Introduction

Both variance and entropy are measures of uncertainty. Variance assumes values vary as points in a space separated by distances. In this document, the variance of a random vector refers to the variance of the distance from its mean (sum of the variances of each component).

Random one-hot vectors are a convenient spacial representation for categorical random variables. A one-hot vector has all components equal to $0$ except one component that equals $1$. This representation has been used in genetics [1]. For genetic loci with only two alleles, a one-hot vector has two redundant components. "Half" of such one-hot vectors are typically used in genetics (e.g. [2] p.40, [3], [4] ). The variance of the "half one-hot vector" is exactly half the variance of its full one-hot vector.

Main Result

Given $N$ independent random one-hot vectors $X_1$, $X_2$, ..., $X_N$, define $$X_* = X_1 \times X_2 \times \dots \times X_N$$ as the Cartesian product.

The variance of $X_*$ can be adjusted to form a lower bound to the collision entropy, $\operatorname{H}_2(X_*)$, and Shannon entropy, $\operatorname{H}(X_*)$: $$- \log_2{\left( 1 - \frac{ \operatorname{Var}({ X_*}) }{N} \right)} \; \le \; \frac{\operatorname{H}_2({ X_*})}{N} \; \le \; \frac{\operatorname{H}({ X_*})}{N}$$

If every $X_i$ takes only two equally likely values, then the lower bounds reach equality: $$-\log_2{\left( 1 - \frac{ \operatorname{Var}({ X_*}) }{N} \right)} = \frac{\operatorname{H}_2({ X_*})}{N} = \frac{\operatorname{H}({ X_*})}{N} = 1$$

Proof

Let $M_i$ be length of $X_i$ (the number of categorical values represented by $X_i$). Let $p_{i,j}$ represent the probability of $X_i$ taking the $j$-th categorical value.

For every $1 \le i \le N$, $$\sum_{j=1}^{M_i} p_{i,j} = 1$$

The expectation and variance of the $i$-th one-hot vector $X_i$ is $$\begin{aligned} \operatorname{E}\!\left({ X_i}\right) & = \left(\; {p}_{i,1} \;,\; {p}_{i,2} \;,\; \dots \;,\; {p}_{i,M_i} \;\right) \\ \operatorname{Var}({ X_i}) & = \sum_{j=1}^{M_i} p_{i,j} \left[ (1 - p_{i,j})^2 + \sum_{k \not= j} (0 - p_{i,k})^2 \right] \\ & = \sum_{j=1}^{M_i} p_{i,j} \left[ 1 - 2 p_{i,j} + \sum_{k=1}^{M_i} p_{i,k}^2 \right] \\ & = 1 - 2 \sum_{j=1}^{M_i} p_{i,j}^2 + \sum_{k=1}^{M_i} p_{i,k}^2 \\ & = 1 - \sum_{j=1}^{M_i} p_{i,j}^2 \end{aligned}$$

Thus the variance of $X_i$ equals the probability of two independent samples from $X_i$ being distinct. This probability of distinction has been called logical entropy [5].

The complement $$1 - \operatorname{Var}({ X_i}) = \sum_{j=1}^{M_i} p_{i,j}^2$$ is the chance of repetition, which is expected probability. Taking the negative log gives Rényi entropy of order 2, also called collision entropy: $$-\log_2{( 1 - \operatorname{Var}({ X_i}))} = -\log_2{\left( \sum_{j=1}^{M_i} p_{i,j}^2 \right)} = \operatorname{H}_2(X_i)$$ Since negative log is a concave function, the negative log of expected probability (collision entropy), is a lower bound to the expected negative log of probability (Shannon entropy) by Jensen's inequality: $$\operatorname{H}_2(X_i) = -\log_2{\left( \sum_{j=1}^{M_i} p_{i,j}^2 \right)} \le \sum_{j=1}^{M_i} p_{i,j} (-\log_2{p_{i,j}}) = \operatorname{H}({ X_i})$$

The total variance, can be adjusted to equal the average probability of one-hot vector repetition (per one-hot vector): $$1 - \frac{ \operatorname{Var}({ X_*}) }{N} = 1 - \frac{1}{N} \sum_{i=1}^N \operatorname{Var}({ X_i}) = \frac{1}{N} \sum_{i=1}^N \sum_{j=1}^{M_i} p_{i,j}^2$$

Negative log with Jensen's inequality can then establish yet another lower bound: $$-\log_2{\left( \frac{1}{N} \sum_{i=1}^N \sum_{j=1}^{M_i} p_{i,j}^2 \right)} \le \frac{1}{N} \sum_{i=1}^N \left( -\log_2{\sum_{j=1}^{M_i} p_{i,j}^2} \right) = \frac{1}{N} \sum_{i=1}^N \operatorname{H}_2(X_i)$$

Collision and Shannon entropy are additive for independent variables. Putting everything together we get $$-\log_2{\left( 1 - \frac{ \operatorname{Var}({ X_*}) }{N} \right)} \; \le \; \frac{\operatorname{H}_2({ X_*})}{N} \; \le \; \frac{\operatorname{H}({ X_*})}{N}$$

References