Statistical and structural bias in birth-death models

Accurate estimation of speciation ({lambda}) and extinction ({micro}) rates from phylogenetic trees is central to studies of diversification, yet it remains unclear whether commonly used estimators are unbiased. Here we examine two sources of error: (1) statistical bias in the estimators themselves, and the (2) structural bias introduced by how small trees are handled in likelihood calculations. For the Yule process, we re-derive the expected bias of the standard estimator, showing that [Formula] underestimates{lambda} by a factor of (n - 2)/(n - 1). Extending to the general birth-death model, we use symbolic regression to find functional forms that minimize the bias in both{lambda} and {micro}. The best-performing correction for{lambda} is identical to the Yule result, while the bias in {micro} depends on both sample size and the estimated extinction fraction ({micro}/{lambda}). Applying these corrections substantially improves the fit between the estimated and generating values. When these corrected estimators are used to derive other diversification-related parameters, turnover is nearly unbiased, but net diversification ({lambda} - {micro}) remains systematically underestimated due to the slight overestimation of {micro}. On the whole, these results clarify the statistical and structural sources of bias in diversification rate estimation and provide a general framework for improving inference under birth-death models.