Flow cytometric measurement of signaling protein abundances has proved particularly useful for elucidation of signaling pathway structure. The single cell nature of the data ensures a very large dataset size, providing a statistically robust dataset for structure learning. Moreover, the approach is easily scaled to many conditions in high throughput. However, the technology suffers from a dimensionality constraint: at the cutting edge, only about 12 protein species can be measured per cell, far from sufficient for most signaling pathways. Because the structure learning algorithm (in practice) requires that all variables be measured together simultaneously, this restricts structure learning to the number of variables that constitute the flow cytometer's upper dimensionality limit. To address this problem, we present here an algorithm that enables structure learning for sparsely distributed data, allowing structure learning beyond the measurement technology's upper dimensionality limit for simultaneously measurable variables. The algorithm assesses pairwise (or n-wise) dependencies, constructs “Markov neighborhoods” for each variable based on these dependencies, measures each variable in the context of its neighborhood, and performs structure learning using a constrained search.
Bayesian networks; flow cytometry; graphical models; proteomics; signaling pathways
Finding latent patterns in high dimensional data is an important research problem with numerous applications. The most well known approaches for high dimensional data analysis are feature selection and dimensionality reduction. Being widely used in many applications, these methods aim to capture global patterns and are typically performed in the full feature space. In many emerging applications, however, scientists are interested in the local latent patterns held by feature subspaces, which may be invisible via any global transformation.
In this paper, we investigate the problem of finding strong linear and nonlinear correlations hidden in feature subspaces of high dimensional data. We formalize this problem as identifying reducible subspaces in the full dimensional space. Intuitively, a reducible subspace is a feature subspace whose intrinsic dimensionality is smaller than the number of features. We present an effective algorithm, REDUS, for finding the reducible subspaces. Two key components of our algorithm are finding the overall reducible subspace, and uncovering the individual reducible subspaces from the overall reducible subspace. A broad experimental evaluation demonstrates the effectiveness of our algorithm.
Reducible Subspace; High Dimensional Data
Investigation of microbial communities, particularly human associated communities, is significantly enhanced by the vast amounts of sequence data produced by high throughput sequencing technologies. However, these data create high-dimensional complex data sets that consist of a large proportion of zeros, non-negative skewed counts, and frequently, limited number of samples. These features distinguish sequence data from other forms of high-dimensional data, and are not adequately addressed by statistical approaches in common use. Ultimately, medical studies may identify targeted interventions or treatments, but lack of analytic tools for feature selection and identification of taxa responsible for differences between groups, is hindering advancement. The objective of this paper is to examine the application of a two-part statistic to identify taxa that differ between two groups. The advantages of the two-part statistic over common statistical tests applied to sequence count datasets are discussed. Results from the t-test, the Wilcoxon test, and the two-part test are compared using sequence counts from microbial ecology studies in cystic fibrosis and from cenote samples. We show superior performance of the two-part statistic for analysis of sequence data. The improved performance in microbial ecology studies was independent of study type and sequence technology used.
DNA arrays permit rapid, large-scale screening for patterns of gene expression and simultaneously yield the expression levels of thousands of genes for samples. The number of samples is usually limited, and such datasets are very sparse in high-dimensional gene space. Furthermore, most of the genes collected may not necessarily be of interest and uncertainty about which genes are relevant makes it difficult to construct an informative gene space. Unsupervised empirical sample pattern discovery and informative genes identification of such sparse high-dimensional datasets present interesting but challenging problems.
A new model called empirical sample pattern detection (ESPD) is proposed to delineate pattern quality with informative genes. By integrating statistical metrics, data mining and machine learning techniques, this model dynamically measures and manipulates the relationship between samples and genes while conducting an iterative detection of informative space and the empirical pattern. The performance of the proposed method with various array datasets is illustrated.
Identifying quantitative trait loci (QTL) for both additive and epistatic effects raises the statistical issue of selecting variables from a large number of candidates using a small number of observations. Missing trait and/or marker values prevent one from directly applying the classical model selection criteria such as Akaike's information criterion (AIC) and Bayesian information criterion (BIC).
We propose a two-step Bayesian variable selection method which deals with the sparse parameter space and the small sample size issues. The regression coefficient priors are flexible enough to incorporate the characteristic of "large p small n" data. Specifically, sparseness and possible asymmetry of the significant coefficients are dealt with by developing a Gibbs sampling algorithm to stochastically search through low-dimensional subspaces for significant variables. The superior performance of the approach is demonstrated via simulation study. We also applied it to real QTL mapping datasets.
