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We present the AdaptSPEC-X method for the joint analysis of a panel of possibly nonstationary time series. The approach is Bayesian and uses a covariate-dependent infinite mixture model to incorporate multiple time series, with mixture components parameterized by a time-varying mean and log spectrum. The mixture components are based on AdaptSPEC, a nonparametric model which adaptively divides the time series into an unknown number of segments and estimates the local log spectra by smoothing splines. AdaptSPEC-X extends AdaptSPEC in three ways. First, through the infinite mixture, it applies to multiple time series linked by covariates. Second, it can handle missing values, a common feature of time series which can cause difficulties for nonparametric spectral methods. Third, it allows for a time-varying mean. Through these extensions, AdaptSPEC-X can estimate time-varying means and spectra at observed and unobserved covariate values, allowing for predictive inference. Estimation is performed by Markov chain Monte Carlo (MCMC) methods, combining data augmentation, reversible jump, and Riemann manifold Hamiltonian Monte Carlo techniques. We evaluate the methodology using simulated data, and describe applications to Australian rainfall data and measles incidence in the United States. Software implementing the method proposed in this article is available in the R package BayesSpec. Supplementary files for this article are available online.

Bayesian inference provides a rigorous approach for neural learning with knowledge representation via the posterior distribution that accounts for uncertainty quantification. Markov Chain Monte Carlo (MCMC) methods typically implement Bayesian inference by sampling from the posterior distribution. This not only provides point estimates of the weights, but the ability to propagate and quantify uncertainty in decision making. However, these techniques face challenges in convergence and scalability, particularly in settings with large datasets and neural network architectures. This paper addresses these challenges in two ways. First, parallel tempering MCMC sampling method is used to explore multiple modes of the posterior distribution and implemented in multi-core computing architecture. Second, we make within-chain sampling scheme more efficient by using Langevin gradient information for creating Metropolis–Hastings proposal distributions. We demonstrate the techniques using time series prediction and pattern classification applications. The results show that the method not only improves the computational time, but provides better decision making capabilities when compared to related methods.

Daily precipitation has an enormous impact on human activity, and the study of how it varies over time and space, and what global indicators influence it, is of paramount importance to Australian agriculture. We analyze over 294 million daily rainfall measurements since 1876, spanning 17,606 sites across continental Australia. The data are not only large but also complex, and the topic would benefit from a common and publicly available statistical framework. We propose a Bayesian hierarchical mixture model that accommodates mixed discrete-continuous data. The observational level describes site-specific temporal and climatic variation via a mixture-of-experts model. At the next level of the hierarchy, spatial variability of the mixture weights’ parameters is modeled by a spatial Gaussian process prior. A parallel and distributed Markov chain Monte Carlo sampler is developed which scales the model to large data sets. We present examples of posterior inference on the mixture weights, monthly intensity levels, daily temporal dependence, offsite prediction of the effects of climate drivers and long-term rainfall trends across the entire continent. Computer code implementing the methods proposed in this paper is available as an R package.

Estimating the impact of environmental processes on vertical reef development in geological time is a very challenging task. pyReef-Core is a deterministic carbonate stratigraphic forward model designed to simulate the key biological and environmental processes that determine vertical reef accretion and assemblage changes in fossil reef drill cores. We present a Bayesian framework called Bayesreef for the estimation and uncertainty quantification of parameters in pyReef-Core that represent environmental conditions affecting the growth of coral assemblages on geological timescales. We demonstrate the existence of multimodal posterior distributions and investigate the challenges of sampling using Markov chain Monte-Carlo (MCMC) methods, which includes parallel tempering MCMC. We use synthetic reef-core to investigate fundamental issues and then apply the methodology to a selected reef-core from the Great Barrier Reef in Australia. The results show that Bayesreef accurately estimates and provides uncertainty quantification of the selected parameters that represent the environment and ecological conditions in pyReef-Core. Bayesreef provides insights into the complex posterior distributions of parameters in pyReef-Core, which provides the groundwork for future research in this area.

