Rényi entropy explained

In information theory, the Rényi entropy is a quantity that generalizes various notions of entropy, including Hartley entropy, Shannon entropy, collision entropy, and min-entropy. The Rényi entropy is named after Alfréd Rényi, who looked for the most general way to quantify information while preserving additivity for independent events. In the context of fractal dimension estimation, the Rényi entropy forms the basis of the concept of generalized dimensions.^[1]

The Rényi entropy is important in ecology and statistics as index 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 can be calculated explicitly because 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 0 < \alpha < inf in and \alpha ≠ 1 , is defined asΗ _{\alpha(X)} = (1)/(1- \alpha) log ( \sum ^{n} _{i =1} p ^{\alpha )} _{i} . It is further defined at \alpha = 0 , 1 , inf in as Η _{\alpha(X)} = \lim _{\gamma} Η _{\gamma(X)} .

Here,

X is a discrete random variable with possible outcomes in the set l{A} = \{x _{1,x} _{2,...,x} _{n\}} and corresponding probabilities p _{i} • eq \Pr(X = x _{i)} for i =1 , ... ,n . The resulting unit of information is determined by the base of the logarithm, e.g. shannon for base 2, or nat for base e.If the probabilities are p _{i=1/n} for all i =1 , ... ,n , then all the Rényi entropies of the distribution are equal: Η _{\alpha(X)=log} n .In general, for all discrete random variables X , Η _{\alpha(X)} is a non-increasing function in \alpha .

Applications often exploit the following relation between the Rényi entropy and the α-norm of the vector of probabilities:

Η _{\alpha(X) = (\alpha)/(1- \alpha)} log \left(\|P\| _{\alpha\right)} .Here, the discrete probability distribution P = (p _{1,...,p} _{n)} is interpreted as a vector in \R ^{n} with p _{i\geq} 0 and $\backslash sum\_^\; p\_i\; =1$.

The Rényi entropy for any

\alpha \geq 0 is Schur concave. Proven by the Schur-Ostrowski criterion.

Special cases

As

\alpha approaches zero, the Rényi entropy increasingly weighs all events with nonzero probability 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 . The limit for \alpha \to 1 is the Shannon entropy. As \alpha approaches infinity, the Rényi entropy is increasingly determined by the events of highest probability.

Hartley or max-entropy

Provided the probabilities are nonzero,^[2]

Η _{0} is the logarithm of the cardinality of the alphabet (l{A} ) of X , sometimes called the Hartley entropy of X ,Η _{0} (X) = log n = log | l{A}|

Shannon entropy

The limiting value of

Η _{\alpha} as \alpha \to 1 is the Shannon entropy:Η _{1} (X) \equiv \lim _{\alpha} Η _{\alpha} (X) = - \sum ^{n} _{i =1} p _{i} log p _{i}

Collision entropy

Collision entropy, sometimes just called "Rényi entropy", refers to the case

\alpha = 2 ,Η _{2} (X) = - log \sum ^{n} _{i =1} p ^{2} _{i} = - log P(X = Y) , where and are independent and identically distributed. The collision entropy is related to the index of coincidence.

Min-entropy

See main article: Min-entropy. In the limit as

\alpha → inf ty , the Rényi entropy Η _{\alpha} converges to the min-entropy Η _{infty} :Η _{infty(X)} • eq min _{i} ( -log p _{i)} = - ( max _{i} log p _{i)} = -log max _{i} p _{i.}

Equivalently, the min-entropy

Η _{infty(X)} is the largest real number such that all events occur with probability at most 2 ^{-b} .

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 for different orders α

That

Η _{\alpha} is non-increasing in \alpha for any given distribution of probabilities p _{i} ,which can be proven by differentiation, as- (d Η _{\alpha})/(d\alpha) = (1)/(( 1- \alpha) ^{2}) \sum ^{n} _{i =1} z _{i} log (z _{i} / p _{i)} = (1)/(( 1- \alpha) ^{2}) D _{KL} (z\| p) which is proportional to Kullback–Leibler divergence (which is always non-negative), wherez _{i} = p ^{\alpha} _{i} / \sum ^{n} _{j =1} p ^{\alpha} _{j} . In particular, it is strictly positive except when the distribution is uniform.

