Selected Ongoing Projects:

  • Global-local shrinkage and the horseshoe+ estimator: The horseshoe estimator of Carvalho, Polson and Scott (2010) was one of the first works to demonstrate the power of "global-local" shrinkage in ultra-sparse Bayesian variable selection problems. Since then, multiple attractive theoretical properties of this estimator have been discovered. In a collaborative work with Nick Polson, Jyotishka Datta and Brandon Willard, we propose a new estimator, termed the "horseshoe+ estimator," that improves upon the horseshoe, both theoretically and empirically. Bhadra et al. (2016a) give the details. It also appears global-local shrinkage priors are good candidates for default priors for low-dimensional, nonlinear functions in a normal means model, where the so-called "flat" priors fail. Bhadra et al. (2016b) demonstrate their use in a few such problems. Bhadra et al. (2016c) demonstrate the use of two integral identities for generating global-local mixtures. Bhadra et al. (2016d) formally demonstrate that the prediction performance for the class of global shrinkage regression methods (ridge regression, principal components regression etc.) can be improved by using local, component-specific shrinkage parameters. Bhadra et al. (2017a) derive fast computational algorithms to perform feature selection using the non-convex horseshoe regularization penalty. Li, Craig and Bhadra (2017) propose the use of the horseshoe prior in estimating the precision matrix for multivariate Gaussian data. Bhadra et al. (2017b) is a review article summarizing the important developments in global-local shrinkage methods between 2010 and 2017.

  • Bayesian models for joint mean-covariance estimation and for mixed discrete-continuous data: Bayesian variable and covariance selections have been treated separately in the statistics literature for a long time. We do a combined analysis in the context of a Gaussian sparse seemingly unrelated regression (SSUR) model to infer jointly the important sparse set of predictors as well as the important sparse set of non-zero partial correlations in the responses. We apply our technique to expression quantitative trait loci (eQTL) analysis where the expression level of a gene (response) is typically affected by a set of important SNPs (predictors) and the responses exhibit conditional dependence among themselves. Both the number of predictors and the number of correlated responses routinely exceed the sample size. We find that a marginalization-based collapsed Gibbs sampler offers a computationally efficient solution. The first ideas appeared in Bhadra and Mallick (2013). Building on that, Feldman, Bhadra and Kirshner (2014) found a way to relax the need to be restricted to decomposable graphs. Bhadra and Baladandayuthapani (2013) is an application of the methodology to brain cancer (glioblastoma) data. Bhadra, Rao and Baladandayuthapani (2017) developed a technique to perform network inference in presence of multivariate data that are of mixed discrete and continuous nature.

  • Selected Past Projects:

  • Iterated filtering and its applications in modeling infectious disease dynamics: Iterated filtering is a simulation-based technique for maximum likelihood inference in hidden Markov models with intractable likelihood. Particle filters (i.e., sequential Monte Carlo filters) are used in iterated filtering to devise a stochastic approximation scheme that converges to the maximum likelihhod estimate of the model parameters. We provide theoretical results on iterated filtering in Ionides et al. (2011), proving the method results in consistent estimates and show it to be a special case of a broad class of stochastic approximation techniques. We apply iterated filtering to estimate parameters in a compartment model of epidemic malaria to capture the spread of the disease in Northwest India and answer scientific questions regarding role of rainfall in the spread of the epidemic in Bhadra et al. (2011) and Laneri et al. (2010). This is joint work with Ed Ionides and Mercedes Pascual, among others.

  • Adaptive particle allocation for off-line iterated particle filters: In many off-line sequential Monte Carlo (SMC) based techniques, the filter is used repeatedly to estimate model parameters or likelihood of the data. Examples include iterated filtering of Ionides et al. or particle MCMC of Andrieu et al . In the off-line setting, we formulate a way to minimize the variance of the likelihood estimate resulting from SMC given a constraint on the total number of particles, i.e., the available computing power. Results in Bhadra and Ionides (2015) indicate up to 55% computational savings.

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