Statistical and Structural Bias in Birth-Death Models

Abstract Accurate estimation of speciation ( λ ) and extinction ( µ ) 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 underestimates λ 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 λ and µ . The best-performing correction for λ is identical to the Yule result, while the bias in µ depends on both sample size and the estimated extinction fraction ( µ/λ ). 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 ( λ − µ ) remains systematically underestimated due to the slight overestimation of µ . 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.