Rényi entropy. 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.

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 , where  and , is defined as
 
It is further defined at  as 
 

Here,  is a discrete random variable with possible outcomes in the set  and corresponding probabilities  for .  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  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 α-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. Proven by the Schur–Ostrowski criterion.

Special cases 

As  approaches zero, the Rényi entropy increasingly weighs all events with nonzero probability 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   is the Shannon entropy. As  approaches infinity, the Rényi entropy is increasingly determined by the events of highest probability.

Hartley or max-entropy 
 is  where  is the number of non-zero probabilities. If the probabilities are all nonzero, it is simply the logarithm of the cardinality of the alphabet () of , sometimes called the Hartley entropy of ,

Shannon entropy 
The limiting value of  as  is the Shannon entropy:

Collision entropy 
Collision entropy, sometimes just called "Rényi entropy", refers to the case ,
 
where  and  are independent and identically distributed. The collision entropy is related to the index of coincidence. It is the negative logarithm of the Simpson diversity index.

Min-entropy 

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 for different orders α 
That  is non-increasing in  for any given distribution of probabilities ,
which can be proven by differentiation, as
 
which is proportional to Kullback–Leibler divergence (which is always non-negative), where
. In particular, it is strictly positive except when the distribution is uniform.

At the  limit, we have .

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 given min-entropy. An example of this is given by the sequence of random variables  for  such that  and  since  but .

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  or alpha-divergence of a distribution  from a distribution  is defined to be
 
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:
 : minus the log probability under  that ;
 : 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.

The Rényi divergence is indeed a divergence, meaning simply that  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
 
where  is the distribution defining the official odds (i.e. the "market") for the game,  is the investor-believed distribution and  is the investor's risk aversion (the Arrow–Pratt relative risk aversion).

If the true distribution is  (not necessarily coinciding with the investor's belief ), the long-term realized rate converges to the true expectation which has a similar mathematical structure

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:
 
for the absolute entropies, and
 
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
 
and
 

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
 
and 
 
where 
 
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  is the von Neumann entropy.

See also 
 Diversity indices
 Tsallis entropy
 Generalized entropy index

Notes

References 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Information theory
Entropy and information