Estimating entropy from empirical samples of finite size is of central importance for information theory as well as the analysis of complex statistical systems. Yet, this delicate task is marred by intrinsic statistical bias. Here we decompose the entropy function into a polynomial approximation function and a remainder function. The approximation function is based on a Taylor expansion of the logarithm. Given n observations, we give an unbiased, linear estimate of the first n power series terms based on counting sets of k coincidences. For the remainder function we use nonlinear Bayesian estimation with a nearly flat prior distribution on the entropy that was developed by Nemenman, Shafee, and Bialek. Our simulations show that the combined entropy estimator has reduced bias in comparison to other available estimators.