Bounds on the entropy of multinomial distribution

The purpose of this study is to derive an upper-bound and a lower-bound of the entropy of a multinomial distribution. In spite of its practicality and versatility, there is no closed-form formula of the entropy of the multinomial distribution. Cichoń derived an asymptotic formula that approximates the entropy, but the approximation is not very useful because it can yield fatal error for non-asymptotic parameters, and there is no clear perspective of the approximation error. This paper proposes an upper-bound and a lower-bound formulas of the entropy of the multinomial distribution. The formulas are effective for arbitrary parameters, and contribute to the quantitative discussion of the entropy of the multinomial distribution.