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The Joint United Nations Programme on HIV/AIDS (UNAIDS) has decided to use Bayesian melding as the basis for its probabilistic projections of HIV prevalence in countries with generalized epidemics. This combines a mechanistic epidemiological model, prevalence data, and expert opinion. Initially, the posterior distribution was approximated by sampling-importance-resampling, which is simple to implement, easy to interpret, transparent to users, and gave acceptable results for most countries. For some countries, however, this is not computationally efficient because the posterior distribution tends to be concentrated around nonlinear ridges and can also be multimodal. We propose instead incremental mixture importance sampling (IMIS), which iteratively builds up a better importance sampling function. This retains the simplicity and transparency of sampling importance resampling, but is much more efficient computationally. It also leads to a simple estimator of the integrated likelihood that is the basis for Bayesian model comparison and model averaging. In simulation experiments and on real data, it outperformed both sampling importance resampling and three publicly available generic Markov chain Monte Carlo algorithms for this kind of problem. [ABSTRACT FROM AUTHOR]

Winter road maintenance is one of the main tasks for the Washington State Department of Transportation. Anti-icing, that is, the preemptive application of chemicals, is often used to keep the roadways free of ice. Given the preventive nature of anti-icing, accurate predictions of road ice are needed. Currently, anti-icing decisions are usually based on deterministic weather forecasts. However, the costs of the two kinds of errors are highly asymmetric because the cost of a road closure due to ice is much greater than that of taking anti-icing measures. As a result, probabilistic forecasts are needed to optimize decision making. We propose two methods for forecasting the probability of ice formation. Starting with deterministic numerical weather predictions, we model temperature and precipitation using distributions centered around the bias-corrected forecasts. This produces a joint predictive probability distribution of temperature and precipitation, which then yields the probability of ice formation, defined here as the occurrence of precipitation when the temperature is below freezing. The first method assumes that temperatures, as well as precipitation, at different spatial locations are conditionally independent given the numerical weather predictions. The second method models the spatial dependence between forecast errors at different locations. The model parameters are estimated using a Bayesian approach via Markov chain Monte Carlo. We evaluate both methods by comparing their probabilistic forecasts with observations of ice formation for Interstate Highway 90 in Washington State for the 2003–2004 and 2004–2005 winter seasons. The use of the probabilistic forecasts reduces costs by about 50% when compared to deterministic forecasts. The spatial method improves the reliability of the forecasts, but does not result in further cost reduction when compared to the first method. [ABSTRACT FROM AUTHOR]

Markov chain Monte Carlo (MCMC) methods for Bayesian computation are mostly used when the dominating measure is the Lebesgue measure, the counting measure, or a product of these. Many Bayesian problems give rise to distributions that are not dominated by the Lebesgue measure or the counting measure alone. In this article we introduce a simple framework for using MCMC algorithms in Bayesian computation with mixtures of mutually singular distributions. The idea is to find a common dominating measure that allows the use of traditional Metropolis--Hastings algorithms. In particular, using our formulation, the Gibbs sampler can be used whenever the full conditionals are available. We compare our formulation with the reversible jump approach and show that the two are closely related. We give results for three examples, involving testing a normal mean, variable selection in regression, and hypothesis testing for differential gene expression under multiple conditions. This allows us to compare the three methods considered: Metropolis--Hastings with mutually singular distributions, Gibbs sampler with mutually singular distributions, and reversible jump. In our examples, we found the Gibbs sampler to be more precise and to need considerably less computer time than the other methods. In addition, the full conditionals used in the Gibbs sampler can be used to further improve the estimates of the model posterior probabilities via Rao--Blackwellization, at no extra cost. [ABSTRACT FROM AUTHOR]

A Bayesian model-based clustering method is proposed for clustering objects on the basis of dissimilarites. This combines two basic ideas. The first is that the objects have latent positions in a Euclidean space, and that the observed dissimilarities are measurements of the Euclidean distances with error. The second idea is that the latent positions are generated from a mixture of multivariate normal distributions, each one corresponding to a cluster. We estimate the resulting model in a Bayesian way using Markov chain Monte Carlo. The method carries out multidimensional scaling and model-based clustering simultaneously, and yields good object configurations and good clustering results with reasonable measures of clustering uncertainties. In the examples we study, the clustering results based on low-dimensional configurations were almost as good as those based on high-dimensional ones. Thus, the method can be used as a tool for dimension reduction when clustering high-dimensional objects, which may be useful especially for visual inspection of clusters. We also propose a Bayesian criterion for choosing the dimension of the object configuration and the number of clusters simultaneously. This is easy to compute and works reasonably well in simulations and real examples. [ABSTRACT FROM AUTHOR]