The two-step procedure coupled with Bayesian classification offers flexibility in modeling "large p small n" data, especially for the sparse and asymmetric parameter space. This approach can be extended to other settings characterized by high dimension and low sample size.
High dimensional statistical problems arise from diverse fields of scientific research and technological development. Variable selection plays a pivotal role in contemporary statistical learning and scientific discoveries. The traditional idea of best subset selection methods, which can be regarded as a specific form of penalized likelihood, is computationally too expensive for many modern statistical applications. Other forms of penalized likelihood methods have been successfully developed over the last decade to cope with high dimensionality. They have been widely applied for simultaneously selecting important variables and estimating their effects in high dimensional statistical inference. In this article, we present a brief account of the recent developments of theory, methods, and implementations for high dimensional variable selection. What limits of the dimensionality such methods can handle, what the role of penalty functions is, and what the statistical properties are rapidly drive the advances of the field. The properties of non-concave penalized likelihood and its roles in high dimensional statistical modeling are emphasized. We also review some recent advances in ultra-high dimensional variable selection, with emphasis on independence screening and two-scale methods.
Variable selection; model selection; high dimensionality; penalized least squares; penalized likelihood; folded-concave penalty; oracle property; dimensionality reduction; LASSO; SCAD; sure screening; sure independence screening; Primary 62J99; secondary 62F12; 68Q32
The incorporation of biological knowledge can enhance the analysis of biomedical data. We present a novel method that uses a proteomic knowledge base to enhance the performance of a rule-learning algorithm in identifying putative biomarkers of disease from high-dimensional proteomic mass spectral data. In particular, we use the Empirical Proteomics Ontology Knowledge Base (EPO-KB) that contains previously identified and validated proteomic biomarkers to select m/zs in a proteomic dataset prior to analysis to increase performance.
We show that using EPO-KB as a pre-processing method, specifically selecting all biomarkers found only in the biofluid of the proteomic dataset, reduces the dimensionality by 95% and provides a statistically significantly greater increase in performance over no variable selection and random variable selection.
Knowledge-based variable selection even with a sparsely-populated resource such as the EPO-KB increases overall performance of rule-learning for disease classification from high-dimensional proteomic mass spectra.
This paper proposed a novel algorithm to sparsely represent a deformable surface (SRDS) with low dimensionality based on spherical harmonic decomposition (SHD) and orthogonal subspace pursuit (OSP). The key idea in SRDS method is to identify the subspaces from a training data set in the transformed spherical harmonic domain and then cluster each deformation into the best-fit subspace for fast and accurate representation. This algorithm is also generalized into applications of organs with both interior and exterior surfaces. To test the feasibility, we first use the computer models to demonstrate that the proposed approach matches the accuracy of complex mathematical modeling techniques and then both ex vivo and in vivo experiments are conducted using 3D magnetic resonance imaging (MRI) scans for verification in practical settings. All results demonstrated that the proposed algorithm features sparse representation of deformable surfaces with low dimensionality and high accuracy. Specifically, the precision evaluated as maximum error distance between the reconstructed surface and the MRI ground truth is better than 3 mm in real MRI experiments.
Clustering the information content of large high-dimensional gene expression datasets has widespread application in "omics" biology. Unfortunately, the underlying structure of these natural datasets is often fuzzy, and the computational identification of data clusters generally requires knowledge about cluster number and geometry.
We integrated strategies from machine learning, cartography, and graph theory into a new informatics method for automatically clustering self-organizing map ensembles of high-dimensional data. Our new method, called AutoSOME, readily identifies discrete and fuzzy data clusters without prior knowledge of cluster number or structure in diverse datasets including whole genome microarray data. Visualization of AutoSOME output using network diagrams and differential heat maps reveals unexpected variation among well-characterized cancer cell lines. Co-expression analysis of data from human embryonic and induced pluripotent stem cells using AutoSOME identifies >3400 up-regulated genes associated with pluripotency, and indicates that a recently identified protein-protein interaction network characterizing pluripotency was underestimated by a factor of four.
