# A measure of non-uniformity of a vector/probability distribution?

In the course of a research project about discrete probability distributions, my coauthors and I keep seeing some quantity appear, and I would like to understand whether it has been studied or has a characterization in the literature (I couldn't find any besides the 3 references below).

Let $$p$$ be a discrete probability distribution over some discrete set $$\mathcal{X}$$, seen as a vector $$(p_i)_{i\in\mathcal{X}}$$. The quantity is then $$s(p) := \frac{\lVert p\rVert_3^3 - \lVert p\rVert_2^4}{\lVert p\rVert_2^3} \tag{\dagger}$$ (one could also consider $$t(p) := \frac{\lVert p\rVert_3^3 - \lVert p\rVert_2^4}{\lVert p\rVert_2^4}$$ if it's more convenient, both are equivalent for my purposes.) Note that the lack of homogeneity is just apparent, as we enforce $$\lVert p\rVert_1=1$$ (it's a probability distribution).

This quantity intuitively captures "how non-uniform" $$p$$ is: for instance, $$s(p)=0$$ iff $$p$$ is uniform on its support, and [1,2] give a (loose) connection between $$t(p)$$ and the total variation distance of $$p$$ to the closest uniform distribution. The numerator also appears in the analysis of the empirical estimator for the collision probability of a distribution [3].

We also have the view that, looking at the random variable $$Y=p(X)$$ for $$X\sim p$$, $$\mathbb{E}[Y] = \lVert p\rVert_2^2, \qquad \operatorname{Var}[Y] = \lVert p\rVert_3^3 - \lVert p\rVert_2^4$$ which maybe may be useful? For instance, by Chebyshev's inequality, $$\mathbb{P}\{ |p(i) - \lVert p\rVert_2^2 |> \alpha\lVert p\rVert_2^2 \} \leq \frac{t(p)}{\alpha^2}$$ thus giving a lower bound on the total mass (under $$p$$) of elements whose probability is within a factor $$\alpha$$ of $$\lVert p\rVert_2^2$$.

Do the quantities $$s(p)$$ (or $$t(p)$$) appear in the literature, and are either of them known to characterize (in some quantitative sense) how "non-uniform" a probability vector is?

[1] Batu, Tuğkan; Canonne, Clément L. Generalized uniformity testing. 58th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2017, 880--889, IEEE Computer Soc..

[2] Diakonikolas, Ilias, Daniel M. Kane, and Alistair Stewart. Sharp bounds for generalized uniformity testing. In Advances in Neural Information Processing Systems, pp. 6201-6210. 2018.

[3] Diakonikolas, Ilias; Gouleakis, Themis; Peebles, John; Price, Eric. Collision-based testers are optimal for uniformity and closeness. Chic. J. Theoret. Comput. Sci. 2019.

• Why the downvote? Is there something wrong with my question, and if so, how can I improve it? – Clement C. Jun 12 '20 at 19:20
• isn't the $L_2$ norm $||p||_2^2$ the typical way to measure nonuniformity of a distribution? – Carlo Beenakker Jun 22 '20 at 13:24
• @CarloBeenakker It is definitely one way. But it has some issues as well, and doesn't quite cut it: for instance, when the support size $k$ is large enough you can have two distributions with same $\ell_2$ norm, but one with all elements within a multiplicative factor say $2$ of $1/k$, and the other where a single (or a constant number of elements) with probabity $\approx 1/\sqrt{k}$ and all the others probability $\approx 1/k$. – Clement C. Jun 22 '20 at 15:00