Bayesian variable selection for high dimensional generalized linear models: convergence rates of the fitted densities

Bayesian variable selection has gained much empirical success recently in a variety of applications when the number $K$ of explanatory variables $(x_1,...,x_K)$ is possibly much larger than the sample size $n$. For generalized linear models, if most of the $x_j$'s have very small effects on the response $y$, we show that it is possible to use Bayesian variable selection to reduce overfitting caused by the curse of dimensionality $K\gg n$. In this approach a suitable prior can be used to choose a few out of the many $x_j$'s to model $y$, so that the posterior will propose probability densities $p$ that are ``often close'' to the true density $p^*$ in some sense. The closeness can be described by a Hellinger distance between $p$ and $p^*$ that scales at a power very close to $n^{-1/2}$, which is the ``finite-dimensional rate'' corresponding to a low-dimensional situation. These findings extend some recent work of Jiang [Technical Report 05-02 (2005) Dept. Statistics, Northwestern Univ.] on consistency of Bayesian variable selection for binary classification.Comment: Published in at http://dx.doi.org/10.1214/009053607000000019 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On Bayesian Oracle Properties

When model uncertainty is handled by Bayesian model averaging (BMA) or Bayesian model selection (BMS), the posterior distribution possesses a desirable "oracle property" for parametric inference, if for large enough data it is nearly as good as the oracle posterior, obtained by assuming unrealistically that the true model is known and only the true model is used. We study the oracle properties in a very general context of quasi-posterior, which can accommodate non-regular models with cubic root asymptotics and partial identification. Our approach for proving the oracle properties is based on a unified treatment that bounds the posterior probability of model mis-selection. This theoretical framework can be of interest to Bayesian statisticians who would like to theoretically justify their new model selection or model averaging methods in addition to empirical results. Furthermore, for non-regular models, we obtain nontrivial conclusions on the choice of prior penalty on model complexity, the temperature parameter of the quasi-posterior, and the advantage of BMA over BMS.Comment: 31 page

The Preliminary Study on the Role of 1-Hexene Monooxygenase in Delayed Fruit Ripening by Rhodococcus rhodochrous DAP 96253

Rhodococcus rhodochrous DAP 96253, a well-known industrial bacterium, had various 1-hexene monooxygenase (1-HMO) activities when grown on YEMEA plates supplemented with eight different carbohydrates. Besides, 1-HMO exhibited different storage temperature preferences. Lactose could induce the highest 1-HMO activity in R. rhodochrous DAP 96253 while the cells showed the lowest 1-HMO activity when trehalose was the supplement. The 1-HMO activity of R. rhodochrous DAP 96253 was not maintained when stored at 37°C as well as at 4°C and 25°C. Trehalose-induced 1-HMO activity of R. rhodochrous DAP 96253 was more stable from Day 0 to Day 21 at all these three temperatures, compared with the other seven carbohydrates. Immobilization of enzymes can maintain enzyme activity longer, offer easier enzyme storage conditions and make some enzymes reusable, much research has been done in this area. In this study, R. rhodochrous DAP 96253, grown on YEMEA plates supplemented by glucose and urea, was investigated using whole bananas as the inducer of 1-HMO activity and different immobilization methods to maintain this enzyme activity. It was shown that calcium-alginate polyvinyl alcohol (PVA) beads could maintain 1-HMO activity of R. rhodochrous DAP 96253 more stable than calcium-alginate beads. Whole bananas exhibited very obvious effects of inducing 1-HMO activity of R. rhodochrous DAP 96253. A number of recent studies have clearly demonstrated that induced cells of R. rhodochrous DAP 96253 can prolong the shelf-life of post-harvested fruits. With USDA estimates of 40% of all harvested produce in the US not being consumed because of loss of quality, the ability to extend the period of ripeness of produce has great potential to improve the quality of nutrition. Modification or degradation of those signals (primary and secondary) associated with ripening in fruit or the perception of those signals represents a potential mode of action for delayed ripening by induced cells of R. rhodochrous DAP 96253. Ethylene and cyanide are the two primary signals in ripening. In this study, the role of 1-HMO from induced cells was investigated by time-course experiments focusing on 1-HMO activity and stability. In addition, fruit volatile organic compounds (VOCs) were detected and compared by GC-FID and GC/MS over the course of fruit ripening. The results show a correlation between 1-HMO activity and stability in delayed fruit ripening. It was further demonstrated that the presence of secondary signal fruit VOCs enhanced 1-HMO activity. Aromatic profiles of treated fruits, by GC-FID and GC/MS, show a consistent picture of VOCs associated with earlier fruit ripening stages

Posterior consistency of nonparametric conditional moment restricted models

This paper addresses the estimation of the nonparametric conditional moment restricted model that involves an infinite-dimensional parameter $g_0$. We estimate it in a quasi-Bayesian way, based on the limited information likelihood, and investigate the impact of three types of priors on the posterior consistency: (i) truncated prior (priors supported on a bounded set), (ii) thin-tail prior (a prior that has very thin tail outside a growing bounded set) and (iii) normal prior with nonshrinking variance. In addition, $g_0$ is allowed to be only partially identified in the frequentist sense, and the parameter space does not need to be compact. The posterior is regularized using a slowly growing sieve dimension, and it is shown that the posterior converges to any small neighborhood of the identified region. We then apply our results to the nonparametric instrumental regression model. Finally, the posterior consistency using a random sieve dimension parameter is studied.Comment: Published in at http://dx.doi.org/10.1214/11-AOS930 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org