We explore probability modelling of discretization uncertainty for system states defined implicitly by ordinary or partial differential equations. Accounting for this uncertainty can avoid posterior under-coverage when likelihoods are constructed from a coarsely discretized approximation to system equations. A formalism is proposed for inferring a fixed but a priori unknown model trajectory through Bayesian updating of a prior process conditional on model information. A one-step-ahead sampling scheme for interrogating the model is described, its consistency and first order convergence properties are proved, and its computational complexity is shown to be proportional to that of numerical explicit one-step solvers. Examples illustrate the flexibility of this framework to deal with a wide variety of complex and large-scale systems. Within the calibration problem, discretization uncertainty defines a layer in the Bayesian hierarchy, and a Markov chain Monte Carlo algorithm that targets this posterior distribution is presented. This formalism is used for inference on the JAK-STAT delay differential equation model of protein dynamics from indirectly observed measurements. The discussion outlines implications for the new field of probabilistic numerics
Motivation: Cancer is the process of accumulating genetic alterations that confer selective advantages to tumor cells. The order in which aberrations occur is not arbitrary, and inferring the order of events is challenging due to the lack of longitudinal samples from tumors. Moreover, a network model of oncogenesis should capture biological facts such as distinct progression trajectories of cancer subtypes and patterns of mutual exclusivity of alterations in the same pathways.
In this paper, we present the disjunctive Bayesian network (DBN), a novel oncogenetic model with a phylogenetic interpretation. DBN is expressive enough to capture cancer subtypes' trajectories and mutually exclusive relations between alterations from unstratified data.
Results: In cases where the number of studied alterations is small (<30), we provide an efficient dynamic programming implementation of an exact structure learning method that finds a best DBN in the superexponential search space of networks. In rare cases that the number of alterations is large, we provided an efficient genetic algorithm in our software package, OncoBN. Through numerous synthetic and real data experiments, we show OncoBN's ability in inferring ground truth networks and recovering biologically meaningful progression networks.
Availability: OncoBN is implemented in R and is available at https://github.com/phillipnicol/OncoBN.
We explore probability modelling of discretization uncertainty for system states defined implicitly by ordinary or partial differential equations. Accounting for this uncertainty can avoid posterior under-coverage when likelihoods are constructed from a coarsely discretized approximation to system equations. A formalism is proposed for inferring a fixed but a priori unknown model trajectory through Bayesian updating of a prior process conditional on model information. A one-step-ahead sampling scheme for interrogating the model is described, its consistency and first order convergence properties are proved, and its computational complexity is shown to be proportional to that of numerical explicit one-step solvers. Examples illustrate the flexibility of this framework to deal with a wide variety of complex and large-scale systems. Within the calibration problem, discretization uncertainty defines a layer in the Bayesian hierarchy, and a Markov chain Monte Carlo algorithm that targets this posterior distribution is presented. This formalism is used for inference on the JAK-STAT delay differential equation model of protein dynamics from indirectly observed measurements. The discussion outlines implications for the new field of probabilistic numerics.
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