By effectively extracting important information from high-dimensional microarray data without prior knowledge or the need for data filtration, AutoSOME can yield systems-level insights from whole genome microarray expression studies. Due to its generality, this new method should also have practical utility for a variety of data-intensive applications, including the results of deep sequencing experiments. AutoSOME is available for download at http://jimcooperlab.mcdb.ucsb.edu/autosome.
Motivation: Combinatorial effects, in which several variables jointly influence an output or response, play an important role in biological systems. In many settings, Boolean functions provide a natural way to describe such influences. However, biochemical data using which we may wish to characterize such influences are usually subject to much variability. Furthermore, in high-throughput biological settings Boolean relationships of interest are very often sparse, in the sense of being embedded in an overall dataset of higher dimensionality. This motivates a need for statistical methods capable of making inferences regarding Boolean functions under conditions of noise and sparsity.
Results: We put forward a statistical model for sparse, noisy Boolean functions and methods for inference under the model. We focus on the case in which the form of the underlying Boolean function, as well as the number and identity of its inputs are all unknown. We present results on synthetic data and on a study of signalling proteins in cancer biology.
Statistical machine learning methods are increasingly used for neuroimaging data analysis. Their main virtue is their ability to model high-dimensional datasets, e.g., multivariate analysis of activation images or resting-state time series. Supervised learning is typically used in decoding or encoding settings to relate brain images to behavioral or clinical observations, while unsupervised learning can uncover hidden structures in sets of images (e.g., resting state functional MRI) or find sub-populations in large cohorts. By considering different functional neuroimaging applications, we illustrate how scikit-learn, a Python machine learning library, can be used to perform some key analysis steps. Scikit-learn contains a very large set of statistical learning algorithms, both supervised and unsupervised, and its application to neuroimaging data provides a versatile tool to study the brain.
machine learning; statistical learning; neuroimaging; scikit-learn; Python
This article describes advances in statistical computation for large-scale data analysis in structured Bayesian mixture models via graphics processing unit (GPU) programming. The developments are partly motivated by computational challenges arising in fitting models of increasing heterogeneity to increasingly large datasets. An example context concerns common biological studies using high-throughput technologies generating many, very large datasets and requiring increasingly high-dimensional mixture models with large numbers of mixture components. We outline important strategies and processes for GPU computation in Bayesian simulation and optimization approaches, give examples of the benefits of GPU implementations in terms of processing speed and scale-up in ability to analyze large datasets, and provide a detailed, tutorial-style exposition that will benefit readers interested in developing GPU-based approaches in other statistical models. Novel, GPU-oriented approaches to modifying existing algorithms software design can lead to vast speed-up and, critically, enable statistical analyses that presently will not be performed due to compute time limitations in traditional computational environments. Supplemental materials are provided with all source code, example data, and details that will enable readers to implement and explore the GPU approach in this mixture modeling context.
Bayesian computation; Desktop parallel computing; Flow cytometry; Graphics processing unit programming; Large datasets; Mixture models
Genomic microarrays are powerful research tools in bioinformatics and modern medicinal research because they enable massively-parallel assays and simultaneous monitoring of thousands of gene expression of biological samples. However, a simple microarray experiment often leads to very high-dimensional data and a huge amount of information, the vast amount of data challenges researchers into extracting the important features and reducing the high dimensionality. In this paper, a nonlinear dimensionality reduction kernel method based locally linear embedding(LLE) is proposed, and fuzzy K-nearest neighbors algorithm which denoises datasets will be introduced as a replacement to the classical LLE's KNN algorithm. In addition, kernel method based support vector machine (SVM) will be used to classify genomic microarray data sets in this paper. We demonstrate the application of the techniques to two published DNA microarray data sets. The experimental results confirm the superiority and high success rates of the presented method.
Manifold learning; Dimensionality reduction; Locally linear embedding; Kernel methods; Support vector machine
We propose a novel method called Partitioning based Adaptive Irrelevant Feature Eliminator (PAIFE) for dimensionality reduction in high-dimensional biomedical datasets. PAIFE evaluates feature-target relationships over not only a whole dataset, but also the partitioned subsets and is extremely effective in identifying features whose relevancies to the target are conditional on certain other features. PAIFE adaptively employs the most appropriate feature evaluation strategy, statistical test and parameter instantiation. We envision PAIFE to be used as a third-party data pre-processing tool for dimensionality reduction of high-dimensional clinical datasets. Experiments on synthetic datasets showed that PAIFE consistently outperformed state-of-the-art feature selection methods in removing irrelevant features while retaining relevant features. Experiments on genomic and proteomic datasets demonstrated that PAIFE was able to remove significant numbers of irrelevant features in real-world biomedical datasets. Classification models constructed from the retained features either matched or improved the classification performances of the models constructed using all features.
