Bayesian Shrinkage Model Research Articles

Trace class Markov chains for the Normal-Gamma Bayesian shrinkage model

High-dimensional data, where the number of variables exceeds or is comparable to the sample size, is now pervasive in many scientific applications. In recent years, Bayesian shrinkage models have been developed as effective and computationally feasible tools to analyze such data, especially in the context of linear regression. In this paper, we focus on the Normal-Gamma shrinkage model developed by Griffin and Brown [7]. This model subsumes the popular Bayesian lasso model, and a three-block Gibbs sampling algorithm to sample from the resulting intractable posterior distribution has been developed in [7]. We consider an alternative two-block Gibbs sampling algorithm, and rigorously demonstrate its advantage over the three-block sampler by comparing specific spectral properties. In particular, we show that the Markov operator corresponding to the two-block sampler is trace class (and hence Hilbert-Schmidt), whereas the operator corresponding to the three-block sampler is not even Hilbert-Schmidt. The trace class property for the two-block sampler implies geometric convergence for the associated Markov chain, which justifies the use of Markov chain CLT’s to obtain practical error bounds for MCMC based estimates. Additionally, it facilitates theoretical comparisons of the two-block sampler with sandwich algorithms which aim to improve performance by inserting inexpensive extra steps in between the two conditional draws of the two-block sampler.

Inclusion of biological knowledge in a Bayesian shrinkage model for joint estimation of SNP effects.

With the aim of improving detection of novel single-nucleotide polymorphisms (SNPs) in genetic association studies, we propose a method of including prior biological information in a Bayesian shrinkage model that jointly estimates SNP effects. We assume that the SNP effects follow a normal distribution centered at zero with variance controlled by a shrinkage hyperparameter. We use biological information to define the amount of shrinkage applied on the SNP effects distribution, so that the effects of SNPs with more biological support are less shrunk toward zero, thus being more likely detected. The performance of the method was tested in a simulation study (1,000 datasets, 500 subjects with ∼200 SNPs in 10 linkage disequilibrium (LD) blocks) using a continuous and a binary outcome. It was further tested in an empirical example on body mass index (continuous) and overweight (binary) in a dataset of 1,829 subjects and 2,614 SNPs from 30 blocks. Biological knowledge was retrieved using the bioinformatics tool Dintor, which queried various databases. The joint Bayesian model with inclusion of prior information outperformed the standard analysis: in the simulation study, the mean ranking of the true LD block was 2.8 for the Bayesian model versus 3.6 for the standard analysis of individual SNPs; in the empirical example, the mean ranking of the six true blocks was 8.5 versus 9.3 in the standard analysis. These results suggest that our method is more powerful than the standard analysis. We expect its performance to improve further as more biological information about SNPs becomes available.

Stock Market Reactions to Customer and Competitor Orientations: The Case of Initial Public Offerings

Recognizing that initial public offerings (IPOs) represent the debut of private firms on the public stage, this study investigates how pre-IPO customer and competitor orientations (CCOs) affect IPO outcomes. Building on information economics, we propose that CCOs influence investors' sentiments toward an IPO and that both IPO-specific variables (which influence the credibility of CCO information) and facets of the organizational institutional and task environments (which influence the appropriateness of CCO information) moderate this influence. We test the framework using data collected from computer-aided text analysis, expert coders, and secondary sources for 543 IPOs across 43 industries between 2000 and 2004. A Bayesian shrinkage model, which accounts for industry-specific effects and uses latent instrumental variables to address CCO endogeneity, shows that CCOs positively influence IPO outcomes. Furthermore, (1) underwriter reputation and venture funding positively moderate the effects of CCOs; (2) technological and market turbulence positively and institutional complexity negatively moderate the effect of customer orientation; and (3) technological turbulence, competitive intensity, and institutional complexity positively moderate the effect of competitor orientation. Also, accounting for endogeneity using latent instrumental variables substantially improves the predictive validity of the model, relative to alternative model specifications.

Forecasting the US real house price index: Structural and non-structural models with and without fundamentals

We employ a 10-variable dynamic structural general equilibrium model to forecast the US real house price index as well as its downturn in 2006:Q2. We also examine various Bayesian and classical time-series models in our forecasting exercise to compare to the dynamic stochastic general equilibrium model, estimated using Bayesian methods. In addition to standard vector-autoregressive and Bayesian vector autoregressive models, we also include the information content of either 10 or 120 quarterly series in some models to capture the influence of fundamentals. We consider two approaches for including information from large data sets — extracting common factors (principle components) in factor-augmented vector autoregressive or Bayesian factor-augmented vector autoregressive models as well as Bayesian shrinkage in a large-scale Bayesian vector autoregressive model. We compare the out-of-sample forecast performance of the alternative models, using the average root mean squared error for the forecasts. We find that the small-scale Bayesian-shrinkage model (10 variables) outperforms the other models, including the large-scale Bayesian-shrinkage model (120 variables). In addition, when we use simple average forecast combinations, the combination forecast using the 10 best atheoretical models produces the minimum RMSEs compared to each of the individual models, followed closely by the combination forecast using the 10 atheoretical models and the DSGE model. Finally, we use each model to forecast the downturn point in 2006:Q2, using the estimated model through 2005:Q2. Only the dynamic stochastic general equilibrium model actually forecasts a downturn with any accuracy, suggesting that forward-looking microfounded dynamic stochastic general equilibrium models of the housing market may prove crucial in forecasting turning points.

A Bayesian Shrinkage Model for Incomplete Longitudinal Binary Data With Application to the Breast Cancer Prevention Trial

We consider inference in randomized longitudinal studies with missing data that is generated by skipped clinic visits and loss to followup. In this setting, it is well known that full data estimands are not identified unless unverified assumptions are imposed. We assume a non-future dependence model for the drop-out mechanism and partial ignorability for the intermittent missingness. We posit an exponential tilt model that links nonidentifiable distributions and distributions identified under partial ignorability. This exponential tilt model is indexed by nonidentified parameters, which are assumed to have an informative prior distribution, elicited from subject-matter experts. Under this model, full data estimands are shown to be expressed as functionals of the distribution of the observed data. To avoid the curse of dimensionality, we model the distribution of the observed data using a Bayesian shrinkage model. In a simulation study, we compare our approach to a fully parametric and a fully saturated model for the distribution of the observed data. Our methodology is motivated by, and applied to, data from the Breast Cancer Prevention Trial.

Bayesian sigmoid shrinkage with improper variance priors and an application to wavelet denoising

The normal Bayesian linear model is extended by assigning a flat prior to the δ th power of the variance components of the regression coefficients 0 < δ ⩽ 1 2 in order to improve prediction accuracy. In the case of orthonormal regressors, easy-to-compute analytic expressions are derived for the posterior distribution of the shrinkage and regression coefficients. The expected shrinkage is a sigmoid function of the squared value of the least-squares estimate divided by its standard error. This gives a small amount of shrinkage for large values and, provided δ is small, heavy shrinkage for small values. The limit behavior for both small and large values approaches that of the ideal coordinatewise shrinker in terms of the expected squared error of prediction, when δ is close to 0. In a simulation study of wavelet denoising, the proposed Bayesian shrinkage model yielded a lower mean squared error than soft thresholding (lasso), and was competitive with two recent wavelet shrinkage methods based on mixture prior distributions.