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The marginal likelihood is a central tool for drawing Bayesian inference about the number of components in mixture models. It is often approximated since the exact form is unavailable. A bias in the approximation may be due to an incomplete exploration by a simulated Markov chain (e.g., a Gibbs sequence) of the collection of posterior modes, a phenomenon also known as lack of label switching, as all possible label permutations must be simulated by a chain in order to converge and hence overcome the bias. In an importance sampling approach, imposing label switching to the importance function results in an exponential increase of the computational cost with the number of components. In this paper, two importance sampling schemes are proposed through choices for the importance function; a MLE proposal and a Rao-Blackwellised importance function. The second scheme is called dual importance sampling. We demonstrate that this dual importance sampling is a valid estimator of the evidence and moreover show that the statistical efficiency of estimates increases. To reduce the induced high demand in computation, the original importance function is approximated but a suitable approximation can produce an estimate with the same precision and with reduced computational workload., 24 pages, 5 figures

While mixtures of Gaussian distributions have been studied for more than a century (Pearson, 1894), the construction of a reference Bayesian analysis of those models still remains unsolved, with a general prohibition of the usage of improper priors (Fruwirth-Schnatter, 2006) due to the ill-posed nature of such statistical objects. This difficulty is usually bypassed by an empirical Bayes resolution (Richardson and Green, 1997). By creating a new parameterisation cantered on the mean and possibly the variance of the mixture distribution itself, we manage to develop here a weakly informative prior for a wide class of mixtures with an arbitrary number of components. We demonstrate that some posterior distributions associated with this prior and a minimal sample size are proper. We provide MCMC implementations that exhibit the expected exchangeability. We only study here the univariate case, the extension to multivariate location-scale mixtures being currently under study. An R package called Ultimixt is associated with this paper., 32 pages, 14 figures, 3 tables

Approximate Bayesian computation (ABC) methods provide an elaborate approach to Bayesian inference on complex models, including model choice. Both theoretical arguments and simulation experiments indicate, however, that model posterior probabilities may be poorly evaluated by standard ABC techniques. We propose a novel approach based on a machine learning tool named random forests to conduct selection among the highly complex models covered by ABC algorithms. We thus modify the way Bayesian model selection is both understood and operated, in that we rephrase the inferential goal as a classification problem, first predicting the model that best fits the data with random forests and postponing the approximation of the posterior probability of the predicted MAP for a second stage also relying on random forests. Compared with earlier implementations of ABC model choice, the ABC random forest approach offers several potential improvements: (i) it often has a larger discriminative power among the competing models, (ii) it is more robust against the number and choice of statistics summarizing the data, (iii) the computing effort is drastically reduced (with a gain in computation efficiency of at least fifty), and (iv) it includes an approximation of the posterior probability of the selected model. The call to random forests will undoubtedly extend the range of size of datasets and complexity of models that ABC can handle. We illustrate the power of this novel methodology by analyzing controlled experiments as well as genuine population genetics datasets. The proposed methodologies are implemented in the R package abcrf available on the CRAN., 39 pages, 15 figures, 6 tables

International audience; This note is commenting on Ronald Gallant’s (2015) reflections on the construction of Bayesian prior distributions from moment conditions. The main conclusion is that the paper does not deliver a working principle that could justify inference based on such priors.

The Adaptive Multiple Importance Sampling algorithm is aimed at an optimal recycling of past simulations in an iterated importance sampling (IS) scheme. The difference with earlier adaptive IS implementations like Population Monte Carlo is that the importance weights of all simulated values, past as well as present, are recomputed at each iteration, following the technique of the deterministic multiple mixture estimator of Owen & Zhou ( J. Amer. Statist. Assoc., 95, 2000, 135). Although the convergence properties of the algorithm cannot be investigated, we demonstrate through a challenging banana shape target distribution and a population genetics example that the improvement brought by this technique is substantial. [ABSTRACT FROM AUTHOR]

In this paper, we show how a complete and exact Bayesian analysis of a parametric mixture model is possible in some cases when components of the mixture are taken from exponential families and when conjugate priors are used. This restricted set-up allows us to show the relevance of the Bayesian approach as well as to exhibit the limitations of a complete analysis, namely that it is impossible to conduct this analysis when the sample size is too large, when the data are not from an exponential family, or when priors that are more complex than conjugate priors are used.

