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 entropy is named after Alfréd Rényi. 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. In theoretical computer science, the min-entropy is used in the context of randomness extractors.
- 1 Definition
- 2 Special cases of the Rényi entropy
- 3 Inequalities between different values of α
- 4 Rényi divergence
- 5 Financial interpretation
- 6 Why α = 1 is special
- 7 Exponential families
- 8 Physical meaning
- 9 See also
- 10 Notes
- 11 References
The Rényi entropy of order , where and , is defined as
Here, is a discrete random variable with possible outcomes and corresponding probabilities for . The logarithm is conventionally taken to be base 2, especially in the context of information theory where bits are used. 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 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-entropy
Collision entropy, sometimes just called "Rényi entropy", refers to the case α = 2,
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 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 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 .
On the other hand, the Shannon entropy can be arbitrarily high for a random variable that has a given min-entropy.
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;
- : 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.
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
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.
is a Jensen difference divergence.
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.
Recently a new replica trick formalism, the so-called multiple parallel world formalism, has revolutionized entropy evaluation for quantum systems. This is one of the first time that quantum information theory is equipped with the notion of time.
- Rényi (1961)
- Franchini (2008)
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- RFC 4086, page 6
- Bromiley, Thacker & Bouhova-Thacker (2004)
- Beck (1993)
- holds because .
- holds because .
- holds because
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