One of the challenges in using Markov chain Monte Carlo for model analysis in studies with very large datasets is the need to scan through the whole data at each iteration of the sampler, which can be computationally prohibitive. Several approaches have been developed to address this, typically drawing computationally manageable subsamples of the data. Here we consider the specific case where most of the data from a mixture model provides little or no information about the parameters of interest, and we aim to select subsamples such that the information extracted is most relevant. The motivating application arises in flow cytometry, where several measurements from a vast number of cells are available. Interest lies in identifying specific rare cell subtypes and characterizing them according to their corresponding markers. We present a Markov chain Monte Carlo approach where an initial subsample of the full dataset is used to guide selection sampling of a further set of observations targeted at a scientificallyinteresting, low probability region. We define a Sequential Monte Carlo strategy in which the targeted subsample is augmented sequentially as estimates improve, and introduce a stopping rule for determining the size of the targeted subsample. An example from flow cytometry illustrates the ability of the approach to increase the resolution of inferences for rare cell subtypes.

We approach a discussion of this paper by asking: to what extent the specific models and computational techniques used succeed in bridging data and research questions? This discussion is mostly seen as an opportunity to pose useful questions rather than providing definitive answers, pointing to relevant literature when necessary.

This paper develops a matrix-variate adaptive Markov chain Monte Carlo (MCMC) methodology for Bayesian Cointegrated Vector Auto Regressions (CVAR). We replace the popular approach to sampling Bayesian CVAR models, involving griddy Gibbs, with an automated efficient alternative, based on the Adaptive Metropolis algorithm of Roberts and Rosenthal (2009). Developing the adaptive MCMC framework for Bayesian CVAR models allows for efficient estimation of posterior parameters in significantly higher dimensional CVAR series than previously possible with existing griddy Gibbs samplers. For a n-dimensional CVAR series, the matrix-variate posterior is in dimension $3n^2 + n$, with significant correlation present between the blocks of matrix random variables. Hence, utilizing a griddy Gibbs sampler for large n becomes computationally impractical as it involves approximating an $n \times n$ full conditional posterior using a spline over a high dimensional $n \times n$ grid. The adaptive MCMC approach is demonstrated to be ideally suited to learning on-line a proposal to reflect the posterior correlation structure, therefore improving the computational efficiency of the sampler.

We also treat the rank of the CVAR model as a random variable and perform joint inference on the rank and model parameters. This is achieved with a Bayesian posterior distribution defined over both the rank and the CVAR model parameters, and inference is made via Bayes Factor analysis of rank.

Practically the adaptive sampler also aids in the development of automated Bayesian cointegration models for algorithmic trading systems considering instruments made up of several assets, such as currency baskets. Previously the literature on financial applications of CVAR trading models typically only considers pairs trading (n=2) due to the computational cost of the griddy Gibbs. We are able to extend under our adaptive framework to $n >> 2$ and demonstrate an example with n = 10, resulting in a posterior distribution with parameters up to dimension 310. By also considering the rank as a random quantity we can ensure our resulting trading models are able to adjust to potentially time varying market conditions in a coherent statistical framework.

We examine a flexible class of nonstationary stochastic models for multivariate spatial data proposed by Majumdar et al. (2010). This covariance model is based on convolutions of spatially varying covariance kernels with centers corresponding to the centers of "local stationarity". A Bayesian method for estimation of the parameters in the model based on a Gibbs sampler is applied to simulated data to check for model sensitivity. The effect of a perturbation of the model in terms of kernel centers is also examined. Finally, the method is applied to a bivariate soil chemistry data from the the Central Arizona Phoenix Long Term Ecological Project (CAP LTER). Prediction bias, prediction standard deviation and predictive coverage are examined for different candidate models. In addition, a comparison with the bivariate stationary coregionalization model introduced by Wackernagel (2003) is carried out. A variant of the model proposed in Majumdar et al. (2010), with random kernel centers, is also examined. The latter model is seen to work much better than the stationary coregionalization model, and to perform comparably with the model with random kernel centers. Simulations indicate that the model is sensitive to under- or over-specification of kernel centers. On the other hand, application to real data seems to indicate that centroids of the regions that are homogeneous can be used as means of the random kernel centers. Cross validation can be used as a way of finding the best model with an appropriate number of kernels.

In this paper we consider the analysis of incomplete tables using correspondence analysis. We focus on a dataset concerning congenital heart disease (Fraser and Hunter, 1975), in which the data forms a square table, but only a symmetrized version of the off- diagonal entries was reported. We use Markov chain Monte Carlo (MCMC) on a hierarchical Bayes model to estimate the underlying rates, and use correspondence analysis to study the relationships in the completed table.

Regularization, e.g. lasso, has been shown to be effective in quantile regression in improving the prediction accuracy (Li and Zhu, 2008; Wu and Liu, 2009). This paper studies regularization in quantile regressions from a Bayesian perspective. By proposing a hierarchical model framework, we give a generic treatment to a set of regularization approaches, including lasso, group lasso and elastic net penalties. Gibbs samplers are derived for all cases. This is the first work to discuss regularized quantile regression with the group lasso penalty and the elastic net penalty. Both simulated and real data examples show that Bayesian regularized quantile regression methods often outperform quantile regression without regularization and their non-Bayesian counterparts with regularization.

The subsurface environment beneath the Municipality of Bologna, Italy, is comprised of a series of alluvial deposits which constitute large and productive aquifer systems. These are separated from the shallow, free surface aquifer by a low permeability barrier called aquitard Alpha. The upper aquifer contains water that shows relevant contamination from industrial pollutants. The deep aquifers are relatively pristine and provide about 80\% of all groundwater used for drinking and industrial purposes in the area of Bologna. Hence, it is imperative that planners understand where along aquitard Alpha there exists potential direct connection between the upper and the deep aquifers, which could lead to contamination of the city's key water supply well fields.

In order to better assess the existence of preferential flow paths between these aquifer systems, we carry out a statistical analysis in which the aquitard is represented as a bivariate spatial process, accounting for dependence between the two spatial components. The first process models its effective thickness. The second process is binary, modeling the presence or absence of direct vertical connections between the aquifers. This map is then cross referenced with other forms of data regarding the hydrology of the region.

Implementing Bayesian variable selection for linear Gaussian regression models for analysing high dimensional data sets is of current interest in many fields. In order to make such analysis operational, we propose a new sampling algorithm based upon Evolutionary Monte Carlo and designed to work under the "large $p$, small $n$" paradigm, thus making fully Bayesian multivariate analysis feasible, for example, in genetics/genomics experiments. Two real data examples in genomics are presented, demonstrating the performance of the algorithm in a space of up to $10,000$ covariates. Finally the methodology is compared with a recently proposed search algorithms in an extensive simulation study.