Online Bayesian Shrinkage Regression

The present work introduces a new online regression method that extends the Shrinkage via Limit of Gibbs sampler (SLOG) in the context of online learning. In particular, we theoretically demonstrate that the proposed Online SLOG (OSLOG) is derived using the Bayesian framework without resorting to the Gibbs sampler. We also state the performance guarantee of OSLOG

Implicit Copulas from Bayesian Regularized Regression Smoothers

We show how to extract the implicit copula of a response vector from a Bayesian regularized regression smoother with Gaussian disturbances. The copula can be used to compare smoothers that employ different shrinkage priors and function bases. We illustrate with three popular choices of shrinkage priors --- a pairwise prior, the horseshoe prior and a g prior augmented with a point mass as employed for Bayesian variable selection --- and both univariate and multivariate function bases. The implicit copulas are high-dimensional, have flexible dependence structures that are far from that of a Gaussian copula, and are unavailable in closed form. However, we show how they can be evaluated by first constructing a Gaussian copula conditional on the regularization parameters, and then integrating over these. Combined with non-parametric margins the regularized smoothers can be used to model the distribution of non-Gaussian univariate responses conditional on the covariates. Efficient Markov chain Monte Carlo schemes for evaluating the copula are given for this case. Using both simulated and real data, we show how such copula smoothing models can improve the quality of resulting function estimates and predictive distributions

Locally adaptive factor processes for multivariate time series

In modeling multivariate time series, it is important to allow time-varying smoothness in the mean and covariance process. In particular, there may be certain time intervals exhibiting rapid changes and others in which changes are slow. If such time-varying smoothness is not accounted for, one can obtain misleading inferences and predictions, with over-smoothing across erratic time intervals and under-smoothing across times exhibiting slow variation. This can lead to mis-calibration of predictive intervals, which can be substantially too narrow or wide depending on the time. We propose a locally adaptive factor process for characterizing multivariate mean-covariance changes in continuous time, allowing locally varying smoothness in both the mean and covariance matrix. This process is constructed utilizing latent dictionary functions evolving in time through nested Gaussian processes and linearly related to the observed data with a sparse mapping. Using a differential equation representation, we bypass usual computational bottlenecks in obtaining MCMC and online algorithms for approximate Bayesian inference. The performance is assessed in simulations and illustrated in a financial application

Shrinkage Estimators in Online Experiments

We develop and analyze empirical Bayes Stein-type estimators for use in the estimation of causal effects in large-scale online experiments. While online experiments are generally thought to be distinguished by their large sample size, we focus on the multiplicity of treatment groups. The typical analysis practice is to use simple differences-in-means (perhaps with covariate adjustment) as if all treatment arms were independent. In this work we develop consistent, small bias, shrinkage estimators for this setting. In addition to achieving lower mean squared error these estimators retain important frequentist properties such as coverage under most reasonable scenarios. Modern sequential methods of experimentation and optimization such as multi-armed bandit optimization (where treatment allocations adapt over time to prior responses) benefit from the use of our shrinkage estimators. Exploration under empirical Bayes focuses more efficiently on near-optimal arms, improving the resulting decisions made under uncertainty. We demonstrate these properties by examining seventeen large-scale experiments conducted on Facebook from April to June 2017

Using VARs and TVP-VARs with many macroeconomic variables

This paper discusses the challenges faced by the empirical macroeconomist and methods for surmounting them. These challenges arise due to the fact that macroeconometric models potentially include a large number of variables and allow for time variation in parameters. These considerations lead to models which have a large number of parameters to estimate relative to the number of observations. A wide range of approaches are surveyed which aim to overcome the resulting problems. We stress the related themes of prior shrinkage, model averaging and model selection. Subsequently, we consider a particular modelling approach in detail. This involves the use of dynamic model selection methods with large TVP-VARs. A forecasting exercise involving a large US macroeconomic data set illustrates the practicality and empirical success of our approach