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In information theory, the Rényi entropy generalizes the Hartley entropy, the Shannon entropy, the collision entropy and the min entropy. Entropies quantify the diversity, uncertainty, or randomness of a system. The Rényi entropy is named after Alfréd Rényi.[1] In the context of fractal dimension estimation, the Rényi entropy forms the basis of the concept of generalized dimensions.

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 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.[2][3] In theoretical computer science, the min-entropy is used in the context of randomness extractors.

Contents

DefinitionEdit

The Rényi entropy of order  , where   and  , is defined as

  .[1]

Here,   is a discrete random variable with possible outcomes   and corresponding probabilities   for  , and the logarithm is base 2. If the probabilities are   for all  , then all the Rényi entropies of the distribution are equal:  . In general, for all discrete random variables  ,   is a non-increasing function in  .

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

  .

Here, the discrete probability distribution   is interpreted as a vector in   with   and  .

The Rényi entropy for any   is Schur concave.

Special cases of the Rényi entropyEdit

 
Rényi entropy of a random variable with two possible outcomes against p1, where P = (p1, 1 − p1). Shown are H0, H1, H2 and H, in units of shannons.

As α approaches zero, the Rényi entropy increasingly weighs all possible events more equally, regardless of their probabilities. In the limit for α → 0, the Rényi entropy is just the logarithm of the size of the support of X. The limit for α → 1 is the Shannon entropy. As α approaches infinity, the Rényi entropy is increasingly determined by the events of highest probability.

Hartley or max-entropyEdit

Provided the probabilities are nonzero,[4]   is the logarithm of the cardinality of X, sometimes called the Hartley entropy of X,

 

Shannon entropyEdit

The limiting value of   as α → 1 is the Shannon entropy:[5]

 

Collision entropyEdit

Collision entropy, sometimes just called "Rényi entropy", refers to the case α = 2,

 

where X and Y are independent and identically distributed.

Min-entropyEdit

In the limit as  , the Rényi entropy   converges to the min-entropy  :

 

Equivalently, the min-entropy   is the largest real number b such that all events occur with probability at most  .

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 αEdit

That   is non-increasing in  , which can be proven by differentiation,[6] as

 

which is proportional to Kullback–Leibler divergence (which is always non-negative), where  .

In particular cases inequalities can be proven also by Jensen's inequality:[7][8]

 

For values of  , inequalities in the other direction also hold. In particular, we have[9][citation needed]

 

On the other hand, the Shannon entropy   can be arbitrarily high for a random variable   that has a given min-entropy.[citation needed]

Rényi divergenceEdit

As well as the absolute Rényi entropies, Rényi also defined a spectrum of divergence measures generalising the Kullback–Leibler divergence.[10]

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

 

when 0 < α < ∞ and α ≠ 1. We can define the Rényi divergence for the special values α = 0, 1, ∞ by taking a limit, and in particular the limit α → 1 gives the Kullback–Leibler divergence.

Some special cases:

  : minus the log probability under Q that pi > 0;
  : minus twice the logarithm of the Bhattacharyya coefficient; (Nielsen & Boltz (2009))
  : the Kullback–Leibler divergence;
  : the log of the expected ratio of the probabilities;
  : the log of the maximum ratio of the probabilities.

The Rényi divergence is indeed a divergence, meaning simply that   is greater than or equal to zero, and zero only when P = Q. For any fixed distributions P and Q, the Rényi divergence is nondecreasing as a function of its order α, and it is continuous on the set of α for which it is finite.[10]

Why α = 1 is specialEdit

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:

 

for the absolute entropies, and

 

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

 

and

 

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 familiesEdit

The Rényi entropies and divergences for an exponential family admit simple expressions [11]

 

and

 

where

 

is a Jensen difference divergence.

Physical meaningEdit

The Renyi entropy in quantum physics is not considered to be an observable, due to its nonlinear dependence on the density matrix. The Shannon entropy shares this nonlinear dependence. Recently, Ansari and Nazarov showed a correspondence that reveals the physical meaning of the Renyi entropy flow in time. His proposal is similar to the fluctuation-dissipation theorem in spirit and allows the measurement of the quantum entropy using the full counting statistics (FCS) of energy transfers.[12][13][14]

See alsoEdit

NotesEdit

  1. ^ a b Rényi (1961)
  2. ^ Franchini (2008)
  3. ^ Its (2010)
  4. ^ RFC 4086, page 6
  5. ^ Bromiley, Thacker & Bouhova-Thacker (2004)
  6. ^ Beck (1993)
  7. ^   holds because  .
  8. ^   holds because  .
  9. ^   holds because  
  10. ^ a b Van Erven, Tim; Harremoës, Peter (2014). "Rényi Divergence and Kullback–Leibler Divergence". IEEE Transactions on Information Theory. 60 (7): 3797–3820. arXiv:1206.2459 . doi:10.1109/TIT.2014.2320500. 
  11. ^ Nielsen & Nock (2011)
  12. ^ Nazarov (2011)
  13. ^ Ansari_Nazarov (2015a)
  14. ^ Ansari_Nazarov (2015b)

ReferencesEdit