Motivated by DNA copy number variation (CNV) analysis based on high-density single nucleotide polymorphism (SNP) data, we consider the problem of detecting and identifying sparse short segments in a long one-dimensional sequence of data with additive Gaussian white noise, where the number, length and location of the segments are unknown. We present a statistical characterization of the identifiable region of a segment where it is possible to reliably separate the segment from noise. An efficient likelihood ratio selection (LRS) procedure for identifying the segments is developed, and the asymptotic optimality of this method is presented in the sense that the LRS can separate the signal segments from the noise as long as the signal segments are in the identifiable regions. The proposed method is demonstrated with simulations and analysis of a real data set on identification of copy number variants based on high-density SNP data. The results show that the LRS procedure can yield greater gain in power for detecting the true segments than some standard signal identification methods.
Likelihood ratio selection; signal detection; multiple testing; DNA copy number
The antibody microarray is a powerful chip-based technology for profiling hundreds of proteins simultaneously and is used increasingly nowadays. To study humoral response in pancreatic cancers, Patwa et al. (2007) developed a two-dimensional liquid separation technique and built a two-dimensional antibody microarray. However, identifying differential expression regions on the antibody microarray requires the use of appropriate statistical methods to fairly assess the large amounts of data generated. In this paper, we propose a permutation-based test using spatial information of the two-dimensional antibody microarray. By borrowing strength from the neighboring differentially expressed spots, we are able to detect the differential expression region with very high power controlling type I error at 0.05 in our simulation studies. We also apply the proposed methodology to a real microarray dataset.
Antibody Microarray; Permutation; Spatial information
Mass spectrometry-based protein identification methods are fundamental to proteomics. Biological experiments are usually performed in replicates and proteomic analyses generate huge datasets which need to be integrated and quantitatively analyzed. The Sequest™ search algorithm is a commonly used algorithm for identifying peptides and proteins from two dimensional liquid chromatography electrospray ionization tandem mass spectrometry (2-D LC ESI MS2) data. A number of proteomic pipelines that facilitate high throughput 'post data acquisition analysis' are described in the literature. However, these pipelines need to be updated to accommodate the rapidly evolving data analysis methods. Here, we describe a proteomic data analysis pipeline that specifically addresses two main issues pertinent to protein identification and differential expression analysis: 1) estimation of the probability of peptide and protein identifications and 2) non-parametric statistics for protein differential expression analysis. Our proteomic analysis workflow analyzes replicate datasets from a single experimental paradigm to generate a list of identified proteins with their probabilities and significant changes in protein expression using parametric and non-parametric statistics.
The input for our workflow is Bioworks™ 3.2 Sequest (or a later version, including cluster) output in XML format. We use a decoy database approach to assign probability to peptide identifications. The user has the option to select "quality thresholds" on peptide identifications based on the P value. We also estimate probability for protein identification. Proteins identified with peptides at a user-specified threshold value from biological experiments are grouped as either control or treatment for further analysis in ProtQuant. ProtQuant utilizes a parametric (ANOVA) method, for calculating differences in protein expression based on the quantitative measure ΣXcorr. Alternatively ProtQuant output can be further processed using non-parametric Monte-Carlo resampling statistics to calculate P values for differential expression. Correction for multiple testing of ANOVA and resampling P values is done using Benjamini and Hochberg's method. The results of these statistical analyses are then combined into a single output file containing a comprehensive protein list with probabilities and differential expression analysis, associated P values, and resampling statistics.
For biologists carrying out proteomics by mass spectrometry, our workflow facilitates automated, easy to use analyses of Bioworks (3.2 or later versions) data. All the methods used in the workflow are peer-reviewed and as such the results of our workflow are compliant with proteomic data submission guidelines to public proteomic data repositories including PRIDE. Our workflow is a necessary intermediate step that is required to link proteomics data to biological knowledge for generating testable hypotheses.
