We introduce a computational efficient data-driven framework suitable for quantifying the uncertainty in physical parameters and model formulation of computer models, represented by differential equations. We construct physics-informed priors, which are multi-output GP priors that encode the model's structure in the covariance function. This is extended into a fully Bayesian framework that quantifies the uncertainty of physical parameters and model predictions. Since physical models often are imperfect descriptions of the real process, we allow the model to deviate from the observed data by considering a discrepancy function. For inference Hamiltonian Monte Carlo is used. Further, approximations for big data are developed that reduce the computational complexity from $\mathcal{O}(N^3)$ to $\mathcal{O}(N\cdot m^2),$ where $m \ll N.$ Our approach is demonstrated in simulation and real data case studies where the physics are described by time-dependent ODEs (cardiovascular models) and space-time dependent PDEs (heat equation). In the studies, it is shown that our modelling framework can recover the true parameters of the physical models in cases where 1) the reality is more complex than our modelling choice and 2) the data acquisition process is biased while also producing accurate predictions. Furthermore, it is demonstrated that our approach is computationally faster than traditional Bayesian calibration methods.

MOTIVATION: DNA methylation has been shown to be spatially dependent across chromosomes. Previous studies have focused on the influence of genomic context on the dependency structure, while not considering differences in dependency structure between individuals. RESULTS: We modeled spatial dependency with a flexible framework to quantify the dependency structure, focusing on inter-individual differences by exploring the association between dependency parameters and technical and biological variables. The model was applied to a subset of the Finnish Twin Cohort study (N = 1611 individuals). The estimates of the dependency parameters varied considerably across individuals, but were generally consistent across chromosomes within individuals. The variation in dependency parameters was associated with bisulfite conversion plate, zygosity, sex and age. The age differences presumably reflect accumulated environmental exposures and/or accumulated small methylation differences caused by stochastic mitotic events, establishing recognizable, individual patterns more strongly seen in older individuals. AVAILABILITY AND IMPLEMENTATION: The twin dataset used in the current study are located in the Biobank of the National Institute for Health and Welfare, Finland. All the biobanked data are publicly available for use by qualified researchers following a standardized application procedure (https://thl.fi/en/web/thl-biobank/for-researchers). A R-script for fitting the dependency structure to publicly available DNA methylation data with the software used in this article is provided in supplementary data. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.