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A note on a Bernstein-type inequality for the log-likelihood function of categorical variables with infinitely many levels

Zhao Yunpeng

Probability and Mathematical Statistics

2024

Vol. 44, Fasc. 1

1--13

We prove a Bernstein-type bound for the difference between the average of the negative log-likelihoods of independent categorical variables with infinitely many levels - that is, a countably infinite number of categories, and its expectation - namely, the Shannon entropy. The result holds for the class of discrete random variables with tails lighter than or of the same order as a discrete power-law distribution. Most commonly used discrete distributions, such as the Poisson distribution, the negative binomial distribution, and the power-law distribution itself, belong to this class. The bound is effective in the sense that we provide a method to compute the constants within it. The new technique we develop allows us to obtain a uniform concentration inequality for categorical variables with a finite number of levels with the same optimal rate as in the literature, but with a much simpler proof.