Acquiring and representing biomedical knowledge is an increasingly important component of contemporary bioinformatics. A critical step of the process is to identify and retrieve relevant documents among the vast volume of modern biomedical literature efficiently. In the real world, many information retrieval tasks are difficult because of high data dimensionality and the lack of annotated examples to train a retrieval algorithm. Under such a scenario, the performance of information retrieval algorithms is often unsatisfactory, therefore improvements are needed.
We studied two approaches that enhance the text categorization performance on sparse and high data dimensionality: (1) semantic-preserving dimension reduction by representing text with semantic-enriched features; and (2) augmenting training data with semi-supervised learning. A probabilistic topic model was applied to extract major semantic topics from a corpus of text of interest. The representation of documents was projected from the high-dimensional vocabulary space onto a semantic topic space with reduced dimensionality. A semi-supervised learning algorithm based on graph theory was applied to identify potential positive training cases, which were further used to augment training data. The effects of data transformation and augmentation on text categorization by support vector machine (SVM) were evaluated.
Results and Conclusion
Semantic-enriched data transformation and the pseudo-positive-cases augmented training data enhance the efficiency and performance of text categorization by SVM.
The ℓ1-penalized method, or the Lasso, has emerged as an important tool for the analysis of large data sets. Many important results have been obtained for the Lasso in linear regression which have led to a deeper understanding of high-dimensional statistical problems. In this article, we consider a class of weighted ℓ1-penalized estimators for convex loss functions of a general form, including the generalized linear models. We study the estimation, prediction, selection and sparsity properties of the weighted ℓ1-penalized estimator in sparse, high-dimensional settings where the number of predictors p can be much larger than the sample size n. Adaptive Lasso is considered as a special case. A multistage method is developed to approximate concave regularized estimation by applying an adaptive Lasso recursively. We provide prediction and estimation oracle inequalities for single- and multi-stage estimators, a general selection consistency theorem, and an upper bound for the dimension of the Lasso estimator. Important models including the linear regression, logistic regression and log-linear models are used throughout to illustrate the applications of the general results.
variable selection; penalized estimation; oracle inequality; generalized linear models; selection consistency; sparsity
Motivation: Eukaryotic gene expression (GE) is subjected to precisely coordinated multi-layer controls, across the levels of epigenetic, transcriptional and post-transcriptional regulations. Recently, the emerging multi-dimensional genomic dataset has provided unprecedented opportunities to study the cross-layer regulatory interplay. In these datasets, the same set of samples is profiled on several layers of genomic activities, e.g. copy number variation (CNV), DNA methylation (DM), GE and microRNA expression (ME). However, suitable analysis methods for such data are currently sparse.
Results: In this article, we introduced a sparse Multi-Block Partial Least Squares (sMBPLS) regression method to identify multi-dimensional regulatory modules from this new type of data. A multi-dimensional regulatory module contains sets of regulatory factors from different layers that are likely to jointly contribute to a local ‘gene expression factory’. We demonstrated the performance of our method on the simulated data as well as on The Cancer Genomic Atlas Ovarian Cancer datasets including the CNV, DM, ME and GE data measured on 230 samples. We showed that majority of identified modules have significant functional and transcriptional enrichment, higher than that observed in modules identified using only a single type of genomic data. Our network analysis of the modules revealed that the CNV, DM and microRNA can have coupled impact on expression of important oncogenes and tumor suppressor genes.
Availability and implementation: The source code implemented by MATLAB is freely available at: http://zhoulab.usc.edu/sMBPLS/.
Supplementary material are available at Bioinformatics online.
Microarray data has a high dimension of variables but available datasets usually have only a small number of samples, thereby making the study of such datasets interesting and challenging. In the task of analyzing microarray data for the purpose of, e.g., predicting gene-disease association, feature selection is very important because it provides a way to handle the high dimensionality by exploiting information redundancy induced by associations among genetic markers. Judicious feature selection in microarray data analysis can result in significant reduction of cost while maintaining or improving the classification or prediction accuracy of learning machines that are employed to sort out the datasets. In this paper, we propose a gene selection method called Recursive Feature Addition (RFA), which combines supervised learning and statistical similarity measures. We compare our method with the following gene selection methods:
Support Vector Machine Recursive Feature Elimination (SVMRFE)Leave-One-Out Calculation Sequential Forward Selection (LOOCSFS)Gradient based Leave-one-out Gene Selection (GLGS)
To evaluate the performance of these gene selection methods, we employ several popular learning classifiers on the MicroArray Quality Control phase II on predictive modeling (MAQC-II) breast cancer dataset and the MAQC-II multiple myeloma dataset. Experimental results show that gene selection is strictly paired with learning classifier. Overall, our approach outperforms other compared methods. The biological functional analysis based on the MAQC-II breast cancer dataset convinced us to apply our method for phenotype prediction. Additionally, learning classifiers also play important roles in the classification of microarray data and our experimental results indicate that the Nearest Mean Scale Classifier (NMSC) is a good choice due to its prediction reliability and its stability across the three performance measurements: Testing accuracy, MCC values, and AUC errors.
