Min-entropy

In information theory, the Rényi entropy generalizes the Shannon entropy, the Hartley entropy, the min-entropy, and the collision entropy. Entropies quantify the diversity, uncertainty, or randomness of a system. The Rényi entropy is named after Alfréd Rényi.

The Rényi entropy is important in ecology and statistics as indices of diversity. The Rényi entropy is also important in quantum information, where it can be used as a measure of entanglement. In the Heisenberg XY spin chain model, the Rényi entropy as a function of α can be calculated explicitly by virtue of the fact that it is an automorphic function with respect to a particular subgroup of the modular group. In theoretical computer science, the min-entropy is used in the context of randomness extractors.

Definition

The Rényi entropy of order $\alpha$, where $\alpha \geq 0$ and $\alpha \neq 1$, is defined as

$H_\alpha\left(X\right) = \frac\left\{1\right\}\left\{1-\alpha\right\}\log\Bigg\left(\sum_\left\{i=1\right\}^n p_i^\alpha\Bigg\right)$ .

Here, $X$ is a discrete random variable with possible outcomes $1,2,...,n$ and corresponding probabilities $p_i \doteq \Pr\left(X=i\right)$ for $i=1,\dots,n$, and the logarithm is base 2. If the probabilities are $p_i=1/n$ for all $i=1,\dots,n$, then all the Rényi entropies of the distribution are equal: $H_\alpha\left(X\right)=\log n$. In general, for all discrete random variables $X$, $H_\alpha\left(X\right)$ is a non-increasing function in $\alpha$.

Applications often exploit the following relation between the Rényi entropy and the p-norm:

$H_\alpha\left(X\right)=\frac\left\{\alpha\right\}\left\{1-\alpha\right\} \log \left\left(\|X\|_\alpha\right\right)$ .

Here, the discrete probability distribution $X$ is interpreted as a vector in $\R^n$ with $X_i=p_i\geq 0$ and $\sum_\left\{i=1\right\}^\left\{n\right\} X_i =1$.

Special cases of the Rényi entropy

As $\alpha$ approaches zero, the Rényi entropy increasingly weighs all possible events more equally, regardless of their probabilities. In the limit for $\alpha\to 0$, the Rényi entropy is just the logarithm of the size of the support of $X$. The limit for $\alpha\to 1$ equals the Shannon entropy, which has special properties. As $\alpha$ approaches infinity, the Rényi entropy is increasingly determined by the events of highest probability.

Hartley entropy

Provided the probabilities are nonzero, $H_0$ is the logarithm of the cardinality of X, sometimes called the Hartley entropy of X:

$H_0 \left(X\right) = \log n = \log |X|.\,$

Shannon entropy

In the limit $\alpha \rightarrow 1$, it can be shown using L'Hôpital's Rule that $H_\alpha$ converges to the Shannon entropy:

$H_1 \left(X\right) = - \sum_\left\{i=1\right\}^n p_i \log p_i.$

Collision entropy

Collision entropy, sometimes just called "Rényi entropy," refers to the case $\alpha = 2$,

$H_2 \left(X\right) = - \log \sum_\left\{i=1\right\}^n p_i^2 = - \log P\left(X = Y\right)$

where X and Y are independent and identically distributed.

Min-entropy

In the limit as $\alpha \rightarrow \infty$, the Rényi entropy $H_\alpha$ converges to the min-entropy $H_\infty$:

$H_\infty\left(X\right) \doteq \min_\left\{i=1\right\}^n \left(-\log p_i\right) = -\left(\max_i \log p_i\right) = -\log \max_i p_i\,.$

Equivalently, the min-entropy $H_\infty\left(X\right)$ is the largest real number $b$ such that all events occur with probability at most $2^\left\{-b\right\}$.

The name min-entropy stems from the fact that it is the smallest entropy measure in the family of Rényi entropies. In this sense, it is the strongest way to measure the information content of a discrete random variable. In particular, the min-entropy is never larger than the Shannon entropy.

The min-entropy has important applications for randomness extractors in theoretical computer science: Extractors are able to extract randomness from random sources that have a large min-entropy; merely having a large Shannon entropy does not suffice for this task.

Inequalities between different values of α

That $H_\alpha$ is non-increasing in $\alpha$, which can be proven by differentiation, as

$-\frac\left\{d H_\alpha\right\}\left\{d\alpha\right\}$

= -\frac{1}{(1-\alpha)^2} \sum_{i=1}^n z_i \log(z_i / p_i), which is proportional to Kullback–Leibler divergence (which is always non-negative), where $z_i = p_i^\alpha / \sum_\left\{j=1\right\}^n p_j^\alpha$.