The book A Treatise on Probability was published by John Maynard Keynes in 1921. It contains a critical assessment of the foundations of probability and of the current statistical methodology. As a modern reader, we review here the aspects that are most related with statistics, avoiding a neophyte's perspective on the philosophical issues. In particular, the book is quite critical of the Bayesian approach and we examine the arguments provided by Keynes, as well as the alternative he proposes. This review does not subsume the scholarly study of Aldrich (2008a) relating Keynes with the statistics community of the time.

Richard CottonHardcover: 196 pagesYear: 2017Publisher: Chapman & Hall/CRCISBN-13: 978-1498763653Preliminary versions of these reviews were posted on xianblog.wordpress.com.When I saw this title by ...

We argue here about the relevance and the ultimate unity of the Bayesian approach in a neutral and agnostic manner. Our main theme is that Bayesian data analysis is an effective tool for handling complex models, as proven by the increasing proportion of Bayesian studies in the applied sciences. We disregard in this essay the philosophical debates on the deeper meaning of probability and on the random nature of parameters as things of the past that do a disservice to the approach and are incomprehensible to most bystanders., Comment: This paper is written in conjunction with the 3rd Bayesian econometrics meeting that took place at the Rimini Centre for Economic Analysis on July 01-02, 2009. A version will eventually be published in the Review of Economic Analysis

The k-nearest-neighbour procedure is a well-known deterministic method used in supervised classification. This paper proposes a reassessment of this approach as a statistical technique derived from a proper probabilistic model; in particular, we modify the assessment made in a previous analysis of this method undertaken by Holmes and Adams (2002,2003), and evaluated by Manocha and Girolami (2007), where the underlying probabilistic model is not completely well-defined. Once a clear probabilistic basis for the k-nearest-neighbour procedure is established, we derive computational tools for conducting Bayesian inference on the parameters of the corresponding model. In particular, we assess the difficulties inherent to pseudo-likelihood and to path sampling approximations of an intractable normalising constant, and propose a perfect sampling strategy to implement a correct MCMC sampler associated with our model. If perfect sampling is not available, we suggest using a Gibbs sampling approximation. Illustrations of the performance of the corresponding Bayesian classifier are provided for several benchmark datasets, demonstrating in particular the limitations of the pseudo-likelihood approximation in this set-up.

Our aim is to analyze the link between optimism and risk aversion in a subjective expected utility setting and to estimate the average level of optimism when weighted by risk tolerance. This quantity is of particular importance since it characterizes the consensus belief in risk-taking situations with heterogeneous beliefs. Its estimation leads to a nontrivial statistical problem. We start from a large lottery survey (1,536 individuals). We assume that individuals have true unobservable characteristics and that their answers in the survey are noisy realizations of these characteristics. We adopt a Bayesian approach for the statistical analysis of this problem and use an hybrid MCMC approximation method to numerically estimate the distributions of the unobservable characteristics. We obtain that individuals are on average pessimistic and that pessimism and risk tolerance are positively correlated. As a consequence, we conclude that the consensus belief is biased towards pessimism.

Variance reduction has always been a central issue in Monte Carlo experiments. Population Monte Carlo can be used to this effect, in that a mixture of importance functions, called a D-kernel, can be iteratively optimized to achieve the minimum asymptotic variance for a function of interest among all possible mixtures. The implementation of this iterative scheme is illustrated for the computation of the price of a European option in the Cox-Ingersoll-Ross model. A Central Limit theorem as well as moderate deviations are established for the D-kernel Population Monte Carlo methodology.

CAT scanning is used in longitudinal animal science experiments to assess possible changes to carcase composition induced by treatment over given time periods. A hierarchical Bayesian mixture model can be used to analyse the CAT scan data in terms of the proportion of each tissue type present in a scan. In this paper we present an extension to the hierarchical Bayesian mixture model in which estimated parameters from neighbouring CAT scans can be incorporated into the current model. These models are demonstrated using two examples.

