The paper proposes Metropolis adjusted Langevin and Hamiltonian Monte Carlo sampling methods defined on the Riemann manifold to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. The methods provide fully automated adaptation mechanisms that circumvent the costly pilot runs that are required to tune proposal densities for MetropolisHastings or indeed Hamiltonian Monte Carlo and Metropolis adjusted Langevin algorithms. This allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the Markov chain.The methodology proposed exploits the Riemann geometry of the parameter space of statistical models and thus automatically adapts to the local structure when simulating paths across this manifold, providing highly efficient convergence and exploration of the target density. The performance of these Riemann manifold Monte Carlo methods is rigorously assessed by performing inference on logistic regression models, log-Gaussian Cox point processes, stochastic volatility models and Bayesian estimation of dynamic systems described by non-linear differential equations. Substantial improvements in the time-normalized effective sample size are reported when compared with alternative sampling approaches. MATLAB code that is available from www.ucl.ac.uk/statistics/ research/rmhmc allows replication of all the results reported.

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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Markov chain Monte Carlo methods (MCMC) are essential tools for solving many modern-day statistical and computational problems; however, a major limitation is the inherently sequential nature of these algorithms. In this paper, we propose a natural generalization of the Metropolis−Hastings algorithm that allows for parallelizing a single chain using existing MCMC methods. We do so by proposing multiple points in parallel, then constructing and sampling from a finite-state Markov chain on the proposed points such that the overall procedure has the correct target density as its stationary distribution. Our approach is generally applicable and straightforward to implement. We demonstrate how this construction may be used to greatly increase the computational speed and statistical efficiency of a variety of existing MCMC methods, including Metropolis-Adjusted Langevin Algorithms and Adaptive MCMC. Furthermore, we show how it allows for a principled way of using every integration step within Hamiltonian Monte Carlo methods; our approach increases robustness to the choice of algorithmic parameters and results in increased accuracy of Monte Carlo estimates with little extra computational cost. S ince its introduction in the 1970s, the Metropolis−Hastings algorithm has revolutionized computational statistics (1). The ability to draw samples from an arbitrary probability distribution, πðXÞ, known only up to a constant, by constructing a Markov chain that converges to the correct stationary distribution has enabled the practical application of Bayesian inference for modeling a huge variety of scientific phenomena, and has resulted in Metropolis−Hastings being noted as one of the top 10 most important algorithms from the 20th century (2). Despite regular increases in available computing power, Markov chain Monte Carlo (MCMC) algorithms can still be computationally very expensive; many thousands of iterations may be necessary to obtain low-variance estimates of the required quantities with an oftentimes complex statistical model being evaluated for each set of proposed parameters. Furthermore, many Metropolis−Hastings algorithms are severely limited by their inherently sequential nature.Many approaches have been proposed for improving the statistical efficiency of MCMC, and although such algorithms are guaranteed to converge asymptotically to the stationary distribution, their performance over a finite number of iterations can vary hugely. Much research effort has therefore focused on developing transition kernels that enable moves to be proposed far from the current point and subsequently accepted with high probability, taking into account, for example, the correlation structure of the parameter space (3, 4), or by using Hamiltonian dynamics (5) or diffusion processes (6). A recent investigation into proposal kernels suggests that more exotic distributions, such as the Bactrian kernel, might also be used to increase the statistical efficiency of MCMC algorithms (7). The efficient exploration of high-dimensional and multimo...

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