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.
The Rényi entropy of order , where and , is defined as
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 entropy
As approaches zero, the Rényi entropy increasingly weighs all possible events more equally, regardless of their probabilities. In the limit for , the Rényi entropy is just the logarithm of the size of the support of . The limit for equals the Shannon entropy, which has special properties. As approaches infinity, the Rényi entropy is increasingly determined by the events of highest probability.
Collision entropy, sometimes just called "Rényi entropy," refers to the case ,
where X and Y are independent and identically distributed.
In the limit as , the Rényi entropy converges to the min-entropy :
Equivalently, the min-entropy is the largest real number 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 α
That is non-increasing in , which can be proven by differentiation, 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:
For values of , inequalities in the other direction also hold. In particular, we have
On the other hand, the Shannon entropy can be arbitrarily high for a random variable that has a constant min-entropy.
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:
Like the Kullback-Leibler divergence, the Rényi divergences are non-negative for α>0. This divergence is also known as the alpha-divergence (-divergence).
Some special cases:
- : minus the log probability under Q that pi>0;
- : minus twice the logarithm of the Bhattacharyya coefficient;
- : the Kullback-Leibler divergence;
- : the log of the expected ratio of the probabilities;
- : the log of the maximum ratio of the probabilities.
Why α=1 is special
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
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.
The Rényi entropies and divergences for an exponential family admit simple expressions (Nielsen & Nock, 2011)
is a Jensen difference divergence.
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