Perplexity, cross/conditional entropy and law of total variance

I was reading about the concept of Perplexity and was thinking whether there's a connexion with the law of law of total variance, but I couldn't find any reference.

The law of total variance is:

$$Var(Y) = E[Var(Y|X)] + Var(E[Y|X])$$

which can be interpreted as saying that the total variance of $Y$ can be decomposed into a part that is explained by $X$ and the part that is left unexplained (see algo Explained Variation for descriptions in terms of "information gained")

Let's say $Y$ is value of the next letter and $X$ contains all the previous letters written. Then a high ratio of explained vs unexplained variation would mean that the previous letters make it very likely that I predict the next letter with accuracy. This sounds very similar to the idea behind perplexity, so can you think of a way to connect them? (formally or not)

$$b^{{-{\frac {1}{N}}\sum _{{i=1}}^{N}\log _{b}q(x_{i})}}$$