Statistical inference in high dimensional dynamical systems is often hindered by the unknown dependency structure of model parameters. In particular, the inference of parameterized differential equations (DEs) via Markov chain Monte Carlo (MCMC) samplers often suffers from high proposal rejection rates and is exacerbated by strong autocorrelation structures within the Markov chains leading to poor mixing properties. In this paper, we develop a novel vine-copula based adaptive MCMC approach for efficient parameter inference in dynamical systems with strong parameter interdependence. We exploit the concept of a vine-copula decomposition of distribution densities in order to generate problem-specific proposals for a hybrid independence/random walk Metropolis-Hastings (MH) sampler. The key advantage of this approach is that the corresponding MH proposals generate independent samples from the posterior distribution more efficiently than common competitors. All copula densities can be updated during the sampling procedure for fine-tuning. The performance of our method is assessed on two small-scale examples and finally evaluated on a delay DE model for the JAK2-STAT5 signaling pathway fitted to time-resolved western blot data. We compare our copula-based approach to an independence sampler, a second-order moment-based random walk MH algorithm, and an adaptive MH sampler.

This paper studies Bayesian inference for $\theta =\mathbb{P}(X<Y)$ in the case where the marginal distributions of $X$ and $Y$ belong to classes of distributions obtained by skewing scale mixtures of normals. We separately address the cases where $X$ and $Y$ are independent or dependent random variables. Dependencies between $X$ and $Y$ are modelled using a Gaussian copula. Noninformative benchmark and vague priors are provided for these scenarios and conditions for the existence of the posterior distribution of $\theta$ are presented. We show that the use of the Bayesian models proposed here is also valid in the presence of set observations. Examples using simulated and real data sets are presented.

We introduce a model for a time series of continuous outcomes, that can be expressed as fully nonparametric regression or density regression on lagged terms. The model is based on a dependent Dirichlet process prior on a family of random probability measures indexed by the lagged covariates. The approach is also extended to sequences of binary responses. We discuss implementation and applications of the models to a sequence of waiting times between eruptions of the Old Faithful Geyser, and to a dataset consisting of sequences of recurrence indicators for tumors in the bladder of several patients.

We derive the asymptotic approximation for the posterior distribution when the data are multinomial and the prior is Dirichlet conditioned on satisfying a finite set of linear equality and inequality constraints so the posterior is also Dirichlet conditioned on satisfying these same constraints. When only equality constraints are imposed, the asymptotic approximation is normal. Otherwise it is normal conditioned on satisfying the inequality constraints. In both cases the posterior is a root-$n$-consistent estimator of the parameter vector of the multinomial distribution. As an application we consider the constrained Polya posterior which is a non-informative stepwise Bayes posterior for finite population sampling which incorporates prior information involving auxiliary variables. The constrained Polya posterior is a root-$n$-consistent estimator of the population distribution, hence yields a root-$n$-consistent estimator of the population mean or any other differentiable function of the vector of population probabilities.

In this paper, we establish some optimality properties of the multiple testing rule induced by the horseshoe estimator due to Carvalho, Polson, and Scott (2010, 2009) from a Bayesian decision theoretic viewpoint. We consider the two-groups model for the data and an additive loss structure such that the total loss is equal to the number of misclassified hypotheses. We use the same asymptotic framework as Bogdan, Chakrabarti, Frommlet, and Ghosh (2011) who introduced the Bayes oracle in the context of multiple testing and provided conditions under which the Benjamini-Hochberg and Bonferroni procedures attain the risk of the Bayes oracle. We prove a similar result for the horseshoe decision rule up to $O(1)$ with the constant in the horseshoe risk close to the constant in the oracle. We use the Full Bayes estimate of the tuning parameter $\tau$. It is worth noting that the Full Bayes estimate cannot be replaced by the Empirical Bayes estimate, which tends to be too small.

