Donglin Han

Xuyuan Wang, Donglin Han, Michael Y. Li

We consider the problem of parameter inference for ordinary differential equation (ODE) models with structural non-identifiability. Such models arise in a wide range of scientific fields, including control theory, systems biology, and public health. Structural non-identifiability occurs when distinct parameter values provide identical model outputs, resulting in lower-dimensional manifolds of observationally equivalent solutions in the parameter space. This poses challenges for Bayesian inference and Markov chain Monte Carlo (MCMC) methods, often leading to poor mixing and slow convergence. We develop two MCMC methods that use information from structural identifiability analysis. The first, Identifiability-Aware Geometric MCMC, constructs proposals that move within and between non-identifiable manifolds. The second, Identifiability-Aware Pseudo-Marginal MCMC, performs inference on the space of identifiable parameter combinations and reconstructs full parameter values. We show that both methods target the correct posterior distribution and are ergodic under standard conditions. Numerical examples demonstrate improved sampling efficiency and convergence compared with standard MCMC methods.

Xuyuan Wang, Donglin Han

Markov chain Monte Carlo (MCMC) methods provide powerful framework for sampling unknown probability measures across a wide range of scientific applications. In some settings, the target distribution is supported on a lower-dimensional submanifold of Euclidean space defined by nonlinear constraints, motivating the development of constrained Hamiltonian Monte Carlo (CHMC) methods. Most existing CHMC algorithms rely on the assumption that the implicit manifold is connected, allowing local constrained integrators such as RATTLE to explore the posterior ergodically. In practice, this assumption is occasionally violated due to complex geometric structures induced by nonlinear constraints of a model. We propose a replica exchange MCMC framework that couples a constrained chain evolving on the implicit manifold with a relaxed auxiliary chain defined in a tubular neighborhood of the constraint. The relaxed chain enables transitions between disconnected components. We show that the resulting algorithm enables sampling from a broader class of implicit manifolds, including those with disconnected components. We prove that the proposed sampler satisfies detailed balance, irreducibility, ergodicity, and convergence. We also demonstrate its effectiveness on examples from molecular and biological dynamical systems.