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 Why α = 1 is special The value , which gives the Shannon entropy and the Kullback–Leibler divergence, is special because it is only at 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 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, 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 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