Regression-Based Bayesian Estimation and Structure Learning for Nonparanormal Graphical Models.

A nonparanormal graphical model is a semiparametric generalization of a Gaussian graphical model for continuous variables in which it is assumed that the variables follow a Gaussian graphical model only after some unknown smooth monotone transformations. We consider a Bayesian approach to inference in a nonparanormal graphical model in which we put priors on the unknown transformations through a random series based on B-splines. We use a regression formulation to construct the likelihood through the Cholesky decomposition on the underlying precision matrix of the transformed variables and put shrinkage priors on the regression coefficients. We apply a plug-in variational Bayesian algorithm for learning the sparse precision matrix and compare the performance to a posterior Gibbs sampling scheme in a simulation study. We finally apply the proposed methods to a microarray data set. The proposed methods have better performance as the dimension increases, and in particular, the variational Bayesian approach has the potential to speed up the estimation in the Bayesian nonparanormal graphical model without the Gaussianity assumption while retaining the information to construct the graph.

Bayesian nonparametric estimation of ROC surface under verification bias.

The receiver operating characteristic (ROC) surface, as a generalization of the ROC curve, has been widely used to assess the accuracy of a diagnostic test for three categories. A common problem is verification bias, referring to the situation where not all subjects have their true classes verified. In this paper, we consider the problem of estimating the ROC surface under verification bias. We adopt a Bayesian nonparametric approach by directly modeling the underlying distributions of the three categories by Dirichlet process mixture priors. We propose a robust computing algorithm by only imposing a missing at random assumption for the verification process but no assumption on the distributions. The method can also accommodate covariates information in estimating the ROC surface, which can lead to a more comprehensive understanding of the diagnostic accuracy. It can be adapted and hugely simplified in the case where there is no verification bias, and very fast computation is possible through the Bayesian bootstrap process. The proposed method is compared with other commonly used methods by extensive simulations. We find that the proposed method generally outperforms other approaches. Applying the method to two real datasets, the key findings are as follows: (1) human epididymis protein 4 has a slightly better diagnosis ability compared to CA125 in discriminating healthy, early stage, and late stage patients of epithelial ovarian cancer. (2) Serum albumin has a prognostic ability in distinguishing different stages of hepatocellular carcinoma.

Fast Translation Invariant Multiscale Image Denoising.

Translation invariant (TI) cycle spinning is an effective method for removing artifacts from images. However, for a method using O(n) time, the exact TI cycle spinning by averaging all possible circulant shifts requires O(n(2)) time where n is the number of pixels, and therefore is not feasible in practice. Existing literature has investigated efficient algorithms to calculate TI version of some denoising approaches such as Haar wavelet. Multiscale methods, especially those based on likelihood decomposition, such as penalized likelihood estimator and Bayesian methods, have become popular in image processing because of their effectiveness in denoising images. As far as we know, there is no systematic investigation of the TI calculation corresponding to general multiscale approaches. In this paper, we propose a fast TI (FTI) algorithm and a more general k-TI (k-TI) algorithm allowing TI for the last k scales of the image, which are applicable to general d-dimensional images (d = 2, 3, …) with either Gaussian or Poisson noise. The proposed FTI leads to the exact TI estimation but only requires O(n log2 n) time. The proposed k-TI can achieve almost the same performance as the exact TI estimation, but requires even less time. We achieve this by exploiting the regularity present in the multiscale structure, which is justified theoretically. The proposed FTI and k-TI are generic in that they are applicable on any smoothing techniques based on the multiscale structure. We demonstrate the FTI and k-TI algorithms on some recently proposed state-of-the-art methods for both Poisson and Gaussian noised images. Both simulations and real data application confirm the appealing performance of the proposed algorithms. MATLAB toolboxes are online accessible to reproduce the results and be implemented for general multiscale denoising approaches provided by the users.

Bayesian ROC curve estimation under verification bias.

Receiver operating characteristic (ROC) curve has been widely used in medical science for its ability to measure the accuracy of diagnostic tests under the gold standard. However, in a complicated medical practice, a gold standard test can be invasive, expensive, and its result may not always be available for all the subjects under study. Thus, a gold standard test is implemented only when it is necessary and possible. This leads to the so-called 'verification bias', meaning that subjects with verified disease status (also called label) are not selected in a completely random fashion. In this paper, we propose a new Bayesian approach for estimating an ROC curve based on continuous data following the popular semiparametric binormal model in the presence of verification bias. By using a rank-based likelihood, and following Gibbs sampling techniques, we compute the posterior distribution of the binormal parameters intercept and slope, as well as the area under the curve by imputing the missing labels within Markov Chain Monte-Carlo iterations. Consistency of the resulting posterior under mild conditions is also established. We compare the new method with other comparable methods and conclude that our estimator performs well in terms of accuracy.

Bayesian bootstrap estimation of ROC curve.

Receiver operating characteristic (ROC) curve is widely applied in measuring discriminatory ability of diagnostic or prognostic tests. This makes the ROC analysis one of the most active research areas in medical statistics. Many parametric and semiparametric estimation methods have been proposed for estimating the ROC curve and its functionals. In this paper, we propose the Bayesian bootstrap (BB), a fully nonparametric estimation method, for the ROC curve and its functionals, such as the area under the curve (AUC). The BB method offers a bandwidth-free smoothing approach to the empirical estimate, and gives credible bounds. The accuracy of the estimate of the ROC curve in the simulation studies is examined by the integrated absolute error. In comparison with other existing curve estimation methods, the BB method performs well in terms of accuracy, robustness and simplicity. We also propose a procedure based on the BB approach to test the binormality assumption.

Nonparametric bayesian estimation of positive false discovery rates.

We propose a Dirichlet process mixture model (DPMM) for the P-value distribution in a multiple testing problem. The DPMM allows us to obtain posterior estimates of quantities such as the proportion of true null hypothesis and the probability of rejection of a single hypothesis. We describe a Markov chain Monte Carlo algorithm for computing the posterior and the posterior estimates. We propose an estimator of the positive false discovery rate based on these posterior estimates and investigate the performance of the proposed estimator via simulation. We also apply our methodology to analyze a leukemia data set.