  Statistical models can involve implicitly defined quantities, such as
solutions to nonlinear ordinary differential equations (ODEs), that unavoidably
need to be numerically approximated in order to evaluate the model. The
approximation error inherently biases statistical inference results, but the
amount of this bias is generally unknown and often ignored in Bayesian
parameter inference. We propose a computationally efficient method for
verifying the reliability of posterior inference for such models, when the
inference is performed using Markov chain Monte Carlo methods. We validate the
efficiency and reliability of our workflow in experiments using simulated and
real data, and different ODE solvers. We highlight problems that arise with
commonly used adaptive ODE solvers, and propose robust and effective
alternatives which, accompanied by our workflow, can now be taken into use
without losing reliability of the inferences.
