This paper describes an approach for variable selection and hypothesis testing in semiparametric additive models using Bayes factors in smoothing spline analysis of variance (SSANOVA) models. Effects can be linear or nonparametric (i.e., smooth or interactions between selected linear and smooth effects). To evaluate the importance of each term in the model, we develop Bayes factors for both linear and nonparametric terms. We compute approximate Bayes factors by Monte Carlo and Laplace integration. These Bayes factors can be computed to compare any two sub-models including one model nested in another. This permits formal tests of any portion or simultaneous portions of an SSANOVA model. We demonstrate this approach with an example.

Modeling nonstationary processes is of paramount importance to many scientific disciplines including environmental science, ecology, and finance, among others. Consequently, flexible methodology that provides accurate estimation across a wide range of processes is a subject of ongoing interest. We propose a novel approach to model-based time–frequency estimation using time-varying autoregressive models. In this context, we take a fully Bayesian approach and allow both the autoregressive coefficients and innovation variance to vary over time. Importantly, our estimation method uses the lattice filter and is cast within the partial autocorrelation domain. The marginal posterior distributions are of standard form and, as a convenient by-product of our estimation method, our approach avoids undesirable matrix inversions. As such, estimation is extremely computationally efficient and stable. To illustrate the effectiveness of our approach, we conduct a comprehensive simulation study that compares our method with other competing methods and find that, in most cases, our approach performs superior in terms of average squared error between the estimated and true time-varying spectral density. Lastly, we demonstrate our methodology through three modeling applications; namely, insect communication signals, environmental data (wind components), and macroeconomic data (US gross domestic product (GDP) and consumption).

In a number of applications, we argue that standard Bayes factor model comparison and selection may be inappropriate for decision making under specific, utility-based, criteria. It has been suggested that the use of scoring rules in this context allows greater flexibility: scores can be customised to a client’s utility and model selection can proceed on the basis of the highest scoring model. We argue here that the approach of comparing the cumulative scores of competing models is not ideal because it tends to ignore a model’s ability to ‘catch up’ through parameter learning. An alternative approach of selecting a model on its maximum posterior score based on a plug in or posterior expected value is problematic in that it uses the data twice in estimation and evaluation. We therefore introduce a new Bayesian posterior score information criterion (BPSIC), which is a generalisation of the Bayesian predictive information criterion proposed by Ando (2007). This allows the analyst both to tailor an appropriate scoring function to the needs of the ultimate decision maker and to correct appropriately for bias in using the data on a posterior basis to revise parameter estimates. We show that this criterion can provide a convenient method of initial model comparison when the number of models under consideration is large or when computational burdens are high. We illustrate the new methods with simulated examples and real data from the UK electricity imbalance market.

A hierarchical Bayesian model for spatial panel data is proposed. The idea behind the proposed method is to analyze spatially dependent panel data by means of a separable covariance matrix. Let us indicate the observations as $y_{it}$, in $i=1,\dots ,N$ regions and at $t=1,\dots ,T$ times, and suppose the covariance matrix of $\mathbf{y}$, given a set of regressors, is written as a Kronecker product of a purely spatial and a purely temporal covariance. On the one hand, the structure of separable covariances dramatically reduces the number of parameters, while on the other hand, the lack of a structured pattern for spatial and temporal covariances permits capturing possible unknown dependencies (both in time and space). The use of the Bayesian approach allows one to overcome some of the difficulties of the classical (MLE or GMM based) approach. We present two illustrative examples: the estimation of cigarette price elasticity and of the determinants of the house price in 120 municipalities in the Province of Rome.