We derive and study SQMC (Sequential Quasi-Monte Carlo), a class of algorithms obtained by introducing QMC point sets in particle filtering. SQMC is related to, and may be seen as an extension of, the array-RQMC algorithm of L'Ecuyer et al. (2006). The complexity of SQMC is $O(N \log N)$, where $N$ is the number of simulations at each iteration, and its error rate is smaller than the Monte Carlo rate $O_P(N^{-1/2})$. The only requirement to implement SQMC is the ability to write the simulation of particle $x_t^n$ given $x_{t-1}^n$ as a deterministic function of $x_{t-1}^n$ and a fixed number of uniform variates. We show that SQMC is amenable to the same extensions as standard SMC, such as forward smoothing, backward smoothing, unbiased likelihood evaluation, and so on. In particular, SQMC may replace SMC within a PMCMC (particle Markov chain Monte Carlo) algorithm. We establish several convergence results. We provide numerical evidence that SQMC may significantly outperform SMC in practical scenarios., 55 pages, 10 figures (final version)

Most of the existing algorithms for approximate Bayesian computation (ABC) assume that it is feasible to simulate pseudo-data from the model at each iteration. However, the computational cost of these simulations can be prohibitive for high dimensional data. An important example is the Potts model, which is commonly used in image analysis. Images encountered in real world applications can have millions of pixels, therefore scalability is a major concern. We apply ABC with a synthetic likelihood to the hidden Potts model with additive Gaussian noise. Using a pre-processing step, we fit a binding function to model the relationship between the model parameters and the synthetic likelihood parameters. Our numerical experiments demonstrate that the precomputed binding function dramatically improves the scalability of ABC, reducing the average runtime required for model fitting from 71 hours to only 7 minutes. We also illustrate the method by estimating the smoothing parameter for remotely sensed satellite imagery. Without precomputation, Bayesian inference is impractical for datasets of that scale., 5th IMS-ISBA joint meeting (MCMSki IV)

This note is a discussion of the article “Bayesian model selection based on proper scoring rules” by A. P. Dawid and M. Musio, to appear in Bayesian Analysis. While appreciating the concepts behind the use of proper scoring rules, we point out here some possible practical difficulties with the advocated approach.

This note is a discussion commenting on the paper by Ly et al. on "Harold Jeffreys's Default Bayes Factor Hypothesis Tests: Explanation, Extension, and Application in Psychology" and on the perceived shortcomings of the classical Bayesian approach to testing, while reporting on an alternative approach advanced by Kamary, Mengersen, Robert and Rousseau (2014. arxiv:1412.2044) as a solution to this quintessential inference problem., Comment: 10 pages, one figure, discussion of Ly, A., Verhagen, A. J., and Wagenmakers, E.-J. (in press). Harold Jeffreys's default Bayes factor hypothesis tests: Explanation, extension, and application in psychology. Journal of Mathematical Psychology

A new piston core from Santa Barbara Basin, California provides evidence of the timing, magnitude, and character of deglaciation, including evidence of warming prior to Termination IA. IMAGES Site MD02-2503 (570 m water depth) consists of intermittently laminated hemipelagic sediments extending to Interstadial (D/O) event 6 (∼34 ka), that accumulated at ∼135 cm/ka. During the deglacial episode (16.75–10 ka), δ 18 O values decreased by 3.2‰ in the planktonic species Globigerina bulloides , indicating a total warming of 8–9 °C recorded by surface-dwelling foraminifera (inferred by removing the 1‰ influence of ice volume change). Similarly, benthic species ( Bolivina argentea and Uvigerina peregrina ) record a 1.65‰ δ 18 O decrease across the deglacial, interpreted as a 2–3 °C warming at upper intermediate depths. δ 18 O values of both planktonic and benthics indicate that surface and intermediate waters began to warm ∼2 ka prior to Termination IA, beginning at ∼16.5 ka. Intermediate water warming exhibits similar structure and synchronous timing with surface waters. These findings are consistent with a growing number of records from around the globe that exhibit pre-Bolling warming prior to Termination IA, and extends the record of such processes to the northern Pacific.

Let ir denote an intractable probability distribution that we would like to explore. Suppose that we have a positive recurrent, irreducible Markov chain that satisfies a minorization condition and has a as its invariant measure. We provide a method of using simulations from the Markov chain to construct a statistical estimate of X from which it is straightforward to sample. We show that this estimate is 'strongly consistent' in the sense that the total variation distance between the estimate and X converges to 0 almost surely as the number of simulations grows. Moreover, we use some recently developed asymptotic results to provide guidance as to how much simulation is necessary. Draws from the estimate can be used to approximate features of X or as intelligent starting values for the original Markov chain. We illustrate our methods with two examples.

