard deviation at each point - $\Sigma(t, t)$. However, I wonder how to distinguish cases with different interpoint correlations $\Sigma(t, t^{'})$. Is there an intuitive way to distinguish the case, when is like $\Sigma(t, t) = \delta_{t t^{'}}$ from the case of long-range correlation between points at $t$ and $t^{'}$.
