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Bayesian analysis provides a convenient setting for the estimation of complex generalized additive regression models (GAMs). Since computational power has tremendously increased in the past decade, it is now possible to tackle complicated inferential problems, for example, with Markov chain Monte Carlo simulation, on virtually any modern computer. This is one of the reasons why Bayesian methods have become increasingly popular, leading to a number of highly specialized and optimized estimation engines and with attention shifting from conditional mean models to probabilistic distributional models capturing location, scale, shape (and other aspects) of the response distribution. To embed many different approaches suggested in literature and software, a unified modeling architecture for distributional GAMs is established that exploits distributions, estimation techniques (posterior mode or posterior mean), and model terms (fixed, random, smooth, spatial,...). It is shown that within this framework implementing algorithms for complex regression problems, as well as the integration of already existing software, is relatively straightforward. The usefulness is emphasized with two complex and computationally demanding application case studies: a large daily precipitation climatology, as well as a Cox model for continuous time with space-time interactions. Supplementary material for this article is available online.

In studies of environmental change of the past few centuries, [Formula omitted]Pb dating is often used to obtain chronologies for sedimentary sequences. One of the most commonly used approaches to estimate the age of material at different depths in a sequence is to assume a constant rate of supply (CRS) or influx of 'unsupported' [Formula omitted]Pb from the atmosphere, together with a constant or varying amount of 'supported' [Formula omitted]Pb. Current [Formula omitted]Pb dating models do not use a proper statistical framework and provide poor estimates of the uncertainties. Here, we develop a new model for [Formula omitted]Pb dating, where ages and values of supported and unsupported [Formula omitted]Pb form part of the parameters. We apply our model to a case study from Canada as well as to some simulated examples. Our model can extend beyond the current CRS approach, deal with asymmetric errors and mix [Formula omitted]Pb with other types of dating, thus obtaining more robust, realistic and statistically better defined age estimates.

We propose a new Monte Carlo method for sampling from multimodal distributions. The idea of this technique is based on splitting the task into two: finding the modes of a target distribution π and sampling, given the knowledge of the locations of the modes. The sampling algorithm relies on steps of two types: local ones, preserving the mode; and jumps to regions associated with different modes. Besides, the method learns the optimal parameters of the algorithm, while it runs, without requiring user intervention. Our technique should be considered as a flexible framework, in which the design of moves can follow various strategies known from the broad MCMC literature. In order to design an adaptive scheme that facilitates both local and jump moves, we introduce an auxiliary variable representing each mode, and we define a new target distribution π̃ on an augmented state space 𝓧 × 𝓘, where 𝓧 is the original state space of π and 𝓘 is the set of the modes. As the algorithm runs and updates its parameters, the target distribution π̃ also keeps being modified. This motivates a new class of algorithms, Auxiliary Variable Adaptive MCMC. We prove general ergodic results for the whole class before specialising to the case of our algorithm.

Perturbation theory for Markov chains addresses the question of how small differences in the transition probabilities of Markov chains are reflected in differences between their distributions. We prove powerful and flexible bounds on the distance of the nth step distributions of two Markov chains when one of them satisfies a Wasserstein ergodicity condition. Our work is motivated by the recent interest in approximate Markov chain Monte Carlo (MCMC) methods in the analysis of big data sets. By using an approach based on Lyapunov functions, we provide estimates for geometrically ergodic Markov chains under weak assumptions. In an autoregressive model, our bounds cannot be improved in general. We illustrate our theory by showing quantitative estimates for approximate versions of two prominent MCMC algorithms, the Metropolis–Hastings and stochastic Langevin algorithms.

Markov Chain Monte–Carlo (MCMC) is an increasingly popular method for obtaining information about distributions, especially for estimating posterior distributions in Bayesian inference. This article provides a very basic introduction to MCMC sampling. It describes what MCMC is, and what it can be used for, with simple illustrative examples. Highlighted are some of the benefits and limitations of MCMC sampling, as well as different approaches to circumventing the limitations most likely to trouble cognitive scientists.

The multispecies coalescent provides a natural framework for accommodating ancestral genetic polymorphism and coalescent processes that can cause different genomic regions to have different genealogical histories. The Bayesian program BPP includes a full-likelihood implementation of the multispecies coalescent, using transmodel Markov chain Monte Carlo to calculate the posterior probabilities of different species trees. BPP is suitable for analyzing multilocus sequence data sets and it accommodates the heterogeneity of gene trees (both the topology and branch lengths) among loci and gene tree uncertainties due to limited phylogenetic information at each locus. Here, we provide a practical guide to the use of BPP in species tree estimation. BPP is a command-line program that runs on linux, macosx, and windows. This protocol shows how to use both BPP 3.4 (http://abacus.gene.ucl.ac.uk/software/) and BPP 4.0 (https://github.com/bpp/).