At the

\alpha \to 1 limit, we have - (d Η _{\alpha})/(d\alpha) \to (12)/(\sum) _{i} p _{i} ( ln p _{i} + H(p)) ^{2} .

In particular cases inequalities can be proven also by Jensen's inequality:^{[3] [4]}

log n =Η _{0\geq} Η _{1} \geq Η _{2} \geq Η _{infty} .

For values of

\alpha> 1 , inequalities in the other direction also hold. In particular, we have^{[5] [6]} Η _{2} \le 2 Η _{infty} .

On the other hand, the Shannon entropy

Η _{1} can be arbitrarily high for a random variable X that has a given min-entropy. An example of this is given by the sequence of random variables X _{n} \sim \{ 0 , \ldots, n\} for n \geq 1 such that P(X _{n} = 0 ) = 1 / 2 and P(X _{n} = x) = 1 /( 2 n) since Η _{infty(X} _{n)} = log 2 but Η _{1(X} _{n)} = ( log 2 + log 2 n)/ 2 .

Rényi divergence

As well as the absolute Rényi entropies, Rényi also defined a spectrum of divergence measures generalising the Kullback–Leibler divergence.^[7]

The Rényi divergence of order or alpha-divergence of a distribution from a distribution is defined to be

D _{\alpha} (P \| Q) = (1)/(\alpha -1) log ( \sum ^{n} _{i =1} (p ^{\alpha} _{i})/(q ^{\alpha -1} _{i}) ) = (1)/(\alpha -1) log E _{i} [(p _{i/q} ^{\alpha -1} ^{i)} ] when and . We can define the Rényi divergence for the special values by taking a limit, and in particular the limit gives the Kullback–Leibler divergence.

Some special cases:

D _{0(P} \| Q) = - log Q(\{i : p _{i} > 0 \}) : minus the log probability under that ;D _{1/2} (P \| Q) = -2 log \sum ^{n} _{i =1} \sqrt{p _{i} q _{i}} : minus twice the logarithm of the Bhattacharyya coefficient; D _{1(P} \| Q) = \sum ^{n} _{i =1} p _{i} log (p _{i})/(q _{i}) : the Kullback–Leibler divergence;D _{2(P} \| Q) = log \ langle (p _{i})/(q _{i}) \ rangle : the log of the expected ratio of the probabilities;D _{infty(P} \| Q) = log \sup _{i} (p _{i})/(q _{i}) : the log of the maximum ratio of the probabilities.

The Rényi divergence is indeed a divergence, meaning simply that

D _{\alpha} (P \| Q) is greater than or equal to zero, and zero only when . For any fixed distributions and, the Rényi divergence is nondecreasing as a function of its order, and it is continuous on the set of for which it is finite, or for the sake of brevity, the information of order obtained if the distribution is replaced by the distribution .

Financial interpretation

A pair of probability distributions can be viewed as a game of chance in which one of the distributions defines official odds and the other contains the actual probabilities. Knowledge of the actual probabilities allows a player to profit from the game. The expected profit rate is connected to the Rényi divergence as follows

{\rm ExpectedRate} = (1)/(R) D _{1(b\|m)} + (R -1)/(R) D _{1/R} (b\|m) , where m is the distribution defining the official odds (i.e. the "market") for the game, b is the investor-believed distribution and R is the investor's risk aversion (the Arrow–Pratt relative risk aversion).

If the true distribution is

p (not necessarily coinciding with the investor's belief b ), the long-term realized rate converges to the true expectation which has a similar mathematical structure{\rm RealizedRate} = (1)/(R) ( D _{1(p\|m)} - D _{1(p\|b)} ) + (R -1)/(R) D _{1/R} (b\|m) .

Properties specific to α = 1

The value, which gives the Shannon entropy and the Kullback–Leibler divergence, is the only value at which the chain rule of conditional probability holds exactly:

Η (A,X) = Η (A) + E _{a} [ Η (X| A = a) ] for the absolute entropies, andD _{KL(p(x|a)p(a)\|m(x,a))} = D _{KL(p(a)\|m(a))} + E _{p(a)} \{D _{KL(p(x|a)\|m(x|a))\},} for the relative entropies.