In particular cases inequalities can be proven also by Jensen's inequality:

$\log n=H_0\geq H_1 \geq H_2 \geq H_\infty$ .,

For values of $\alpha>1$, inequalities in the other direction also hold. In particular, we have

$H_2 \le 2H_\infty$ .

On the other hand, the Shannon entropy $H_1$ can be arbitrarily high for a random variable $X$ that has a constant min-entropy.

Rényi divergence

As well as the absolute Rényi entropies, Rényi also defined a spectrum of divergence measures generalising the Kullback–Leibler divergence.

The Rényi divergence of order α, where α > 0, of a distribution P from a distribution Q is defined to be:

$D_\alpha \left(P \| Q\right) = \frac\left\{1\right\}\left\{\alpha-1\right\}\log\Bigg\left(\sum_\left\{i=1\right\}^n \frac\left\{p_i^\alpha\right\}\left\{q_i^\left\{\alpha-1\right\}\right\}\Bigg\right) = \frac\left\{1\right\}\left\{\alpha-1\right\}\log \sum_\left\{i=1\right\}^n p_i^\alpha q_i^\left\{1-\alpha\right\}.\,$

Like the Kullback-Leibler divergence, the Rényi divergences are non-negative for α>0. This divergence is also known as the alpha-divergence ($\alpha$-divergence).

Some special cases:

: minus the log probability under Q that pi>0;
$D_\left\{1/2\right\}\left(P \| Q\right) = -2 \log \sum_\left\{i=1\right\}^n \sqrt\left\{p_i q_i\right\}$ : minus twice the logarithm of the Bhattacharyya coefficient;
$D_1\left(P \| Q\right) = \sum_\left\{i=1\right\}^n p_i \log \frac\left\{p_i\right\}\left\{q_i\right\}$ : the Kullback-Leibler divergence;
$D_2\left(P \| Q\right) = \log \Big\langle \frac\left\{p_i\right\}\left\{q_i\right\} \Big\rangle \,$ : the log of the expected ratio of the probabilities;
$D_\infty\left(P \| Q\right) = \log \sup_i \frac\left\{p_i\right\}\left\{q_i\right\}$ : the log of the maximum ratio of the probabilities.

Why α=1 is special

The value α = 1, which gives the Shannon entropy and the Kullback–Leibler divergence, is special because it is only at α=1 that the chain rule of conditional probability holds exactly:

$H\left(A,X\right) = H\left(A\right) + \mathbb\left\{E\right\}_\left\{a \sim A\right\} \big\left[ H\left(X| A=a\right) \big\right]$

for the absolute entropies, and

$D_\mathrm\left\{KL\right\}\left(p\left(x|a\right)p\left(a\right)||m\left(x,a\right)\right) = D_\mathrm\left\{KL\right\}\left(p\left(a\right)||m\left(a\right)\right) + \mathbb\left\{E\right\}_\left\{p\left(a\right)\right\}\\left\{D_\mathrm\left\{KL\right\}\left(p\left(x|a\right)||m\left(x|a\right)\right)\\right\},$

for the relative entropies.

The latter in particular means that if we seek a distribution p(x,a) which minimizes the divergence from some underlying prior measure m(x,a), and we acquire new information which only affects the distribution of a, then the distribution of p(x|a) remains m(x|a), unchanged.

The other Rényi divergences satisfy the criteria of being positive and continuous; being invariant under 1-to-1 co-ordinate transformations; and of combining additively when A and X are independent, so that if p(A,X) = p(A)p(X), then

$H_\alpha\left(A,X\right) = H_\alpha\left(A\right) + H_\alpha\left(X\right)\;$

and

$D_\alpha\left(P\left(A\right)P\left(X\right)\|Q\left(A\right)Q\left(X\right)\right) = D_\alpha\left(P\left(A\right)\|Q\left(A\right)\right) + D_\alpha\left(P\left(X\right)\|Q\left(X\right)\right).$

The stronger properties of the α = 1 quantities, which allow the definition of conditional information and mutual information from communication theory, may be very important in other applications, or entirely unimportant, depending on those applications' requirements.

Exponential families

The Rényi entropies and divergences for an exponential family admit simple expressions (Nielsen & Nock, 2011)



H_\alpha(p_F(x;\theta)) = \frac{1}{1-\alpha} \left(F(\alpha\theta)-\alpha F(\theta)+\log E_p[e^{(\alpha-1)k(x)}]\right)

and



D_\alpha(p:q) = \frac{J_{F,\alpha}(\theta:\theta')}{1-\alpha} where



J_{F,\alpha}(\theta:\theta')= \alpha F(\theta)+(1-\alpha) F(\theta')- F(\alpha\theta+(1-\alpha)\theta') is a Jensen difference divergence.