The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators suggests that formal uncertainty quantification can also be performed in this context. Competing statistical paradigms can be considered and Bayesian probabilistic numerical methods (PNMs) are obtained when Bayesian statistical principles are deployed. Bayesian PNM have the appealing property of being closed under composition, such that uncertainty due to different sources of discretisation in a numerical method can be jointly modelled and rigorously propagated. Despite recent attention, no exact Bayesian PNM for the numerical solution of ordinary differential equations (ODEs) has been proposed. This raises the fundamental question of whether exact Bayesian methods for (in general nonlinear) ODEs even exist. The purpose of this paper is to provide a positive answer for a limited class of ODE. To this end, we work at a foundational level, where a novel Bayesian PNM is proposed as a proof-of-concept. Our proposal is a synthesis of classical Lie group methods, to exploit underlying symmetries in the gradient field, and non-parametric regression in a transformed solution space for the ODE. The procedure is presented in detail for first and second order ODEs and relies on a certain strong technical condition – existence of a solvable Lie algebra – being satisfied. Numerical illustrations are provided.

Hamiltonian Monte Carlo (HMC) and related algorithms have become routinely used in Bayesian computation. In this article, we present a simple and provably accurate method to improve the efficiency of HMC and related algorithms with essentially no extra computational cost. This is achieved by recycling the intermediate states along simulated trajectories of Hamiltonian dynamics. Standard algorithms use only the end points of trajectories, wastefully discarding all the intermediate states. Compared to the alternative methods for utilizing the intermediate states, our algorithm is simpler to apply in practice and requires little programming effort beyond the usual implementations of HMC and related algorithms. Our algorithm applies straightforwardly to the no-U-turn sampler, arguably the most popular variant of HMC. Through a variety of experiments, we demonstrate that our recycling algorithm yields substantial computational efficiency gains.

Variance parameters in additive models are typically assigned independent priors that do not account for model structure. We present a new framework for prior selection based on a hierarchical decomposition of the total variance along a tree structure to the individual model components. For each split in the tree, an analyst may be ignorant or have a sound intuition on how to attribute variance to the branches. In the former case a Dirichlet prior is appropriate to use, while in the latter case a penalised complexity (PC) prior provides robust shrinkage. A bottom-up combination of the conditional priors results in a proper joint prior. We suggest default values for the hyperparameters and offer intuitive statements for eliciting the hyperparameters based on expert knowledge. The prior framework is applicable for R packages for Bayesian inference such as INLA and RStan.

Three simulation studies show that, in terms of the application-specific measures of interest, PC priors improve inference over Dirichlet priors when used to penalise different levels of complexity in splits. However, when expressing ignorance in a split, Dirichlet priors perform equally well and are preferred for their simplicity. We find that assigning current state-of-the-art default priors for each variance parameter individually is less transparent and does not perform better than using the proposed joint priors. We demonstrate practical use of the new framework by analysing spatial heterogeneity in neonatal mortality in Kenya in 2010–2014 based on complex survey data.

Longitudinal cohorts are a valuable resource for studying HIV disease progression; however, dropout is common in these studies. Subjects often fail to return for visits due to disease progression, loss to follow-up, or death. When dropout depends on unobserved outcomes, data are missing not at random, and results from standard longitudinal data analyses can be biased. Several methods have been proposed to adjust for non-ignorable dropout; however, many of these approaches rely on parametric assumptions about the distribution of dropout times and the functional form of the relationship between the outcome and dropout time. More flexible approaches may be needed when the distribution of dropout times does not follow a known distribution or violates proportional hazards assumptions, or when the relationship between the outcome and dropout times does not have a simple polynomial form. We propose a Bayesian semi-parametric Dirichlet process mixture model to flexibly model the relationship between dropout time and the outcome and show that more accurate inference can be obtained by non-parametrically modeling the distribution of subject-specific effects as well as the distribution of dropout times. Results from simulation studies as well as an application to a longitudinal HIV cohort study database illustrate the strengths of our Bayesian semi-parametric approach.

Optimal Bayesian feature filtering (OBF) is a supervised screening method designed for biomarker discovery. In this article, we prove two major theoretical properties of OBF. First, optimal Bayesian feature selection under a general family of Bayesian models reduces to filtering if and only if the underlying Bayesian model assumes all features are mutually independent. Therefore, OBF is optimal if and only if one assumes all features are mutually independent, and OBF is the only filter method that is optimal under at least one model in the general Bayesian framework. Second, OBF under independent Gaussian models is consistent under very mild conditions, including cases where the data is non-Gaussian with correlated features. This result provides conditions where OBF is guaranteed to identify the correct feature set given enough data, and it justifies the use of OBF in non-design settings where its assumptions are invalid.

Modeling correlation (and covariance) matrices can be challenging due to the positive-definiteness constraint and potential high-dimensionality. Our approach is to decompose the covariance matrix into the correlation and variance matrices and propose a novel Bayesian framework based on modeling the correlations as products of unit vectors. By specifying a wide range of distributions on a sphere (e.g. the squared-Dirichlet distribution), the proposed approach induces flexible prior distributions for covariance matrices (that go beyond the commonly used inverse-Wishart prior). For modeling real-life spatio-temporal processes with complex dependence structures, we extend our method to dynamic cases and introduce unit-vector Gaussian process priors in order to capture the evolution of correlation among components of a multivariate time series. To handle the intractability of the resulting posterior, we introduce the adaptive $\Delta$-Spherical Hamiltonian Monte Carlo. We demonstrate the validity and flexibility of our proposed framework in a simulation study of periodic processes and an analysis of rat’s local field potential activity in a complex sequence memory task.

