Statistical and Structural Bias in Birth-Death Models

Accurate estimation of speciation ( $\lambda$ ) and extinction ( $\mu$ ) 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 $\hat{\lambda }$ 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 $\mu$ . The best-performing correction for $\lambda$ is identical to the Yule result, while the bias in $\mu$ depends on both sample size and the estimated extinction fraction ( $\mu /\lambda$ ). 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 ( $\lambda -\mu$ ) remains systematically underestimated due to the slight overestimation of $\mu$ . On the whole, these results begin to clarify the statistical and structural sources of bias in diversification rate estimation and provide a general framework for improving inference under birth-death models.