This paper reviews the literature on sparse high dimensional models and discusses some applications in economics and finance. Recent developments of theory, methods, and implementations in penalized least squares and penalized likelihood methods are highlighted. These variable selection methods are proved to be effective in high dimensional sparse modeling. The limits of dimensionality that regularization methods can handle, the role of penalty functions, and their statistical properties are detailed. Some recent advances in ultra-high dimensional sparse modeling are also briefly discussed.
Variable selection; independence screening; sparsity; oracle properties; penalized least squares; penalized likelihood; spurious correlation; sparse VAR; factor models; volatility estimation; portfolio selection
Motivation: The study of complex biological relationships is aided by large and high-dimensional data sets whose analysis often involves dimension reduction to highlight representative or informative directions of variation. In principle, information theory provides a general framework for quantifying complex statistical relationships for dimension reduction. Unfortunately, direct estimation of high-dimensional information theoretic quantities, such as entropy and mutual information (MI), is often unreliable given the relatively small sample sizes available for biological problems. Here, we develop and evaluate a hierarchy of approximations for high-dimensional information theoretic statistics from associated low-order terms, which can be more reliably estimated from limited samples. Due to a relationship between this metric and the minimum spanning tree over a graph representation of the system, we refer to these approximations as MIST (Maximum Information Spanning Trees).
Results: The MIST approximations are examined in the context of synthetic networks with analytically computable entropies and using experimental gene expression data as a basis for the classification of multiple cancer types. The approximations result in significantly more accurate estimates of entropy and MI, and also correlate better with biological classification error than direct estimation and another low-order approximation, minimum-redundancy–maximum-relevance (mRMR).
Availability: Software to compute the entropy approximations described here is available as Supplementary Material.
Supplementary information: Supplementary data are available at Bioinformatics online.
Large point referenced datasets occur frequently in the environmental and natural sciences. Use of Bayesian hierarchical spatial models for analyzing these datasets is undermined by onerous computational burdens associated with parameter estimation. Low-rank spatial process models attempt to resolve this problem by projecting spatial effects to a lower-dimensional subspace. This subspace is determined by a judicious choice of “knots” or locations that are fixed a priori. One such representation yields a class of predictive process models (e.g., Banerjee et al., 2008) for spatial and spatial-temporal data. Our contribution here expands upon predictive process models with fixed knots to models that accommodate stochastic modeling of the knots. We view the knots as emerging from a point pattern and investigate how such adaptive specifications can yield more flexible hierarchical frameworks that lead to automated knot selection and substantial computational benefits.
Bayesian hierarchical models; Gaussian process; Intensity surfaces; Low-rank models; Markov chain Monte Carlo; Predictive process
High-dimensional and highly correlated data leading to non- or weakly identified effects are commonplace. Maximum likelihood will typically fail in such situations and a variety of shrinkage methods have been proposed. Standard techniques, such as ridge regression or the lasso, shrink estimates toward zero, with some approaches allowing coefficients to be selected out of the model by achieving a value of zero. When substantive information is available, estimates can be shrunk to nonnull values; however, such information may not be available. We propose a Bayesian semiparametric approach that allows shrinkage to multiple locations. Coefficients are given a mixture of heavy-tailed double exponential priors, with location and scale parameters assigned Dirichlet process hyperpriors to allow groups of coefficients to be shrunk toward the same, possibly nonzero, mean. Our approach favors sparse, but flexible, structure by shrinking toward a small number of random locations. The methods are illustrated using a study of genetic polymorphisms and Parkinson’s disease.
Dirichlet process; Hierarchical model; Lasso; MCMC; Mixture model; Nonparametric; Regularization; Shrinkage prior