equals unity by definition, the denominator equals the weighted geometric mean of the values, with the values themselves being used as the weights (exponents in the equation). The term within the parentheses hence equals true diversity , and {{math|H}} equals . When all types in the dataset of interest are equally common, all values equal , and the Shannon index hence takes the value . The more unequal the abundances of the types, the larger the weighted geometric mean of the values, and the smaller the corresponding Shannon entropy. If practically all abundance is concentrated to one type, and the other types are very rare (even if there are many of them), Shannon entropy approaches zero. When there is only one type in the dataset, Shannon entropy exactly equals zero (there is no uncertainty in predicting the type of the next randomly chosen entity). In machine learning the Shannon index is also called as Information gain. Rényi entropy The Rényi entropy is a generalization of the Shannon entropy to other values of than unity. It can be expressed: which equals This means that taking the logarithm of true diversity based on any value of gives the Rényi entropy corresponding to the same value of . Simpson index The Simpson index was introduced in 1949 by Edward H. Simpson to measure the degree of concentration when individuals are classified into types. The same index was rediscovered by Orris C. Herfindahl in 1950. The square root of the index had already been introduced in 1945 by the economist Albert O. Hirschman. As a result, the same measure is usually known as the Simpson index in ecology, and as the Herfindahl index or the Herfindahl–Hirschman index (HHI) in economics. The measure equals the probability that two entities taken at random from the dataset of interest represent the same type. It equals: , where is richness (the total number of types in the dataset). This equation is also equal to the weighted arithmetic mean of the proportional abundances of the types of interest, with the proportional abundances themselves being used as the weights. Proportional abundances are by definition constrained to values between zero and unity, but it is a weighted arithmetic mean, hence , which is reached when all types are equally abundant. By comparing the equation used to calculate λ with the equations used to calculate true diversity, it can be seen that