Abstract Inferring the infinitesimal rates of continuous-time Markov chains (CTMCs) is a central challenge in many scientific domains. This task is hindered by 3 factors: quadratic growth in the number of rates as the CTMC state space expands, strong dependencies among rates, and incomplete information for many transitions. We introduce a new Bayesian framework that flexibly models CTMC rates by incorporating covariates through Gaussian processes. This approach improves inference by integrating new information and deepens our understanding of CTMC stochastic behaviour by shedding light on potential external drivers. Unlike previous approaches limited to linear covariate effects, our method captures complex nonlinear relationships, enabling fuller use of covariate information and more accurate characterization of their influence. For efficient posterior inference, we employ Hamiltonian Monte Carlo. To avoid the prohibitive cost of repeated matrix-exponential derivatives under existing approaches, we develop 2 scalable strategies for exact and approximate gradient computation. Given N observations and K CTMC states, these strategies reduce the cost from O(NK3), with a substantial constant, to O(K3+NK2), with cheaper constants. We demonstrate the method in Bayesian phylogenetic and phylogeographic inference, showing strong performance on synthetic and real datasets and empirical quadratic scaling in K, even when N
We provide a novel characterization of augmented balancing weights, also known as automatic debiased machine learning. These popular doubly robust estimators combine outcome modelling with balancing weights-weights that achieve covariate balance directly instead of estimating and inverting the propensity score. When the outcome and weighting models are both linear in some (possibly infinite) basis, we show that the augmented estimator is equivalent to a single linear model with coefficients that combine those of the original outcome model with those from unpenalized ordinary least-squares (OLS). Under certain choices of regularization parameters, the augmented estimator in fact collapses to the OLS estimator alone. We then extend these results to specific outcome and weighting models. We first show that the augmented estimator that uses (kernel) ridge regression for both outcome and weighting models is equivalent to a single, undersmoothed (kernel) ridge regression-implying a novel analysis of undersmoothing. When the weighting model is instead lasso-penalized, we demonstrate a familiar 'double selection' property. Our framework opens the black box on this increasingly popular class of estimators, bridges the gap between existing results on the semiparametric efficiency of undersmoothed and doubly robust estimators, and provides new insights into the performance of augmented balancing weights.
