# Intuition about the Gaussian processes

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 $$t^{'}$$.

• What exactly do you mean by “distinguishing” them?
– Tim
Commented Feb 10, 2021 at 21:03
• @Tim to look at the figure and say something about the behaviour of $\Sigma(t_1,t_2)$ for $t_1 \neq t_2$ Commented Feb 10, 2021 at 21:07
• Nothing, since this shows only a marginal plot. To show a covariance matrix, you would need to plot something it as a heatmap etc.
– Tim
Commented Feb 11, 2021 at 9:00