The latter in particular means that if we seek a distribution which minimizes the divergence from some underlying prior measure, and we acquire new information which only affects the distribution of, then the distribution of remains, 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 and are independent, so that if, then

Η _{\alpha(A,X)} = Η _{\alpha(A)} + Η _{\alpha(X)} andD _{\alpha(P(A)P(X)\|Q(A)Q(X))} = D _{\alpha(P(A)\|Q(A))} + D _{\alpha(P(X)\|Q(X)).}

The stronger properties of the quantities allow the definition of conditional information and mutual information from communication theory.

Exponential families

The Rényi entropies and divergences for an exponential family admit simple expressions

Η _{\alpha(p} _{F(x;\theta))} = (1)/(1- \alpha) \left(F(\alpha\theta) - \alpha F(\theta) +log E ^{(\alpha -1 )k(x)} _{p[e} ]\right) and D _{\alpha(p:q)} = (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.

Physical meaning

The Rényi entropy in quantum physics is not considered to be an observable, due to its nonlinear dependence on the density matrix. (This nonlinear dependence applies even in the special case of the Shannon entropy.) It can, however, be given an operational meaning through the two-time measurements (also known as full counting statistics) of energy transfers.

The limit of the quantum mechanical Rényi entropy as

\alpha\to 1 is the von Neumann entropy.

See also

• Diversity indices
• Tsallis entropy
• Generalized entropy index

References

• Book: Beck . Christian . Schlögl . Friedrich . 1993 . Thermodynamics of chaotic systems: an introduction . Cambridge University Press . 0521433673.
• Jizba . P. . Arimitsu . T. . 2004 . The world according to Rényi: Thermodynamics of multifractal systems . Annals of Physics . 312 . 1 . 17–59 . cond-mat/0207707 . 2004AnPhy.312...17J . 10.1016/j.aop.2004.01.002 . 119704502.
• Jizba . P. . Arimitsu . T. . 2004 . On observability of Rényi's entropy . Physical Review E . 69 . 2 . 026128 . cond-mat/0307698 . 2004PhRvE..69b6128J . 10.1103/PhysRevE.69.026128. 14995541 . 39231939.
  • Franchini . F. . Its . A. R. . Korepin . V. E. . 2008 . Rényi entropy as a measure of entanglement in quantum spin chain . Journal of Physics A: Mathematical and Theoretical . 41 . 25302 . 025302 . 0707.2534 . 2008JPhA...41b5302F . 10.1088/1751-8113/41/2/025302 . 119672750.
  • Hero . A. O. . Michael . O. . Gorman . J. . 2002 . Alpha-divergence for Classification, Indexing and Retrieval . Technical Report CSPL-328 . Communications and Signal Processing Laboratory, University of Michigan . 10.1.1.373.2763.
• Its . A. R. . Korepin . V. E. . 2010 . Generalized entropy of the Heisenberg spin chain . Theoretical and Mathematical Physics . 164 . 3 . 1136–1139 . 10.1007/s11232-010-0091-6 . 2010TMP...164.1136I . 119525704.
• Nielsen . F. . Boltz . S. . 2010 . The Burbea-Rao and Bhattacharyya centroids . IEEE Transactions on Information Theory . 57 . 8 . 5455–5466 . 1004.5049 . 10.1109/TIT.2011.2159046 . 14238708.
• Nielsen . Frank . Nock . Richard . 2012 . A closed-form expression for the Sharma–Mittal entropy of exponential families . Journal of Physics A . 45 . 3 . 032003 . 1112.4221 . 2012JPhA...45c2003N . 10.1088/1751-8113/45/3/032003 . 8653096.
• Nielsen . Frank . Nock . Richard . 2011 . On Rényi and Tsallis entropies and divergences for exponential families . Journal of Physics A . 45 . 3 . 032003 . 1105.3259 . 2012JPhA...45c2003N . 10.1088/1751-8113/45/3/032003 . 8653096.
• Alfréd . Rényi . Alfréd Rényi . On measures of information and entropy . 1961 . Proceedings of the fourth Berkeley Symposium on Mathematics, Statistics and Probability 1960 . 547–561.
• Rosso . O. A. . 2006 . EEG analysis using wavelet-based information tools . Journal of Neuroscience Methods . 153 . 2 . 163–182 . 10.1016/j.jneumeth.2005.10.009 . 16675027 . 7134638.
• Zachos . C. K. . 2007 . A classical bound on quantum entropy . Journal of Physics A . 40 . 21 . F407–F412 . 10.1088/1751-8113/40/21/F02 . 2007JPhA...40..407Z . hep-th/0609148. 1619604.
• Nazarov . Y. . 2011 . Flows of Rényi entropies . Physical Review B . 84 . 10 . 205437 . 1108.3537 . 2015PhRvB..91j4303A . 10.1103/PhysRevB.91.104303 . 40312624.
• Ansari . Mohammad H. . Nazarov . Yuli V. . 2015 . Rényi entropy flows from quantum heat engines . Physical Review B . 91 . 10 . 104303 . 1408.3910 . 2015PhRvB..91j4303A . 10.1103/PhysRevB.91.104303 . 40312624 . .
• Ansari . Mohammad H. . Nazarov . Yuli V. . 2015 . Exact correspondence between Rényi entropy flows and physical flows . Physical Review B . 91 . 17 . 174307 . 1502.08020 . 2015PhRvB..91q4307A . 10.1103/PhysRevB.91.174307 . 36847902 . .
• Soklakov . A. N. . 2020 . Economics of Disagreement—Financial Intuition for the Rényi Divergence . Entropy . 22 . 8 . 860 . 10.3390/e22080860 . 33286632 . 7517462 . 1811.08308 . 2020Entrp..22..860S . . free.
• Ansari . Mohammad H. . van Steensel . Alwin . Nazarov . Yuli V. . 2019 . Entropy Production in Quantum is Different . Entropy . 21 . 9 . 854 . 1907.09241 . 10.3390/e21090854 . 198148019 . . free.
• Book: Rioul . Olivier . This is IT: A Primer on Shannon's Entropy and Information . 2021 . Information Theory . Progress in Mathematical Physics . 78 . Birkhäuser . 49–86 . 10.1007/978-3-030-81480-9_2 . 978-3-030-81479-3 . 204783328 . https://hal.telecom-paris.fr/hal-02287963/file/201811rioul.pdf.pdf.

