Sampling from probability distributions is a critical problem in statistics and machine learning, particularly in Bayesian inference, where direct integration over the posterior distribution is often infeasible, making sampling from the posterior essential for inference. This paper introduces the Schrödinger-Föllmer sampler (SFS), a novel approach for sampling from potentially unnormalized distributions. The SFS leverages the Schrödinger-Föllmer diffusion process on the unit interval, incorporating a time-dependent drift term that evolves the distribution from a degenerate form at time zero to the target distribution at time one. Unlike existing Markov chain Monte Carlo methods that rely on ergodicity, SFS operates independently of ergodicity. Computationally, SFS is straightforward to implement using the Euler-Maruyama discretization. In our theoretical analysis, we derive non-asymptotic error bounds for the SFS sampling distribution in the Wasserstein distance, subject to reasonable conditions. Numerical experiments demonstrate that SFS generates higher-quality samples than several established methods.

Deep nonparametric regression, characterized by the utilization of deep neural networks to learn target functions, has emerged as a focus of research attention in recent years. Despite considerable progress in understanding convergence rates, the absence of asymptotic properties hinders rigorous statistical inference. To address this gap, we propose a novel framework that transforms the deep estimation paradigm into a platform conducive to conditional mean estimation, leveraging the conditional diffusion model. Theoretically, we develop an end-to-end convergence rate for the conditional diffusion model and establish the asymptotic normality of the generated samples. Consequently, we are equipped to construct confidence regions, facilitating robust statistical inference. Furthermore, through numerical experiments, we empirically validate the efficacy of our proposed methodology.

The goal of supervised representation learning is to construct effective data representations for prediction. Among all the characteristics of an ideal nonparametric representation of high-dimensional complex data, sufficiency, low dimensionality and disentanglement are some of the most essential ones. We propose a deep dimension reduction approach to learning representations with these characteristics. The proposed approach is a nonparametric generalization of the sufficient dimension reduction method. We formulate the ideal representation learning task as that of finding a nonparametric representation that minimizes an objective function characterizing conditional independence and promoting disentanglement at the population level. We then estimate the target representation at the sample level nonparametrically using deep neural networks. We show that the estimated deep nonparametric representation is consistent in the sense that its excess risk converges to zero. Our extensive numerical experiments using simulated and real benchmark data demonstrate that the proposed methods have better performance than several existing dimension reduction methods and the standard deep learning models in the context of classification and regression.

Improving the generalization of neural networks under domain shift is an important and challenging task in computer vision. Obtaining an invariant representation across domains is a benchmark method in the literature. In this paper, we propose an invariant and sufficient supervised representation learning (ISSRL) approach to learn a domain invariant representation which is also preserving information used for downstream tasks. To this end, we formulate ISSRL by finding a nonlinear map$\boldsymbol{g}$such that$Y\perp X\vert \boldsymbol{g}(X)$and$(Y,\boldsymbol{g}(X))\perp D$at the population level, where D is the label of the domains and$(X, Y)$is the paired data sampled from domains with label. We use distance correlation to characterize the (conditional) independence. At the sample level, we construct a novel loss function through an unbiased empirical version of distance correlation. We train the representation map by parameterizing it with deep neural networks. Both simulation study and real data evaluation show that ISSRL outperforms the state-of-the-art on out-of-distribution generalization. The PyTorch code for ISSRL is available at https://github.com/CaC033/ISSRL.