This article proposes a method for approximating integrated likelihoods in finite mixture models. We formulate the model in terms of the unobserved group memberships, z, and make them the variables of integration. The integral is then evaluated using importance sampling over the z. We propose an adaptive importance sampling function which is itself a mixture, with two types of component distributions, one concentrated and one diffuse. The more concentrated type of component serves the usual purpose of an importance sampling function, sampling mostly group assignments of high posterior probability. The less concentrated type of component allows for the importance sampling function to explore the space in a controlled way to find other, unvisited assignments with high posterior probability. Components are added adaptively, one at a time, to cover areas of high posterior probability not well covered by the current importance sampling function. The method is called incremental mixture importance sampling (IMIS). IMIS is easy to implement and to monitor for convergence. It scales easily for higher dimensional mixture distributions when a conjugate prior is specified for the mixture parameters. The simulated values on which the estimate is based are independent, which allows for straightforward estimation of standard errors. The self-monitoring aspects of the method make it easier to adjust tuning parameters in the course of estimation than standard Markov chain Monte Carlo algorithms. With only small modifications to the code, one can use the method for a wide variety of mixture distributions of different dimensions. The method performed well in simulations and in mixture problems in astronomy and medical research. [ABSTRACT FROM AUTHOR]

We consider robust estimation of gene intensities from cDNA microarray data with replicates. Several statistical methods for estimating gene intensities from microarrays have been proposed, but little work has been done on robust estimation. This is particularly relevant for experiments with replicates, because even one outlying replicate can have a disastrous effect on the estimated intensity for the gene concerned. Because of the many steps involved in the experimental process from hybridization to image analysis, cDNA microarray data often contain outliers. For example, an outlying data value could occur because of scratches or dust on the surface, imperfections in the glass, or imperfections in the array production. We develop a Bayesian hierarchical model for robust estimation of cDNA microarray intensities. Outliers are modeled explicitly using a f-distribution, and our model also addresses such classical issues as design effects, normalization, transformation, and nonconstant variance. Parameter estimation is carried out using Markov chain Monte Carlo. By identifying potential outliers, the method provides automatic quality control of replicate, array, and gene measurements. The method is applied to three publicly available gene expression datasets and compared with three other methods: ANOVA-normalized log ratios, the median log ratio, and estimation after the removal of outliers based on Dixon's test. We find that the between-replicate variability of the intensity estimates is lower for our method than for any of the others. We also address the issue of whether the background should be subtracted when estimating intensities. It has been argued that this should not be done because it increases variability, whereas the arguments for doing so are that there is a physical basis for the image background, and that not doing so will bias downward the estimated log ratios of differentially expressed genes. We show that the arguments on both sides of this debate are correct for our data, but that by using our model one can have the best of both worlds: One can subtract the background without increasing variability by much. [ABSTRACT FROM AUTHOR]

Motivated by the problem of minefield detection, we investigate the problem of classifying mixtures of spatial point processes. In particular we are interested in testing the hypothesis that a given dataset was generated by a Poisson process versus a mixture of a Poisson process and a hard-core Strauss process. We propose testing this hypothesis by comparing the evidence for each model by using partial Bayes factors. We use the term partial Bayes factor to describe a Bayes factor, a ratio of integrated likelihoods, based on only part of the available information, namely that information contained in a small number of functionals of the data. We applied our method to both real and simulated data, and considering the difficulty of classifying these point patterns by eye, our approach overall produced good results. [ABSTRACT FROM AUTHOR]

Network models are widely used to represent relational information among interacting units. In studies of social networks, recent emphasis has been placed on random graph models where the nodes usually represent individual social actors and the edges represent the presence of a specified relation between actors. We develop a class of models where the probability of a relation between actors depends on the positions of individuals in an unobserved "social space." We make inference for the social space within maximum likelihood and Bayesian frameworks, and propose Markov chain Monte Carlo procedures for making inference on latent positions and the effects of observed covariates. We present analyses of three standard datasets from the social networks literature, and compare the method to an alternative stochastic blockmodeling approach. In addition to improving on model fit for these datasets, our method provides a visual and interpretable model-based spatial representation of social relationships and improves on existing methods by allowing the statistical uncertainty in the social space to be quantified and graphically represented. [ABSTRACT FROM AUTHOR]