The toolbox available in Bayesian Statistics has increased considerably in the past decade and it has opened new avenues for Bayesian inference, the first and foremost being Bayesian model choice. The MCMC and particle filter technologies have hugely increased the potential for Bayesian applications, in particular in missing variable models, as illustrated in this short tutorial. We will also mention a new direction in this field, namely the development of adaptive algorithms that avoid a lenghty tuning to fit the problem at hand by automatically modifying the parameters of the algorithm.

Summary The choice of the summary statistics that are used in Bayesian inference and in particular in approximate Bayesian computation algorithms has bearings on the validation of the resulting inference. Those statistics are nonetheless customarily used in approximate Bayesian computation algorithms without consistency checks. We derive necessary and sufficient conditions on summary statistics for the corresponding Bayes factor to be convergent, namely to select the true model asymptotically. Those conditions, which amount to the expectations of the summary statistics differing asymptotically under the two models, are quite natural and can be exploited in approximate Bayesian computation settings to infer whether or not a choice of summary statistics is appropriate, via a Monte Carlo validation.

A late Miocene to early Pliocene sequence drilled on the continental slope of southwest Africa off the Orange river mouth (ODP Site 1085) has been investigated. Clay mineral assemblages, coarse siliciclastics and benthic foraminifer accumulation rates (BFAR) unravel a step by step evolution of marine and continental environments closely related to sea level variations, ocean circulation and global climate: (1) smectite is a typical tracer of the Orange river load, whereas illite is mostly transported by the Benguela current (like chlorite) and winds, and kaolinite is derived from low latitudes by the poleward undercurrent and the North Atlantic Deep Water (NADW); (2) increased erosion and influence of the Orange river after 9.6 Ma is linked to a sea level drop at a time of Antarctic ice-growth. This has been followed by an increased seasonality of precipitation and high productivity, but low oxygen content and associated dissolution of carbonates; (3) increased productivity and dissolution of carbonates, and coeval increase of continental aridity after 8.9 Ma express a further development of the Benguela current and upwelling; (4) better preservation of carbonates and increased contribution of terrigenous material from northern sources at 6.9 Ma are related to increased circulation of NADW after an early stage of northern hemisphere glaciation; (5) increased erosion and contribution from the Orange river and westward shift of the area of higher productivity from 5.9–5.8 Ma to 5.3–5.2 Ma are related to a significant fall of sea level, and encompass the time of the entire Mediterranean salinity crisis; (6) short-term variations of the smectite/illite ratio (S/I) and BFAR suggest a major control of productivity by wind and current activities (and related upwelling), but may express brief variations of sea level in specific intervals before 8.9 Ma and during the late Messinian especially.

We consider the choice of an optimal sample size for multiple-comparison problems. The motivating application is the choice of the number of microarray experiments to be carried out when learning about differential gene expression. However, the approach is valid in any application that involves multiple comparisons in a large number of hypothesis tests. We discuss two decision problems in the context of this setup: the sample size selection and the decision about the multiple comparisons. We adopt a decision-theoretic approach, using loss functions that combine the competing goals of discovering as many differentially expressed genes as possible, while keeping the number of false discoveries manageable. For consistency, we use the same loss function for both decisions. The decision rule that emerges for the multiple-comparison problem takes the exact form of the rules proposed in the recent literature to control the posterior expected falsediscovery rate. For the sample size selection, we combine the expe...

Johnstone, Kerkyacharian, Picard and Raimondo Johnstone, Kerkyacharian, Picard and Raimondo are interested in the inverse problem of estimating f where f has been convolved with g and then contaminated with white noise. This popular problem has been tackled by a wide variety of procedures and wavelet methods have recently generated considerable interest. Donoho’s (1995) seminal wavelet–vaguelette paper introduced the notion that wavelets would be a good choice for the representation of f since real life objects, such as images, are more likely to be efficiently represented using wavelets when compared with, for example, Fourier representations. Johnstone and his colleagues have moved the field on significantly. In particular, their procedure is more direct than wavelet–vaguelette or Abramovich and Silverman’s (1998) vaguelette–wavelet method; it can handle boxcar blur theoretically and practically, they have rates of convergence forp =2 (p defines the type of loss) and the paper innovates through use of the new maxiset approach. For me, the most appealing of these innovations is that of enabling the treatment of boxcar blur which is one of the most common types of inverse problem. However, is it really, really, the case that for rational a nothing can be done? Formula (4) compels us to say no, nothing can, but naively it still feels wrong. Formula (19) is the popular ‘signal-plus-noise’ model but here it is a little different from what normally appears in the literature because the quantities are complex-valued random variables. More specifically, the zl are zero-mean Gaussian variables which are complex valued and satisfy E.zlzk/= δlk. One question is why threshold the βk and not the yl directly? The covariance of the βk is given by cov.βk, βl/=n ∑ m ΨkmΨ l m

This article shows how Bayesian inference for switching regression models and their generalizations can be achieved by the specification of loss functions which overcome the label switching problem common to all mixture models. We also derive an extension to models where the number of components in the mixture is unknown, based on the birthand-death technique developed in recent literature. The methods are illustrated on various real datasets.