MOTIVATION: The findings from genome-wide association studies (GWASs) have greatly helped us to understand the genetic basis of human complex traits and diseases. Despite the tremendous progress, much effects are still needed to address several major challenges arising in GWAS. First, most GWAS hits are located in the non-coding region of human genome, and thus their biological functions largely remain unknown. Second, due to the polygenicity of human complex traits and diseases, many genetic risk variants with weak or moderate effects have not been identified yet. RESULTS: To address the above challenges, we propose a powerful and adaptive latent model (PALM) to integrate cell-type/tissue-specific functional annotations with GWAS summary statistics. Unlike existing methods, which are mainly based on linear models, PALM leverages a tree ensemble to adaptively characterize non-linear relationship between functional annotations and the association status of genetic variants. To make PALM scalable to millions of variants and hundreds of functional annotations, we develop a functional gradient-based expectation-maximization algorithm, to fit the tree-based non-linear model in a stable manner. Through comprehensive simulation studies, we show that PALM not only controls false discovery rate well, but also improves statistical power of identifying risk variants. We also apply PALM to integrate summary statistics of 30 GWASs with 127 cell type/tissue-specific functional annotations. The results indicate that PALM can identify more risk variants as well as rank the importance of functional annotations, yielding better interpretation of GWAS results. AVAILABILITY AND IMPLEMENTATION: The source code is available at https://github.com/YangLabHKUST/PALM. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Spatial transcriptomics has been emerging as a powerful technique for resolving gene expression profiles while retaining tissue spatial information. These spatially resolved transcriptomics make it feasible to examine the complex multicellular systems of different microenvironments. To answer scientific questions with spatial transcriptomics and expand our understanding of how cell types and states are regulated by microenvironment, the first step is to identify cell clusters by integrating the available spatial information. Here, we introduce SC-MEB, an empirical Bayes approach for spatial clustering analysis using a hidden Markov random field. We have also derived an efficient expectation-maximization algorithm based on an iterative conditional mode for SC-MEB. In contrast to BayesSpace, a recently developed method, SC-MEB is not only computationally efficient and scalable to large sample sizes but is also capable of choosing the smoothness parameter and the number of clusters. We performed comprehensive simulation studies to demonstrate the superiority of SC-MEB over some existing methods. We applied SC-MEB to analyze the spatial transcriptome of human dorsolateral prefrontal cortex tissues and mouse hypothalamic preoptic region. Our analysis results showed that SC-MEB can achieve a similar or better clustering performance to BayesSpace, which uses the true number of clusters and a fixed smoothness parameter. Moreover, SC-MEB is scalable to large 'sample sizes'. We then employed SC-MEB to analyze a colon dataset from a patient with colorectal cancer (CRC) and COVID-19, and further performed differential expression analysis to identify signature genes related to the clustering results. The heatmap of identified signature genes showed that the clusters identified using SC-MEB were more separable than those obtained with BayesSpace. Using pathway analysis, we identified three immune-related clusters, and in a further comparison, found the mean expression of COVID-19 signature genes was greater in immune than non-immune regions of colon tissue. SC-MEB provides a valuable computational tool for investigating the structural organizations of tissues from spatial transcriptomic data.

MOTIVATION: Mendelian randomization (MR) is a valuable tool to examine the causal relationships between health risk factors and outcomes from observational studies. Along with the proliferation of genome-wide association studies, a variety of two-sample MR methods for summary data have been developed to account for horizontal pleiotropy (HP), primarily based on the assumption that the effects of variants on exposure (γ) and HP (α) are independent. In practice, this assumption is too strict and can be easily violated because of the correlated HP. RESULTS: To account for this correlated HP, we propose a Bayesian approach, MR-Corr2, that uses the orthogonal projection to reparameterize the bivariate normal distribution for γ and α, and a spike-slab prior to mitigate the impact of correlated HP. We have also developed an efficient algorithm with paralleled Gibbs sampling. To demonstrate the advantages of MR-Corr2 over existing methods, we conducted comprehensive simulation studies to compare for both type-I error control and point estimates in various scenarios. By applying MR-Corr2 to study the relationships between exposure-outcome pairs in complex traits, we did not identify the contradictory causal relationship between HDL-c and CAD. Moreover, the results provide a new perspective of the causal network among complex traits. AVAILABILITY AND IMPLEMENTATION: The developed R package and code to reproduce all the results are available at https://github.com/QingCheng0218/MR.Corr2. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: Although genome-wide association studies (GWAS) have deepened our understanding of the genetic architecture of complex traits, the mechanistic links that underlie how genetic variants cause complex traits remains elusive. To advance our understanding of the underlying mechanistic links, various consortia have collected a vast volume of genomic data that enable us to investigate the role that genetic variants play in gene expression regulation. Recently, a collaborative mixed model (CoMM) was proposed to jointly interrogate genome on complex traits by integrating both the GWAS dataset and the expression quantitative trait loci (eQTL) dataset. Although CoMM is a powerful approach that leverages regulatory information while accounting for the uncertainty in using an eQTL dataset, it requires individual-level GWAS data and cannot fully make use of widely available GWAS summary statistics. Therefore, statistically efficient methods that leverages transcriptome information using only summary statistics information from GWAS data are required. RESULTS: In this study, we propose a novel probabilistic model, CoMM-S2, to examine the mechanistic role that genetic variants play, by using only GWAS summary statistics instead of individual-level GWAS data. Similar to CoMM which uses individual-level GWAS data, CoMM-S2 combines two models: the first model examines the relationship between gene expression and genotype, while the second model examines the relationship between the phenotype and the predicted gene expression from the first model. Distinct from CoMM, CoMM-S2 requires only GWAS summary statistics. Using both simulation studies and real data analysis, we demonstrate that even though CoMM-S2 utilizes GWAS summary statistics, it has comparable performance as CoMM, which uses individual-level GWAS data. AVAILABILITY AND IMPLEMENTATION: The implement of CoMM-S2 is included in the CoMM package that can be downloaded from https://github.com/gordonliu810822/CoMM. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: The results from Genome-Wide Association Studies (GWAS) on thousands of phenotypes provide an unprecedented opportunity to infer the causal effect of one phenotype (exposure) on another (outcome). Mendelian randomization (MR), an instrumental variable (IV) method, has been introduced for causal inference using GWAS data. Due to the polygenic architecture of complex traits/diseases and the ubiquity of pleiotropy, however, MR has many unique challenges compared to conventional IV methods. RESULTS: We propose a Bayesian weighted Mendelian randomization (BWMR) for causal inference to address these challenges. In our BWMR model, the uncertainty of weak effects owing to polygenicity has been taken into account and the violation of IV assumption due to pleiotropy has been addressed through outlier detection by Bayesian weighting. To make the causal inference based on BWMR computationally stable and efficient, we developed a variational expectation-maximization (VEM) algorithm. Moreover, we have also derived an exact closed-form formula to correct the posterior covariance which is often underestimated in variational inference. Through comprehensive simulation studies, we evaluated the performance of BWMR, demonstrating the advantage of BWMR over its competitors. Then we applied BWMR to make causal inference between 130 metabolites and 93 complex human traits, uncovering novel causal relationship between exposure and outcome traits. AVAILABILITY AND IMPLEMENTATION: The BWMR software is available at https://github.com/jiazhao97/BWMR. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: A large number of recent genome-wide association studies (GWASs) for complex phenotypes confirm the early conjecture for polygenicity, suggesting the presence of large number of variants with only tiny or moderate effects. However, due to the limited sample size of a single GWAS, many associated genetic variants are too weak to achieve the genome-wide significance. These undiscovered variants further limit the prediction capability of GWAS. Restricted access to the individual-level data and the increasing availability of the published GWAS results motivate the development of methods integrating both the individual-level and summary-level data. How to build the connection between the individual-level and summary-level data determines the efficiency of using the existing abundant summary-level resources with limited individual-level data, and this issue inspires more efforts in the existing area. RESULTS: In this study, we propose a novel statistical approach, LEP, which provides a novel way of modeling the connection between the individual-level data and summary-level data. LEP integrates both types of data by LEveraging Pleiotropy to increase the statistical power of risk variants identification and the accuracy of risk prediction. The algorithm for parameter estimation is developed to handle genome-wide-scale data. Through comprehensive simulation studies, we demonstrated the advantages of LEP over the existing methods. We further applied LEP to perform integrative analysis of Crohn's disease from WTCCC and summary statistics from GWAS of some other diseases, such as Type 1 diabetes, Ulcerative colitis and Primary biliary cirrhosis. LEP was able to significantly increase the statistical power of identifying risk variants and improve the risk prediction accuracy from 63.39% (±0.58%) to 68.33% (±0.32%) using about 195 000 variants. AVAILABILITY AND IMPLEMENTATION: The LEP software is available at https://github.com/daviddaigithub/LEP. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: In genome-wide association studies (GWASs) where multiple correlated traits have been measured on participants, a joint analysis strategy, whereby the traits are analyzed jointly, can improve statistical power over a single-trait analysis strategy. There are two questions of interest to be addressed when conducting a joint GWAS analysis with multiple traits. The first question examines whether a genetic loci is significantly associated with any of the traits being tested. The second question focuses on identifying the specific trait(s) that is associated with the genetic loci. Since existing methods primarily focus on the first question, this article seeks to provide a complementary method that addresses the second question. RESULTS: We propose a novel method, Variational Inference for Multiple Correlated Outcomes (VIMCO) that focuses on identifying the specific trait that is associated with the genetic loci, when performing a joint GWAS analysis of multiple traits, while accounting for correlation among the multiple traits. We performed extensive numerical studies and also applied VIMCO to analyze two datasets. The numerical studies and real data analysis demonstrate that VIMCO improves statistical power over single-trait analysis strategies when the multiple traits are correlated and has comparable performance when the traits are not correlated. AVAILABILITY AND IMPLEMENTATION: The VIMCO software can be downloaded from: https://github.com/XingjieShi/VIMCO. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: Genome-wide association studies (GWASs) have been successful in identifying many genetic variants associated with complex traits. However, the mechanistic links between these variants and complex traits remain elusive. A scientific hypothesis is that genetic variants influence complex traits at the organismal level via affecting cellular traits, such as regulating gene expression and altering protein abundance. Although earlier works have already presented some scientific insights about this hypothesis and their findings are very promising, statistical methods that effectively harness multilayered data (e.g. genetic variants, cellular traits and organismal traits) on a large scale for functional and mechanistic exploration are highly demanding. RESULTS: In this study, we propose a collaborative mixed model (CoMM) to investigate the mechanistic role of associated variants in complex traits. The key idea is built upon the emerging scientific evidence that genetic effects at the cellular level are much stronger than those at the organismal level. Briefly, CoMM combines two models: the first model relating gene expression with genotype and the second model relating phenotype with predicted gene expression using the first model. The two models are fitted jointly in CoMM, such that the uncertainty in predicting gene expression has been fully accounted. To demonstrate the advantages of CoMM over existing methods, we conducted extensive simulation studies, and also applied CoMM to analyze 25 traits in NFBC1966 and Genetic Epidemiology Research on Aging (GERA) studies by integrating transcriptome information from the Genetic European in Health and Disease (GEUVADIS) Project. The results indicate that by leveraging regulatory information, CoMM can effectively improve the power of prioritizing risk variants. Regarding the computational efficiency, CoMM can complete the analysis of NFBC1966 dataset and GERA datasets in 2 and 18 min, respectively. AVAILABILITY AND IMPLEMENTATION: The developed R package is available at https://github.com/gordonliu810822/CoMM. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Motivation: Thousands of risk variants underlying complex phenotypes (quantitative traits and diseases) have been identified in genome-wide association studies (GWAS). However, there are still two major challenges towards deepening our understanding of the genetic architectures of complex phenotypes. First, the majority of GWAS hits are in non-coding region and their biological interpretation is still unclear. Second, accumulating evidence from GWAS suggests the polygenicity of complex traits, i.e. a complex trait is often affected by many variants with small or moderate effects, whereas a large proportion of risk variants with small effects remain unknown. Results: The availability of functional annotation data enables us to address the above challenges. In this study, we propose a latent sparse mixed model (LSMM) to integrate functional annotations with GWAS data. Not only does it increase the statistical power of identifying risk variants, but also offers more biological insights by detecting relevant functional annotations. To allow LSMM scalable to millions of variants and hundreds of functional annotations, we developed an efficient variational expectation-maximization algorithm for model parameter estimation and statistical inference. We first conducted comprehensive simulation studies to evaluate the performance of LSMM. Then we applied it to analyze 30 GWAS of complex phenotypes integrated with nine genic category annotations and 127 cell-type specific functional annotations from the Roadmap project. The results demonstrate that our method possesses more statistical power than conventional methods, and can help researchers achieve deeper understanding of genetic architecture of these complex phenotypes. Availability and implementation: The LSMM software is available at https://github.com/mingjingsi/LSMM. Supplementary information: Supplementary data are available at Bioinformatics online.