The selection of appropriate hyperpriors for variance parameters is an important and sensible topic in all kinds of Bayesian regression models involving the specification of (conditionally) Gaussian prior structures where the variance parameters determine a data-driven, adaptive amount of prior variability or precision. We consider the special case of structured additive distributional regression where Gaussian priors are used to enforce specific properties such as smoothness or shrinkage on various effect types combined in predictors for multiple parameters related to the distribution of the response. Relying on a recently proposed class of penalised complexity priors motivated from a general set of construction principles, we derive a hyperprior structure where prior elicitation is facilitated by assumptions on the scaling of the different effect types. The posterior distribution is assessed with an adaptive Markov chain Monte Carlo scheme and conditions for its propriety are studied theoretically. We investigate the new type of scale-dependent priors in simulations and two challenging applications, in particular in comparison to the standard inverse gamma priors but also alternatives such as half-normal, half-Cauchy and proper uniform priors for standard deviations.

In the context of robust Bayesian analysis, we introduce a new class of prior distributions based on stochastic orders and distortion functions. We provide the new definition, its interpretation and the main properties and we also study the relationship with other classical classes of prior beliefs. We also consider Kolmogorov and Kantorovich metrics to measure the uncertainty induced by such a class, as well as its effect on the set of corresponding Bayes actions. Finally, we conclude the paper with some numerical examples.

Regression models with varying coefficients changing over certain underlying covariates offer great flexibility in capturing a functional relationship between the response and other covariates. This article extends such regression models to include random effects and to account for correlation and heteroscedasticity in error terms, and proposes an efficient new data-driven method to estimate varying regression coefficients via reparameterization and partial collapse. The proposed methodology is illustrated with a simulated study and longitudinal data from a study of soybean growth.

Chain Event Graphs (CEGs) are a rich and provenly useful class of graphical models. The class contains discrete Bayesian Networks as a special case and is able to depict directly the asymmetric context-specific statements in the model. But bespoke efficient algorithms now need to be developed to search the enormous CEG model space. In different contexts Bayes Factor scored search algorithm using non-local priors (NLPs) has recently proved very successful for searching other huge model spaces. Here we define and explore three different types of NLP that we customise to search CEG spaces. We demonstrate how one of these candidate NLPs provides a framework for search which is both robust and computationally efficient. It also avoids selecting an overfitting model as the standard conjugate methods sometimes do. We illustrate the efficacy of our methods with two examples. First we analyse a previously well-studied 5-year longitudinal study of childhood hospitalisation. The second much larger example selects between competing models of prisoners’ radicalisation in British prisons: because of its size an application beyond the scope of earlier Bayes Factor search algorithms.

Bayesian analysis of continuous time, discrete state space time series is an important and challenging problem, where incomplete observation and large parameter sets call for user-defined priors based on known properties of the process. Generalized linear models have a largely unexplored potential to construct such prior distributions. We show that an important challenge with Bayesian generalized linear modelling of continuous time Markov chains is that classical Markov chain Monte Carlo techniques are too ineffective to be practical in that setup. We address this issue using an auxiliary variable construction combined with an adaptive Hamiltonian Monte Carlo algorithm. This sampling algorithm and model make it efficient both in terms of computation and analyst’s time to construct stochastic processes informed by prior knowledge, such as known properties of the states of the process. We demonstrate the flexibility and scalability of our framework using synthetic and real phylogenetic protein data, where a prior based on amino acid physicochemical properties is constructed to obtain accurate rate matrix estimates.

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.

The authors present an ingenious probabilistic numerical solver for deterministic differential equations (DEs). The true solution is progressively identified via model interrogations, in a formal framework of Bayesian updating. I have attempted to extend the authors’ ideas to stochastic differential equations (SDEs), and discuss two challenges encountered in this endeavor: (i) the non-differentiability of SDE sample paths, and (ii) the sampling of diffusion bridges, typically required of solutions to the SDE inverse problem.

This note is a discussion of the article “Bayesian Solution Uncertainty Quantification for Differential Equations” by Chkrebtii, Campbell, Calderhead, and Girolami. The authors propose stochastic models for differential equation discretizations. While appreciating the main concepts, we point out some possible extensions and modifications.