Standard MCMC methods can scale poorly to big data settings due to the need to evaluate the likelihood at each iteration. There have been a number of approximate MCMC algorithms that use sub-sampling ideas to reduce this computational burden, but with the drawback that these algorithms no longer target the true posterior distribution. We introduce a new family of Monte Carlo methods based upon a multidimensional version of the Zig-Zag process of [Ann. Appl. Probab. 27 (2017) 846–882], a continuous-time piecewise deterministic Markov process. While traditional MCMC methods are reversible by construction (a property which is known to inhibit rapid convergence) the Zig-Zag process offers a flexible nonreversible alternative which we observe to often have favourable convergence properties. We show how the Zig-Zag process can be simulated without discretisation error, and give conditions for the process to be ergodic. Most importantly, we introduce a sub-sampling version of the Zig-Zag process that is an example of an exact approximate scheme, that is, the resulting approximate process still has the posterior as its stationary distribution. Furthermore, if we use a control-variate idea to reduce the variance of our unbiased estimator, then the Zig-Zag process can be super-efficient: after an initial preprocessing step, essentially independent samples from the posterior distribution are obtained at a computational cost which does not depend on the size of the data.

The total-evidence approach to divergence time dating uses molecular and morphological data from extant and fossil species to infer phylogenetic relationships, species divergence times, and macroevolutionary parameters in a single coherent framework. Current model-based implementations of this approach lack an appropriate model for the tree describing the diversification and fossilization process and can produce estimates that lead to erroneous conclusions. We address this shortcoming by providing a total-evidence method implemented in a Bayesian framework. This approach uses a mechanistic tree prior to describe the underlying diversification process that generated the tree of extant and fossil taxa. Previous attempts to apply the total-evidence approach have used tree priors that do not account for the possibility that fossil samples may be direct ancestors of other samples, that is, ancestors of fossil or extant species or of clades. The fossilized birth–death (FBD) process explicitly models the diversification, fossilization, and sampling processes and naturally allows for sampled ancestors. This model was recently applied to estimate divergence times based on molecular data and fossil occurrence dates. We incorporate the FBD model and a model of morphological trait evolution into a Bayesian total-evidence approach to dating species phylogenies. We apply this method to extant and fossil penguins and show that the modern penguins radiated much more recently than has been previously estimated, with the basal divergence in the crown clade occurring at ∼12.7 Ma and most splits leading to extant species occurring in the last 2 myr. Our results demonstrate that including stem-fossil diversity can greatly improve the estimates of the divergence times of crown taxa. The method is available in BEAST2 (version 2.4) software www.beast2.org with packages SA (version at least 1.1.4) and morph-models (version at least 1.0.4) installed.

We describe NIMBLE, a system for programming statistical algorithms for general model structures within R. NIMBLE is designed to meet three challenges: flexible model specification, a language for programming algorithms that can use different models, and a balance between high-level programmability and execution efficiency. For model specification, NIMBLE extends the BUGS language and creates model objects, which can manipulate variables, calculate log probability values, generate simulations, and query the relationships among variables. For algorithm programming, NIMBLE provides functions that operate with model objects using two stages of evaluation. The first stage allows specialization of a function to a particular model and/or nodes, such as creating a Metropolis-Hastings sampler for a particular block of nodes. The second stage allows repeated execution of computations using the results of the first stage. To achieve efficient second-stage computation, NIMBLE compiles models and functions via C++, using the Eigen library for linear algebra, and provides the user with an interface to compiled objects. The NIMBLE language represents a compilable domain-specific language (DSL) embedded within R. This article provides an overview of the design and rationale for NIMBLE along with illustrative examples including importance sampling, Markov chain Monte Carlo (MCMC) and Monte Carlo expectation maximization (MCEM). Supplementary materials for this article are available online.

When conducting Bayesian inference, delayed-acceptance (DA) Metropolis-Hastings (MH) algorithms and DA pseudo-marginal MH algorithms can be applied when it is computationally expensive to calculate the true posterior or an unbiased estimate thereof, but a computationally cheap approximation is available. A first accept-reject stage is applied, with the cheap approximation substituted for the true posterior in the MH acceptance ratio. Only for those proposals that pass through the first stage is the computationally expensive true posterior (or unbiased estimate thereof) evaluated, with a second accept-reject stage ensuring that detailed balance is satisfied with respect to the intended true posterior. In some scenarios, there is no obvious computationally cheap approximation. A weighted average of previous evaluations of the computationally expensive posterior provides a generic approximation to the posterior. If only the k-nearest neighbors have nonzero weights then evaluation of the approximate posterior can be made computationally cheap provided that the points at which the posterior has been evaluated are stored in a multi-dimensional binary tree, known as a KD-tree. The contents of the KD-tree are potentially updated after every computationally intensive evaluation. The resulting adaptive, delayed-acceptance [pseudo-marginal] Metropolis-Hastings algorithm is justified both theoretically and empirically. Guidance on tuning parameters is provided and the methodology is applied to a discretely observed Markov jump process characterizing predator-prey interactions and an ODE system describing the dynamics of an autoregulatory gene network. Supplementary material for this article is available online.