MOTIVATION: Results from genome-wide association studies (GWAS) suggest that a complex phenotype is often affected by many variants with small effects, known as 'polygenicity'. Tens of thousands of samples are often required to ensure statistical power of identifying these variants with small effects. However, it is often the case that a research group can only get approval for the access to individual-level genotype data with a limited sample size (e.g. a few hundreds or thousands). Meanwhile, summary statistics generated using single-variant-based analysis are becoming publicly available. The sample sizes associated with the summary statistics datasets are usually quite large. How to make the most efficient use of existing abundant data resources largely remains an open question. RESULTS: In this study, we propose a statistical approach, IGESS, to increasing statistical power of identifying risk variants and improving accuracy of risk prediction by i ntegrating individual level ge notype data and s ummary s tatistics. An efficient algorithm based on variational inference is developed to handle the genome-wide analysis. Through comprehensive simulation studies, we demonstrated the advantages of IGESS over the methods which take either individual-level data or summary statistics data as input. We applied IGESS to perform integrative analysis of Crohns Disease from WTCCC and summary statistics from other studies. IGESS was able to significantly increase the statistical power of identifying risk variants and improve the risk prediction accuracy from 63.2% ( ±0.4% ) to 69.4% ( ±0.1% ) using about 240 000 variants. AVAILABILITY AND IMPLEMENTATION: The IGESS software is available at https://github.com/daviddaigithub/IGESS . CONTACT: [email protected] or [email protected] or [email protected]. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: Genome-wide association studies (GWAS), which genotype millions of single nucleotide polymorphisms (SNPs) in thousands of individuals, are widely used to identify the risk SNPs underlying complex human phenotypes (quantitative traits or diseases). Most conventional statistical methods in GWAS only investigate one phenotype at a time. However, an increasing number of reports suggest the ubiquity of pleiotropy, i.e. many complex phenotypes sharing common genetic bases. This motivated us to leverage pleiotropy to develop new statistical approaches to joint analysis of multiple GWAS. RESULTS: In this study, we propose a latent low-rank (LLR) approach to colocalizing genetic risk variants using summary statistics. In the presence of pleiotropy, there exist risk loci that affect multiple phenotypes. To leverage pleiotropy, we introduce a low-rank structure to modulate the probabilities of the latent association statuses between loci and phenotypes. Regarding the computational efficiency of LLR, a novel expectation-maximization-path (EM-path) algorithm has been developed to greatly reduce the computational cost and facilitate model selection and inference. We demonstrate the advantages of LLR over competing approaches through simulation studies and joint analysis of 18 GWAS datasets. AVAILABILITY AND IMPLEMENTATION: The LLR software is available on https://sites.google.com/site/liujin810822. CONTACT: [email protected]. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: Researchers worldwide have generated a huge volume of genomic data, including thousands of genome-wide association studies (GWAS) and massive amounts of gene expression data from different tissues. How to perform a joint analysis of these data to gain new biological insights has become a critical step in understanding the etiology of complex diseases. Due to the polygenic architecture of complex diseases, the identification of risk genes remains challenging. Motivated by the shared risk genes found in complex diseases and tissue-specific gene expression patterns, we propose as an Empirical Bayes approach to integrating Pleiotropy and Tissue-Specific information (EPS) for prioritizing risk genes. RESULTS: As demonstrated by extensive simulation studies, EPS greatly improves the power of identification for disease-risk genes. EPS enables rigorous hypothesis testing of pleiotropy and tissue-specific risk gene expression patterns. All of the model parameters can be adaptively estimated from the developed expectation-maximization (EM) algorithm. We applied EPS to the bipolar disorder and schizophrenia GWAS from the Psychiatric Genomics Consortium, along with the gene expression data for multiple tissues from the Genotype-Tissue Expression project. The results of the real data analysis demonstrate many advantages of EPS. AVAILABILITY AND IMPLEMENTATION: The EPS software is available on https://sites.google.com/site/liujin810822 CONTACT: [email protected] SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Gene expression profiling has been extensively conducted in cancer research. The analysis of multiple independent cancer gene expression datasets may provide additional information and complement single-dataset analysis. In this study, we conduct multi-dataset analysis and are interested in evaluating the similarity of cancer-associated genes identified from different datasets. The first objective of this study is to briefly review some statistical methods that can be used for such evaluation. Both marginal analysis and joint analysis methods are reviewed. The second objective is to apply those methods to 26 Gene Expression Omnibus (GEO) datasets on five types of cancers. Our analysis suggests that for the same cancer, the marker identification results may vary significantly across datasets, and different datasets share few common genes. In addition, datasets on different cancers share few common genes. The shared genetic basis of datasets on the same or different cancers, which has been suggested in the literature, is not observed in the analysis of GEO data.