Renyi

Renyi, often written Rényi, is a Hungarian surname. The best-known bearer is Alfréd Rényi (1921–1970), a mathematician and probabilist who made influential contributions to probability theory, information theory, and combinatorics. His work helped shape modern approaches to randomness, entropy, and statistical modeling, and his ideas have influenced various areas of mathematics, statistics, and theoretical computer science.

In information theory, two widely used concepts bear the Rényi name: Rényi entropy and Rényi divergence. Rényi entropy is a one-parameter family of entropy measures for a discrete distribution P, defined as H_α(P) = (1/(1−α)) log ∑ p_i^α for α > 0, α ≠ 1. As α approaches 1, Rényi entropy converges to the Shannon entropy; as α → 0, H_α(P) approaches log of the support size, and as α → ∞, it relates to the logarithm of the inverse of the maximum probability. This family provides a flexible framework for assessing uncertainty and information content in diverse contexts, including coding, statistics, and dynamical systems.

Rényi divergence, or Rényi relative entropy, generalizes the Kullback–Leibler divergence. For distributions P and Q, D_α(P||Q) = (1/(α−1)) log ∑ p_i^α q_i^(1−α), defined for α > 0, α ≠ 1. It reduces to KL divergence as α → 1 and is used in hypothesis testing, machine learning, and information-theoretic analysis where robustness to tail behavior or alternative risk measures is desirable.

The spelling Renyi without diacritics appears in some publications and transliterations; Rényi remains the standard Hungarian rendering. The term Renyi/Rényi may refer to the person or to the associated entropy and divergence measures named in his honor.