The variable selection method proposed in the paper is based on the evaluation of the Kullback–Leibler distance between the full (or encompassing) model and its submodels. The Bayesian implementation of the method does not require a separate prior modeling on the submodels since the corresponding parameters for the submodels are defined as the Kullback–Leibler projections of the full model parameters. The result of the selection procedure is the submodel with the smallest number of covariates which is at an acceptable distance of the full model. We introduce the notion of explanatory power of a model and scale the maximal acceptable distance in terms of the explanatory power of the full model. Moreover, an additivity property between embedded submodels shows that our selection procedure is equivalent to select the submodel with the smallest number of covariates which has a sufficient explanatory power. We illustrate the performances of this method on a breast cancer dataset

We provide an exact simulation algorithm that produces variables from truncated Gaussian distributions on (\Bbb R+)p via a perfect sampling scheme, based on stochastic ordering and slice sampling, since accept-reject algorithms like the one of Geweke (1991) and Robert (1995) are difficult to extend to higher dimensions.

Paperback: 276 pages Publisher: Cambridge University Press and Institute of Mathematical Statistics Year: 2010 Language: English ISBN-13: 978-0-5211-9249-1 Large-Scale Inference: Empirical Bayes Me...

Hardcover: 516 pages Publisher: CRC Press, first edition (June 2010) Language: English ISBN-10: 1439803544 Here is another Bayesian textbook that appeared recently. I read it within a few days and,...

Detailed investigation (630 samples) of clay mineral assemblages and benthic foraminifer accumulation rates (BFAR) as a proxy for paleoproductivity has been conducted on two late Eocene–Oligocene sequences drilled at intermediate water depths of the Kerguelen Plateau and Maud Rise (Southern Ocean). Distinct changes in clay assemblages and BFAR – that are in general a factor two higher on the Maud Rise compared to the Kerguelen Plateau – unravel a step-by-step evolution of marine and continental Antarctic environments that eventually resulted in permanent ice coverage and dense cold water formation: (1) late Eocene dominance of chemical weathering was associated with intermittent erosion of Antarctic soils and substrates. Main erosional events are recorded at 36 and 34 Ma, prior to the major earliest Oligocene glacial event, and are especially noticeable in the Indian sector of the Southern Ocean; (2) physical weathering progressed in continental areas adjacent to the Maud Rise and intermediate water production intensified in the Indian sector of the Southern Ocean throughout the Eocene–Oligocene transition, as the Tasmanian seaway developed. Environmental changes and a strong 600 kyr lasting peak of paleoproductivity seem closely associated with the earliest Oligocene isotope shift and the accumulation of ice-rafted detritus on the Kerguelen Plateau; (3) aridity and physical weathering progressed in East Antarctica near the Maud Rise (Site 689) at 33.2–33.15 Ma, as polar transfer of heat within the deep ocean increased slightly. This may have prevented extensive ice development during at least part of the early Oligocene; (4) increased productivity, vertical mixing and turnover at Site 744 on the Kerguelen Plateau, followed by a two-step increase of physical weathering on the adjacent continent from 32 to 31.2 Ma, suggest progressive homogenization of climates and circulation as the passageway South of Tasmania widened and deepened; (5) physical weathering reached a maximum in East Antarctic areas adjacent to the Maud Rise and a hiatus developed on the Kerguelen Plateau during the mid-Oligocene Oi2–Oi2a interval (31–26.5 Ma), in possible relation with opening processes of the modern Drake Passage and Scotia Sea and a further step in the development of oceanic circumpolar circulation; (6) near-400-kyr cycles of productivity and clay mineral composition throughout the two sequences suggest that orbital frequencies played a role in the small-scale variations of both continental and marine Antarctic environments of the late Paleogene.

