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We propose a Monte Carlo algorithm to sample from high dimensional probability distributions that combines Markov chain Monte Carlo and importance sampling. We provide a careful theoretical analysis, including guarantees on robustness to high dimensionality, explicit comparison with standard Markov chain Monte Carlo methods and illustrations of the potential improvements in efficiency. Simple and concrete intuition is provided for when the novel scheme is expected to outperform standard schemes. When applied to Bayesian variable-selection problems, the novel algorithm is orders of magnitude more efficient than available alternative sampling schemes and enables fast and reliable fully Bayesian inferences with tens of thousand regressors.

Although hydrological model forecasts aid water management decisions, they normally have predictive uncertainties. Generalized likelihood uncertainty estimation (GLUE) is popular for constructing predictive uncertainty bounds (PUBs). GLUE is based on simple Monte Carlo sampling (SMCS), a technique known to be ineffective in establishing behavioural simulations. This study introduced randomized block quasi-Monte Carlo sampling (RBMC). In RBMC, each parameter's range is divided into a stipulated number of sub-blocks (Snb). Parameters' values are separately generated in each sub-block. Finally, the sub-blocks are shuffled while maintaining the sequence of generated values in each sub-block. When Snb is equal to the number of simulations, RBMC reduces to SMCS. Otherwise, each Snb leads to a separate RBMC configuration or sampling scheme. The number of RBMC-based behavioural solutions was often found to be greater than that of SMCS, in some cases, by up to 33.6%. The width of the 90% confidence interval on 95th percentile flow based on SMCS was often larger than those of RBMC, sometimes by up to 23.4%. PUBs were found to vary in widths among RBMC configurations, thereby revealing the influence of the choice of a sampling scheme. Thus, GLUE based on RBMC is recommended to take into account the said influence.

An evolutionary reconstruction technique (ERT) was developed for three-dimensional (3D) reconstruction of luminescent objects, in particular turbulent flames for the first time. The computed tomography (CT) algorithm is comprised of a genetic algorithm (GA) and a ray-tracing software. To guide the reconstruction process, a mask is introduced. It uses a Metropolis algorithm (MA) to sample locations where specific genetic operators can be applied. Based on an extensive parameter study, performed on several types of phantoms, the ability of our algorithm for 3D reconstructions of fields with varying complexities is demonstrated. Furthermore, it was applied to three experiments, to reconstruct the instantaneous chemiluminescence field of a bunsen flame, a highly turbulent swirl flame and the turbulent Cambridge-Sandia stratified flame. Additionally, we show direct and quantitative comparison to an advanced computed tomography of chemiluminescence (CTC) method that is based on an algebraic reconstruction technique (ART). The results showed good agreement between CTC and ERT using both phantom data from flame simulations, and experimental data.

We study the properties of points in [0, 1]d generated by applying Hilbert's space filling curve to uniformly distributed points in [0, 1]. For deterministic sampling we obtain a discrepancy of O(n–1/d) for d ≥ 2. For random stratified sampling, and scrambled van der Corput points, we derive a mean-squared error of O(n–1–2/d) for integration of Lipschitz continuous integrands, when d ≥ 3. These rates are the same as those obtained by sampling on d-dimensional grids and they show a deterioration with increasing d. The rate for Lipschitz functions is, however, the best possible at that level of smoothness and is better than plain independent and identically distributed sampling. Unlike grids, space filling curve sampling provides points at any desired sample size, and the van der Corput version is extensible in n. We also introduce a class of piecewise Lipschitz functions whose discontinuities are in rectifiable sets described via Minkowski content. Although these functions may have infinite variation in the sense of Hardy and Krause, they can be integrated with a mean-squared error of O(n–1–1/d). It was previously known only that the rate was o(n–1). Other space filling curves, such as those due to Sierpinski and Peano, also attain these rates, whereas upper bounds for the Lebesgue curve are somewhat worse, as if the dimension were log2(3) times as high.

We generalize stochastic subgradient descent methods to situations in which we do not receive independent samples from the distribution over which we optimize, instead receiving samples coupled over time. We show that as long as the source of randomness is suitably ergodic---it converges quickly enough to a stationary distribution---the method enjoys strong convergence guarantees, both in expectation and with high probability. This result has implications for stochastic optimization in high-dimensional spaces, peer-to-peer distributed optimization schemes, decision problems with dependent data, and stochastic optimization problems over combinatorial spaces. [PUBLICATION ABSTRACT]

We consider Bayesian inference for stochastic differential equation mixed effects models (SDEMEMs) exemplifying tumour response to treatment and regrowth in mice. We produce an extensive study on how an SDEMEM can be fitted by using both exact inference based on pseudo-marginal Markov chain Monte Carlo sampling and approximate inference via Bayesian synthetic likelihood (BSL). We investigate a two-compartments SDEMEM, corresponding to the fractions of tumour cells killed by and survived on a treatment. Case-study data consider a tumour xenography study with two treatment groups and one control, each containing 5–8 mice. Results from the case-study and from simulations indicate that the SDEMEM can reproduce the observed growth patterns and that BSL is a robust tool for inference in SDEMEMs. Finally, we compare the fit of the SDEMEM with a similar ordinary differential equation model. Because of small sample sizes, strong prior information is needed to identify all model parameters in the SDEMEM and it cannot be determined which of the two models is the better in terms of predicting tumour growth curves. In a simulation study we find that with a sample of 17 mice per group BSL can identify all model parameters and distinguish treatment groups.

