As the title of the post suggests, this week I will discuss a geometric intuition for Markov’s inequality, which for a nonnegative random variable, $X$, states
P(X \geq a) \leq E[X]/a.
This is a simple result in basic probability that still felt surprising every time I used it… until very recently. (Warning: Basic measure theoretic probability lies ahead. These notes look like they provide sufficient background if this post is confusing and you are sufficiently motivated!)
Continue reading “A Geometric Intuition for Markov’s Inequality”
Mutual information is a quantification of the dependency between random variables. It is sometimes contrasted with linear correlation since mutual information captures nonlinear dependence. In this short note I will discuss the relationship between these quantities in the case of a bivariate Gaussian distribution, and I will explore two implications of that relationship.
Continue reading “Correlation and Mutual Information”
In my last blog post I wrote about the asymptotic equipartition principle. This week I will write about something completely unrelated.
This blog post evolved from a discussion with Brendan O’Connor about science and evidence. The back story is as follows. Continue reading “It Depends on the Model”
The Asymptotic Equipartition Property/Principle (AEP) is a well-known result that is likely covered in any introductory information theory class. Nevertheless, when I first learned about it in such a course, I did not appreciate the implications of its general form. In this post I will review this beautiful, classic result and offer the mental picture I have of its implications. I will frame my discussion in terms of Markov chains with discrete state spaces, but note that the AEP holds even more generally. My treatment will be relatively informal, and I will assume basic familiarity with Markov chains. See the references for more details.
Continue reading “Asymptotic Equipartition of Markov Chains”