more binomial MADness

I posted earlier about the mean absolute deviation (MAD) of a binomial variable S_n with parameters (n,p). Here’s a little follow-up with plots. This is a plot of \mathbb{E}|S_n - np| versus p for different values of n.

The first is for n = 10. Looks beautifully scalloped, no? As we’d expect, the MAD is symmetric about p = 1/2 and monotonically increasing for the first half of the unit interval. Unfortunately, it’s clearly not concave (although it is piecewise concave), which means I have to do a bit more algebra later on.

When $n = 100$ the scallops turn into a finely serrated dome.

By the time you get to $n = 1000$ the thing might as well be concave for all that your eye can tell. But you would be deceived. Like a shark’s skin, the tiny denticles can abrade your proof, damaging it beyond repair.

Why do I care about this? If you take n samples from a Bernoulli variable with parameter p, then the empirical distribution (unnormalized) is (n - S_n, S_n). So \frac{1}{n} \mathbb{E}|S_n - np| is the expected total variational distance between the empirical distribution and its mean. More generally, the expected total variational distance for finite-alphabet distributions is a sum of MAD terms.



Some interesting stuff has passed my way while being in India (and one or two things from before). Might as well post them before I forget, no?

Slavoj Žižek may be a curmudgeonly Marxist, but the animation helps soften it, I think. I don’t think I fully agree with him, but there’s stuff in there to chew on.

The Purdue anonymization project won a big NSF award.

Tips for tasks related to graduating (h/t Bobak).

Some interesting news about the future of the textbook market. It’s doubly interesting since I am in Pune, a treasure-trove of cheaper editions of technical books.

Apparently I sometimes wear a lab coat.