Notes and References

1. Barros . Vanessa . Rousseau . Jérôme . 2021-06-01 . Shortest Distance Between Multiple Orbits and Generalized Fractal Dimensions . Annales Henri Poincaré . en . 22 . 6 . 1853–1885 . 10.1007/s00023-021-01039-y . 209376774 . 1424-0661. 1912.07516 . 2021AnHP...22.1853B .
2. http://tools.ietf.org/html/rfc4086.html#page-6 RFC 4086, page 6
3. Η _{1} \ge Η _{2} holds because \sum\limits ^{M} _{i =1} {p _{i} log p _{i} } \le log \sum\limits ^{M} _{i =1} {p ^{2} _{i} } .
4. Η _{infty} \le Η _{2} holds because log \sum\limits ^{n} _{i =1} {p ^{2} _{i} } \le log \sup _{i} p _{i} \left( {\sum\limits ^{n} _{i =1} {p _{i} } } \right) = log \sup _{i} p _{i} .
5. Η _{2} \le 2 Η _{infty} holds because log \sum\limits ^{n} _{i =1} {p ^{2} _{i} } \ge log \sup _{i} p ^{2} _{i} = 2log \sup _{i} p _{i}
6. Book: Devroye . Luc . A Probabilistic Theory of Pattern Recognition . Györfi . Laszlo . Lugosi . Gabor . 1996-04-04 . Springer . 978-0-387-94618-4 . Corrected . New York, NY . English.
7. Van Erven. Tim. Harremoës. Peter. Rényi Divergence and Kullback–Leibler Divergence. IEEE Transactions on Information Theory. 2014. 60. 7. 3797–3820. 10.1109/TIT.2014.2320500. 1206.2459. 17522805.