### What is the Uniform Distribution?

#### Posted by Tom Leinster

Today I gave the Statistics and Data Science seminar at Queen Mary University of London, at the kind invitation of Nina Otter. There I explained an idea that arose in work with Emily Roff. It’s an answer to this question:

What is the “canonical” or “uniform” probability distribution on a metric space?

You can see my slides here, and I’ll give a lightning summary of the ideas now.

Let $X$ be a compact metric space.

**Step 1**The uniform probability distribution (or more formally, probability measure) on $X$ should be one that’s highly spread out. So, we need to be able to quantify the “spread” of a probability distribution on a metric space.There are many such measures of spread — a whole one-parameter family of them, in fact. They’re the

**diversities**$(D_q)_{q \in \mathbb{R}^+}$. Or if you prefer, you can work with the entropies $\log D_q$; it makes little difference.**Step 2**We now appear to have a problem. Different values of $q$ give different diversity measures $D_q$, so it seems to be hoping for*way*too much for there to be a probability measure on $X$ that maximizes $D_q$ for all uncountably many $q$s at once.But miraculously, there is! Call it the

**maximizing measure**on $X$.**Step 3**Statisticians are very familiar with the idea of a maximum entropy distribution as being somehow canonical or preferable. But it’s*not*what we should call the uniform measure, as it’s not scale-invariant. For example, converting our metric from centimetres to inches would change the maximizing measure, and that’s not good.The idea now is to take the large-scale limit. In other words, for each scale factor $t \gt 0$, write $\mathbb{P}_t$ for the maximizing measure on the scaled space $t X$, and define the

**uniform measure**on $X$ to be $\lim_{t \to \infty} \mathbb{P}_t$. This*is*scale-invariant.**Step 4**Let’s check this gives sensible results. We already know what “uniform distribution” should mean when $X$ is finite, or homogeneous (it should mean Haar measure), or a subset of Euclidean space (it should mean normalized Lebesgue measure). Does our general definition of uniform measure give the right thing in these cases? Yes, it does!There’s also a connection between uniform measures and the Jeffreys prior, an “objective” or “noninformative” prior derived from Fisher information.

You can find all this and more in the slides.

## Re: What is the Uniform Distribution?

Very nice! I’d be very interested to hear if you get any good suggested answers to your questions on the last slide, especially