# What is the entropy of a Gaussian Process?

I have a very basic understanding of Gaussian Processes. From what I understand, a Gaussian process for a set $$X$$, is the assignment of a Gaussian distribution to every element of the set. This is meant to expand the idea of a function to the case where we don't have total information about a function.

We can define the entropy of a probability distribution $$p(x)$$ as follows:

$$S = \int_X p(x) \log p(x)$$

with $$x \in X$$. How do we compute the entropy of a Gaussian Process?

The reason for this guess is that a GP is defined as a collection of random variables $$X_{t_i}$$ indexed on some set $$T$$, such that, for every finite sequence of indices $$(t_1, ..., t_m) \in T^m$$ you pick, the joint distribution of $$(X_{t_1}, ..., X_{t_m})$$ is a multivariate normal distribution.
However, if you fix the indices (i.e. focus only on a set of points $$(t_1, ..., t_m)$$), then the entropy can be calculated, as the distribution $$(X_{t_1}, ..., X_{t_m}) \sim \mathcal N(\mu, \Sigma_{t_1, ..., t_m})$$, and the entropy of this multivariate normal is:
$$H(t_1, ..., t_m) = \frac{1}{2}\log|\Sigma_{t_1, ..., t_m}| + \frac{m}{2}\log (2 \pi e)$$