SummaryWe consider the construction of perfect samplers for posterior distributions associated with mixtures of exponential families and conjugate priors, starting with a perfect slice sampler in the spirit of Mira and co-workers. The methods rely on a marginalization akin to Rao–Blackwellization and illustrate the duality principle of Diebolt and Robert. A first approximation embeds the finite support distribution on the latent variables within a continuous support distribution that is easier to simulate by slice sampling, but we later demonstrate that the approximation can be very poor. We conclude by showing that an alternative perfect sampler based on a single backward chain can be constructed. This alternative can handle much larger sample sizes than the slice sampler first proposed.

Markov chain Monte Carlo (MCMC) methods, while facilitating the solution of many complex problems in Bayesian inference, are not currently well adapted to the problem of marginal maximum a posteriori (MMAP) estimation, especially when the number of parameters is large. We present here a simple and novel MCMC strategy, called State-Augmentation for Marginal Estimation (SAME), which leads to MMAP estimates for Bayesian models. We illustrate the simplicity and utility of the approach for missing data interpolation in autoregressive time series and blind deconvolution of impulsive processes.

International audience; Johnson proposes replacing the usual P = 0.05 standard for significance with the more stringent P = 0.005. This might be good advice in practice, but we remain troubled by Johnson's logic because it seems to dodge the essential nature of any such rule, which is that it expresses a tradeoff between the risks of publishing misleading results and of important results being left unpublished. Ultimately such decisions should depend on costs, benefits, and probabilities of all outcomes.

This paper discusses the dual interpretation of the Jeffreys--Lindley's paradox associated with Bayesian posterior probabilities and Bayes factors, both as a differentiation between frequentist and Bayesian statistics and as a pointer to the difficulty of using improper priors while testing. We stress the considerable impact of this paradox on the foundations of both classical and Bayesian statistics. While assessing existing resolutions of the paradox, we focus on a critical viewpoint of the paradox discussed by Spanos (2013) in Philosophy of Science., 15 pages (second revision)

Following the critical review of Seaman et al. (2012), we reflect on what is presumably the most essential aspect of Bayesian statistics, namely the selection of a prior density. In some cases, Bayesian inference remains fairly stable under a large range of noninformative prior distributions. However, as discussed by \citet{Hd}, there may also be unintended consequences of a choice of a noninformative prior and, these authors consider this problem ignored in Bayesian studies. As they based their argumentation on four examples, we reassess these examples and their Bayesian processing via different prior choices. Our conclusion is to lower the degree of worry about the impact of the prior, exhibiting an overall stability of the posterior distributions. We thus consider that the warnings of Seaman et al. (2012), while commendable, do not jeopardize the use of most noninformative priors., 15 pages, 8 figures, 5 tables

The purpose of this book is to provide a self-contained entry into practical and computational Bayesian statistics using generic examples from the most common models for a class duration of about seven blocks that roughly correspond to 13–15 weeks of teaching (with three hours of lectures per week), depending on the intended level and the prerequisites imposed on the students. (That estimate does not include practice—i.e., R programming labs, writing data reports—since those may have a variable duration, also depending on the students’ involvement and their programming abilities.)

International audience; This Bayesian modeling book provides a self-contained entry to computational Bayesian statistics. Focusing on the most standard statistical models and backed up by real datasets and an all-inclusive R (CRAN) package called bayess, the book provides an operational methodology for conducting Bayesian inference, rather than focusing on its theoretical and philosophical justifications. Readers are empowered to participate in the real-life data analysis situations depicted here from the beginning. The stakes are high and the reader determines the outcome. Special attention is paid to the derivation of prior distributions in each case and specific reference solutions are given for each of the models. Similarly, computational details are worked out to lead the reader towards an effective programming of the methods given in the book. In particular, all R codes are discussed with enough detail to make them readily understandable and expandable. This works in conjunction with the bayess package.Bayesian Essentials with R can be used as a textbook at both undergraduate and graduate levels, as exemplified by courses given at Université Paris Dauphine (France), University of Canterbury (New Zealand), and University of British Columbia (Canada). It is particularly useful with students in professional degree programs and scientists to analyze data the Bayesian way. The text will also enhance introductory courses on Bayesian statistics. Prerequisites for the book are an undergraduate background in probability and statistics, if not in Bayesian statistics. A strength of the text is the noteworthy emphasis on the role of models in statistical analysis.This is the new, fully-revised edition to the book Bayesian Core: A Practical Approach to Computational Bayesian Statistics.