This article describes an interesting application of Markov chain Monte Carlo (MCMC). The authors found that MCMC worked well for estimating discrete time event history models with unobserved heterogeneity included. Although the authors found only a small amount of unobserved heterogeneity in the Iran data, it is reassuring to know that it was accounted for and that the significance of the main effects was real. There is another benefit to using MCMC to get a sample from the posterior distribution. The sample can be used to examine marginal posterior distributions of the fixed effects graphically, permitting the researcher to check for any asymmetry or nonnormality of any of the marginal distributions. This can be done using a nonparametric density estimation technique. During the investigation, the authors explored many models including other possible covariates. They fitted models using variables such as knowledge and use of various forms of contraception, duration of breast-feeding, or kind of work the woman was employed in, as well as a number of different family-planning variables. Inclusion of one or more of these variables did not improve the model, based on the Bayes factor comparing the model presented in the article with any larger model.

We consider the problem of accounting for model uncertainty in linear regression models. Conditioning on a single selected model ignores model uncertainty, and thus leads to the underestimation of uncertainty when making inferences about quantities of interest. A Bayesian solution to this problem involves averaging over all possible models (i.e., combinations of predictors) when making inferences about quantities of interest. This approach is often not practical. In this article we offer two alternative approaches. First, we describe an ad hoc procedure, "Occam's window," which indicates a small set of models over which a model average can be computed. Second, we describe a Markov chain Monte Carlo approach that directly approximates the exact solution. In the presence of model uncertainty, both of these model averaging procedures provide better predictive performance than any single model that might reasonably have been selected. In the extreme case where there are many candidate predictors but no relationship between any of them and the response, standard variable selection procedures often choose some subset of variables that yields a high R 2 and a highly significant overall F value. In this situation, Occam's window usually indicates the null model (or a small number of models including the null model) as the only one (or ones) to be considered thus largely resolving the problem of selecting significant models when there is no signal in the data. Software to implement our methods is available from StatLib. [ABSTRACT FROM AUTHOR]

We consider adaptive importance sampling techniques that use kernel density estimates at each iteration as importance sampling functions. These can provide more nearly constant importance weights and more precise estimates of quantities of interest than the sampling importance resampling algorithm when the initial importance sampling function is diffuse relative to the target. We propose a new method that adapts to the varying local structure of the target. When the target has unusual structure, such as strong nonlinear relationships between variables, this method provides estimates with smaller mean squared error than alternative methods. [ABSTRACT FROM AUTHOR]

We propose adaptive incremental mixture Markov chain Monte Carlo (AIMM), a novel approach to sample from challenging probability distributions defined on a general state-space. While adaptive MCMC methods usually update a parametric proposal kernel with a global rule, AIMM locally adapts a semiparametric kernel. AIMM is based on an independent Metropolis–Hastings proposal distribution which takes the form of a finite mixture of Gaussian distributions. Central to this approach is the idea that the proposal distribution adapts to the target by locally adding a mixture component when the discrepancy between the proposal mixture and the target is deemed to be too large. As a result, the number of components in the mixture proposal is not fixed in advance. Theoretically, we prove that there exists a stochastic process that can be made arbitrarily close to AIMM and that converges to the correct target distribution. We also illustrate that it performs well in practice in a variety of challenging situations, including high-dimensional and multimodal target distributions. Finally, the methodology is successfully applied to two real data examples, including the Bayesian inference of a semiparametric regression model for the Boston Housing dataset. for this article are available online. [ABSTRACT FROM AUTHOR]

We describe a Bayesian projection model to produce country-specific projections of the total fertility rate (TFR) for all countries. The model decomposes the evolution of TFR into three phases: pre-transition high fertility, the fertility transition, and post-transition low fertility. The model for the fertility decline builds on the United Nations Population Division's current deterministic projection methodology, which assumes that fertility will eventually fall below replacement level. It models the decline in TFR as the sum of two logistic functions that depend on the current TFR level, and a random term. A Bayesian hierarchical model is used to project future TFR based on both the country's TFR history and the pattern of all countries. It is estimated from United Nations estimates of past TFR in all countries using a Markov chain Monte Carlo algorithm. The post-transition low fertility phase is modeled using an autoregressive model, in which long-term TFR projections converge toward and oscillate around replacement level. The method is evaluated using out-of-sample projections for the period since 1980 and the period since 1995, and is found to be well calibrated. [ABSTRACT FROM AUTHOR]