Gaussian processes

There is no lecture this week!

Exercises take place as usual.

We introduced the notion of discrete entropy and discussed some of its motivations: the asymptotic equipartition property and optimal coding of random variables. We also discussed shortly the maximum entropy principle for the uniform measure. Thereafter we discussed differential entropy, calculated it for Gaussians and mentioned it is the maximal entropy distribution for fixed variance. We discussed the connection between discrete and differential entropy.

This week we will continue to think about entropy and Gaussians. In particular, we try to find a nicer way to look at why Gaussians maximize entropy using a dynamics approach. We also discuss entropy and its relation to CLT and try to prove the relevant entropy power inequality via Fisher information.

I am unfortunately ill, but I came up with the following plan:

The idea is to prove the theorem we stated last time for 4 equivalent definitions of the discrete Gaussian free field.

I suggest you come to class and try to do it together. The statement is given again in the document here and also some tips - first some vague tips on p.2 and then more detailed ones in the footnotes. Xiao will come at 11h15 to help you move forward when you got stuck, but it is much more useful if you try to first prove it yourself and maybe manage too. Doing it together would be nice as well.