In this study, we investigate the sensitivity of simulated shallow cumulus and stratocumulus to selected tunable parameters of Cloud Layers Unified by Binormals (CLUBB) in the single‐column version of Community Atmosphere Model version 5 (SCAM5). A quasi‐Monte Carlo (QMC) sampling approach is adopted to effectively explore the high‐dimensional parameter space and a generalized linear model is adopted to study the responses of simulated cloud fields to tunable parameters. One stratocumulus and two shallow cumulus cases are configured at both coarse and fine vertical resolutions in this study. Our results show that most of the variance in simulated cloud fields can be explained by a small number of tunable parameters. The parameters related to Newtonian and buoyancy‐damping terms of total water flux are found to be the most influential parameters for stratocumulus. For shallow cumulus, the most influential parameters are those related to skewness of vertical velocity, reflecting the strong coupling between cloud properties and dynamics in this regime. The influential parameters in the stratocumulus case are sensitive to the vertical resolution while little sensitivity is found for the shallow cumulus cases, as eddy mixing length (or dissipation time scale) plays a more important role and depends more strongly on the vertical resolution in stratocumulus than in shallow convections. The influential parameters remain almost unchanged when the number of tunable parameters increases from 16 to 35. This study improves understanding of the CLUBB behavior associated with parameter uncertainties and provides valuable insights for other high‐order turbulence closure schemes. Key Points Most variances in cloud fields can be explained by a small number of parameters Results for stratocumulus are sensitive to vertical resolution Critical parameters in shallow cumulus are related to vertical velocity skewness

We consider the generic problem of performing sequential Bayesian inference in a state space model with observation process y, state process x and fixed parameter θ. An idealized approach would be to apply the iterated batch importance sampling algorithm of Chopin. This is a sequential Monte Carlo algorithm in the θ-dimension, that samples values of θ, reweights iteratively these values by using the likelihood increments p(yt|y1:t–1,θ) and rejuvenates the θ-particles through a resampling step and a Markov chain Monte Carlo update step. In state space models these likelihood increments are intractable in most cases, but they may be unbiasedly estimated by a particle filter in the x-dimension, for any fixed θ. This motivates the SMC2 algorithm that is proposed in the paper: a sequential Monte Carlo algorithm, defined in the θ-dimension, which propagates and resamples many particle filters in the x-dimension. The filters in the x-dimension are an example of the random weight particle filter. In contrast, the particle Markov chain Monte Carlo framework that has been developed by Andrieu and colleagues allows us to design appropriate Markov chain Monte Carlo rejuvenation steps. Thus, the θ-particles target the correct posterior distribution at each iteration t, despite the intractability of the likelihood increments. We explore the applicability of our algorithm in both sequential and non-sequential applications and consider various degrees of freedom, as for example increasing dynamically the number of x-particles. We contrast our approach with various competing methods, both conceptually and empirically through a detailed simulation study, and based on particularly challenging examples.

Very often when studying non-equilibrium systems one is interested in analysing dynamical behaviour that occurs with very low probability, so called rare events. In practice, since rare events are by definition atypical, they are often difficult to access in a statistically significant way. What are required are strategies to 'make rare events typical' so that they can be generated on demand. Here we present such a general approach to adaptively construct a dynamics that efficiently samples atypical events. We do so by exploiting the methods of reinforcement learning (RL), which refers to the set of machine learning techniques aimed at finding the optimal behaviour to maximise a reward associated with the dynamics. We consider the general perspective of dynamical trajectory ensembles, whereby rare events are described in terms of ensemble reweighting. By minimising the distance between a reweighted ensemble and that of a suitably parametrised controlled dynamics we arrive at a set of methods similar to those of RL to numerically approximate the optimal dynamics that realises the rare behaviour of interest. As simple illustrations we consider in detail the problem of excursions of a random walker, for the case of rare events with a finite time horizon; and the problem of a studying current statistics of a particle hopping in a ring geometry, for the case of an infinite time horizon. We discuss natural extensions of the ideas presented here, including to continuous-time Markov systems, first passage time problems and non-Markovian dynamics.

We review some advances of the particle filtering (PF) algorithm that have been achieved in the last decade in the context of target tracking, with regard to either a single target or multiple targets in the presence of false or missing data. The first part of our review is on remarkable achievements that have been made for the single-target PF from several aspects including importance proposal, computing efficiency, particle degeneracy/impoverishment and constrained/multi-modal systems. The second part of our review is on analyzing the intractable challenges raised within the general multitarget (multi-sensor) tracking due to random target birth and termination, false alarm, misdetection, measurement-to-track (M2T) uncertainty and track uncertainty. The mainstream multitarget PF approaches consist of two main classes, one based on M2T association approaches and the other not such as the finite set statistics-based PF. In either case, significant challenges remain due to unknown tracking scenarios and integrated tracking management.