Objectives This paper describes the use of machine learning techniques to implement a Bayesian approach to modelling the dependency between offence data and environmental factors such as demographic characteristics and spatial location. The main goal of this paper is to provide a fully probabilistic approach to modelling crime which reflects all uncertainties in the prediction of offences as well as the uncertainties surrounding model parameters. Methods The proposed method is based on a Bayesian framework, with a Gaussian Process prior and MCMC, allowing uncertainties in prediction and inference to be quantified via the posterior distributions of interest. By using Bayesian updating, these predictions and inferences are dynamic in the sense that they change as new information becomes available. Results We applied the proposed methodology to particular offence data, such as domestic violence-related assaults, burglary and motor vehicle theft, in the state of New South Wales (NSW), Australia. Our results demonstrate the strength of the technique by validating the factors that are associated with high and low criminal activity, including bounds on the degree of the relation. Conclusions We argue that this fully probabilistic approach will improve prediction, in the sense that the uncertainties are more accurately quantified, with attendant benefits to policymakers and policing organisations seeking to deploy limited criminal justice resources to prevent and control crime. While limitations and areas for potential improvement are identified, the success of the Bayesian approach, implemented using machine learning techniques, in a criminological context represents an exciting development.

Bayesian inference provides a rigorous methodology for estimation and uncertainty quantification of parameters in geophysical forward models. Badlands (basin and landscape dynamics model) is a landscape evolution model that simulates topography development at various space and time scales. Badlands consists of a number of geophysical parameters that needs estimation with appropriate uncertainty quantification; given the observed present-day ground truth such as surface topography and the stratigraphy of sediment deposition through time. The inference of unknown parameters is challenging due to the scarcity of data, sensitivity of the parameter setting and complexity of the Badlands model. In this paper, we take a Bayesian approach to provide inference using Markov chain Monte Carlo sampling (MCMC). We present Bayeslands ; a Bayesian framework for Badlands that fuses information obtained from complex forward models with observational data and prior knowledge. As a proof-of-concept, we consider a synthetic and real-world topography with two parameters for Bayeslands inference, namely precipitation and erodibility. The results of the experiments show that Bayeslands yields a promising distribution of the parameters. Moreover, we demonstrate the challenge in sampling irregular and multi-modal posterior distributions using a likelihood surface that has a range of sub-optimal modes.

The Bayesian paradigm is becoming an increasingly popular framework for estimation and uncertainty quantification of unknown parameters in geo-physical inversion problems. Badlands is a basin and landscape evolution forward model for simulating topography evolution at a large range of spatial and time scales. Our previous work presented Bayeslands that used the Bayesian paradigm to make inference for unknown parameters in the Badlands model using Markov chain Monte Carlo (MCMC) sampling. Bayeslands faced challenges in convergence due to multi-modal posterior distributions in the selected parameters of Badlands. Parallel tempering is an advanced MCMC method suited for irregular and multi-modal posterior distributions. In this paper, we extend Bayeslands using parallel tempering (PT-Bayeslands) with high performance computing to address previous limitations in parameter space exploration in the context of the computationally expensive Badlands model. Our results show that PT-Bayeslands not only reduces the computation time, but also provides an improvement of the sampling for multi-modal posterior distributions. This provides an improvement over Bayeslands which used single chain MCMC that face difficulties in convergence and can lead to misleading inference. This motivates its usage in large-scale basin and landscape evolution models.

Although neural networks have been very promising tools for chaotic time series prediction, they lack methodology for uncertainty quantification. Bayesian inference using Markov Chain Mont-Carlo (MCMC) algorithms have been popular for uncertainty quantification for linear and non-linear models. Langevin dynamics refer to a class of MCMC algorithms that incorporate gradients with Gaussian noise in parameter updates. In the case of neural networks, the parameter updates refer to the weights of the network. We apply Langevin dynamics in neural networks for chaotic time series prediction. The results show that the proposed method improves the MCMC random-walk algorithm for majority of the problems considered. In particular, it gave much better performance for the real-world problems that featured noise.