The link between smectite composition in sediments from the northern North Atlantic and Labrador Sea, and deep circulation is being further investigated through detailed studies of the X-ray pattern of smectites and cation saturations. This allows clear distinction of dominant terrigenous sources associated to the main components of the modern Western Boundary Undercurrent. Time variations of smectite characteristics in two piston cores from the inlet and outlet of the Western Boundary Undercurrent gyre in the Labrador Sea indicate: (1) a more southern circulation of North East Atlantic Deep Water during the Late Glacial; (2) a step by step transition to the modern pattern of deep circulation during the Late Glacial/Holocene transition, with intensification of North East Atlantic Deep Water and Davis Strait Overflow; (3) an expansion of Davis Strait Overflow and Labrador Sea Water circulation in relation to ice surges and deposition of detrital layers; (4) an intensified circulation of North East Atlantic Deep Water during the Younger Dryas; and (5) a very recent increased influence of Denmark Strait Overflow Water beginning between 4.4 and

This paper develops an extension of the Riemann sum techniques of Philippe (J. Statist. Comput. Simul. 59: 295–314) in the setup of MCMC algorithms. It shows that these techniques apply equally well to the output of these algorithms, with similar speeds of convergence which improve upon the regular estimator. The restriction on the dimension associated with Riemann sums can furthermore be overcome by Rao–Blackwellization methods. This approach can also be used as a control variate technique in convergence assessment of MCMC algorithms, either by comparing the values of alternative versions of Riemann sums, which estimate the same quantity, or by using genuine control variate, that is, functions with known expectations, which are available in full generality for constants and scores.

This book review analyses the strengths and weaknesses of Gentle's book in terms of computational statistics and of statistical computing.

This chapter surveys advances in the field of Bayesian computation over the past twenty years, with missing data. It also contains some novel computational entries on the double-exponential model that may be of interest per se., 26 pages, 10 figures, revision of a paper written as a chapter for the Annual Review of Statistics and Its Applications

In this work we apply the methodology of integral priors to handle Bayesian model selection in binomial regression models with a general link function. These models are very often used to investigate associations and risks in epidemiological studies where one goal is to exhibit whether or not an exposure is a risk factor for developing a certain disease; the purpose of the current paper is to test the effect of specific exposure factors. We formulate the problem as a Bayesian model selection case and solve it using objective Bayes factors. To construct the reference prior distributions on the regression coefficients of the binomial regression models, we rely on the methodology of integral priors that is nearly automatic as it only requires the specification of estimation reference priors and it does not depend on tuning parameters or on hyperparameters within these priors., 17 pages, 2 figures, 3 tables

Switching ARMA processes have recently appeared as an efficient modelling to nonlinear time-series models, because they can represent multiple or heterogeneous dynamics through simple components. The levels of dependence between the observations are double: at a first level, the parameters of the model are selected by a Markovian procedure. At a second level, the next observation is generated according to a standard time-series model. When the model involves a moving average structure, the complexity of the resulting likelihood function is such that simulation techniques, like those proposed by Shephard (1994, Biometrika 81, 115–131) and Billio and Monfort (1998, Journal of Statistical Planning and Inference 68, 65–103), are necessary to derive an inference on the parameters of the model. We propose in this paper a Bayesian approach with a non-informative prior distribution developed in Mengersen and Robert (1996, Bayesian Statistics 5. Oxford University Press, Oxford, pp. 255–276) and Robert and Titterington (1998, Statistics and Computing 8(2), 145–158) in the setup of mixtures of distributions and hidden Markov models, respectively. The computation of the Bayes estimates relies on MCMC techniques which iteratively simulate missing states, innovations and parameters until convergence. The performances of the method are illustrated on several simulated examples. This work also extends the papers by Chib and Greenberg (1994, Journal of Econometrics 64, 183–206) and Chib (1996, Journal of Econometrics 75(1), 79–97) which deal with ARMA and hidden Markov models, respectively.

In complex models like hidden Markov chains, the convergence of the MCMC algorithms used to approximate the posterior distribution and the Bayes estimates of the parameters of interest must be controlled in a robust manner. We propose in this paper a series of online controls, which rely on classical non-parametric tests, to evaluate independence from the start-up distribution, stability of the Markov chain, and asymptotic normality. These tests lead to graphical control spreadsheets which arepresentedin the set-up of normalmixture hidden Markov chains to compare the full Gibbs sampler with an aggregated Gibbs sampler based on the forward – backward formulas.