We propose a model of brain atrophy as a function of high-dimensional genetic information and low-dimensional covariates such as gender, age, APOE gene, and disease status. A nonparametric single-index Bayesian model of high-dimension is proposed to model the relationship using B-spline series prior on the unknown functions and Dirichlet process scale mixture of centered normal prior to the distributions of the random effects. The posterior rate of contraction without the random effect is established for a fixed number of regions and time points with increasing sample size. We implement an efficient computation algorithm through a Hamiltonian Monte Carlo (HMC) algorithm. The performance of the proposed Bayesian method is compared with the corresponding least square estimator in the linear model with horseshoe prior, Least Absolute Shrinkage and Selection Operator (LASSO) and Smoothly Clipped Absolute Deviation (SCAD) penalization on the high-dimensional covariates. The proposed Bayesian method is applied to a dataset on volumes of brain regions recorded over multiple visits of 748 individuals using 620,901 SNPs and 6other covariates for each individual, to identify factors associated with brain atrophy.

Expert assessments are routinely used to inform management and other decision making. However, often these assessments contain considerable biases and uncertainties for which reason they should be calibrated if possible. Moreover, coherently combining multiple expert assessments into one estimate poses a long-standing problem in statistics since modeling expert knowledge is often difficult. Here, we present a hierarchical Bayesian model for expert calibration in a task of estimating a continuous univariate parameter. The model allows experts’ biases to vary as a function of the true value of the parameter and according to the expert’s background. We follow the fully Bayesian approach (the so-called supra-Bayesian approach) and model experts’ bias functions explicitly using hierarchical Gaussian processes. We show how to use calibration data to infer the experts’ observation models with the use of bias functions and to calculate the bias corrected posterior distributions for an unknown system parameter of interest. We demonstrate and test our model and methods with simulated data and a real case study on data-limited fisheries stock assessment. The case study results show that experts’ biases vary with respect to the true system parameter value and that the calibration of the expert assessments improves the inference compared to using uncalibrated expert assessments or a vague uniform guess. Moreover, the bias functions in the real case study show important differences between the reliability of alternative experts. The model and methods presented here can be also straightforwardly applied to other applications than our case study.

In Bayesian hypothesis testing and model selection, prior distributions must be chosen carefully. For example, setting arbitrarily large prior scales for location parameters, which is common practice in estimation problems, can lead to undesirable behavior in testing (see Lindley’s paradox; Lindley (1957)). We study the properties of some restricted type II maximum likelihood (type II ML) priors on regression coefficients. In type II ML, hyperparameters are “estimated” by maximizing the marginal likelihood of a model. In this article, we define priors by estimating their variances or covariance matrices, adding restrictions which ensure that the resulting priors are at least as vague as conventional proper priors for model uncertainty. We find that these type II ML priors typically yield results that are close to answers obtained with the Bayesian Information Criterion (BIC; Schwarz (1978)).

In Bayesian statistics, if the distribution of the data is unknown, then each plausible distribution of the data is indexed by a parameter value, and the prior distribution of the parameter is specified. To the extent that more complicated data distributions tend to require more coincidences for their construction than simpler data distributions, default prior distributions should be transformed to assign additional prior probability or probability density to the parameter values that refer to simpler data distributions. The proposed transformation of the prior distribution relies on the entropy of each data distribution as the relevant measure of complexity. The transformation is derived from a few first principles and extended to stochastic processes.

Markov chain Monte Carlo (MCMC) methods are ubiquitous tools for simulation-based inference in many fields but designing and identifying good MCMC samplers is still an open question. This paper introduces a novel MCMC algorithm, namely, Nested Adaptation MCMC. For sampling variables or blocks of variables, we use two levels of adaptation where the inner adaptation optimizes the MCMC performance within each sampler, while the outer adaptation explores the space of valid kernels to find the optimal samplers. We provide a theoretical foundation for our approach. To show the generality and usefulness of the approach, we describe a framework using only standard MCMC samplers as candidate samplers and some adaptation schemes for both inner and outer iterations. In several benchmark problems, we show that our proposed approach substantially outperforms other approaches, including an automatic blocking algorithm, in terms of MCMC efficiency and computational time.

Objective prior distributions represent an important tool that allows one to have the advantages of using a Bayesian framework even when information about the parameters of a model is not available. The usual objective approaches work off the chosen statistical model and in the majority of cases the resulting prior is improper, which can pose limitations to a practical implementation, even when the complexity of the model is moderate. In this paper we propose to take a novel look at the construction of objective prior distributions, where the connection with a chosen sampling distribution model is removed. We explore the notion of defining objective prior distributions which allow one to have some degree of flexibility, in particular in exhibiting some desirable features, such as being proper, or log-concave, convex etc. The basic tool we use are proper scoring rules and the main result is a class of objective prior distributions that can be employed in scenarios where the usual model based priors fail, such as mixture models and model selection via Bayes factors. In addition, we show that the proposed class of priors is the result of minimising the information it contains, providing solid interpretation to the method.