We describe a unique application of modularization in the context of a Bayesian meta–analysis of quantitative information obtained from the literature. Incomplete reporting, resulting in large amounts of missing data, is common in many meta–analyses, and, in this study, it led to poor mixing and identifiability problems in a fully Bayesian meta–analysis model. As an alternative to the full Bayesian approach, we modularized model components (e.g., modules of covariates, sample sizes, and standard errors) to prevent missing covariate data in these modules from allowing feedback that would otherwise affect parameters in the covariate module (direct feedback control) or affect covariate effects parameters in the mean model for the response (indirect feedback control). The combination of direct and indirect feedback control greatly improves mixing and facilitates convergence of Markov chain Monte Carlo (MCMC), yielding realistic pseudo–posteriors. Thus, our modularization approach allowed us to address important limitations of existing meta–analytic methods by accommodating incomplete reporting and by considering all model quantities as stochastic, including the response variable of interest (e.g., a sample mean) and sample sizes, standard errors, and all covariates, reported or not. We illustrate our approach using data summaries extracted from literature on specific leaf area (SLA) of trees, an important functional trait linked to tree growth and forest dynamics and a key parameter in models of forest responses to climate change. A hierarchical model based on taxonomic relationships allows borrowing of strength to infer SLA for 305 tree species in the United States based on information for 158 of those species. In the context of the SLA meta–analysis, we discuss problems that arise from feedback among model components and provide ecological arguments for modularization—for “cutting feedback.” We anticipate that our approach may be applied to meta–analyses of other important tree traits and to similar meta–analytical studies in general.

We introduce a strategy for quantifying and synthesising uncertainty about elements of a risk assessment using Bayes linear methods. We view the population of subjective belief structures and the use of Bayes linear adjustments as a flexible and transparent tool for risk assessors who want to quantify their uncertainty about hazard based on disparate sources of information. For motivation, we use an application of the strategy to human skin sensitisation risk assessment where there are many competing sources of information available.

Conflicting information, arising from prior misspecification or outlying observations, may contaminate the posterior inference in Bayesian modelling. The use of densities with sufficiently heavy tails usually leads to robust posterior inference, as the influence of the conflicting information decreases with the importance of the conflict. In this paper, we study full robustness in Bayesian modelling of a scale parameter. The log-slowly, log-regularly and log-exponentially varying functions as well as log-exponential credence (LE-credence) are introduced in order to characterize the tail behaviour of a density. The asymptotic behaviour of the marginal and the posterior is described and we find that the scale parameter given the complete information converges in distribution to the scale given the non-conflicting information, as the conflicting values (outliers and/or prior’s scale) tend to 0 or $\infty$, at any given rate. We propose a new family of densities defined on $\mathbb{R}$ with a large spectrum of tail behaviours, called generalized exponential power of the second form (GEP₂), and its exponential transformation defined on $(0,\infty )$, called log-GEP₂, which proves to be helpful for robust modelling. Practical considerations are addressed through a case of combination of experts’ opinions, where non-robust and robust models are compared.

In the context of Bayesian sample size determination in clinical trials, a quantity of interest is the marginal probability that the posterior probability of an alternative hypothesis of interest exceeds a specified threshold. This marginal probability is the same as “average power”; that is, the average of the power function with respect to the prior distribution when using a test based on a Bayesian rejection region. We give conditions under which this marginal probability (or average power) converges to the prior probability of the alternative hypothesis as the sample size increases. This same large sample behavior also holds for the average power of a (frequentist) consistent test. We also examine the limiting behavior of “conditional average power”; that is, power averaged with respect to the prior distribution conditional on the alternative hypothesis being true.

A model is presented for analysis of multivariate binary data with spatio-temporal dependencies, and applied to congressional roll call data from the United States House of Representatives and Senate. The model considers each legislator’s constituency (location), the congressional session (time) of each vote, and the details (text) of each piece of legislation. The model can predict votes of new legislation from only text, while imposing smooth temporal evolution of legislator latent features, and correlation of legislators with adjacent constituencies. Additionally, the model estimates the number of latent dimensions required to represent the data. A Gibbs sampler is developed for posterior inference. The model is demonstrated as an exploratory tool of legislation and it performs well in quantitative comparisons to a traditional ideal-point model.