Pub Date : 2024-06-01Epub Date: 2024-04-09DOI: 10.1214/23-BA1364
Qian Zhang, Faming Liang
The exponential random graph model (ERGM) is a popular model for social networks, which is known to have an intractable likelihood function. Sampling from the posterior for such a model is a long-standing problem in statistical research. We analyze the performance of the stochastic gradient Langevin dynamics (SGLD) algorithm (also known as noisy Longevin Monte Carlo) in tackling this problem, where the stochastic gradient is calculated via running a short Markov chain (the so-called inner Markov chain in this paper) at each iteration. We show that if the model size grows with the network size slowly enough, then SGLD converges to the true posterior in 2-Wasserstein distance as the network size and iteration number become large regardless of the length of the inner Markov chain performed at each iteration. Our study provides a scalable algorithm for analyzing large-scale social networks with possibly high-dimensional ERGMs.
{"title":"Bayesian Analysis of Exponential Random Graph Models Using Stochastic Gradient Markov Chain Monte Carlo.","authors":"Qian Zhang, Faming Liang","doi":"10.1214/23-BA1364","DOIUrl":"10.1214/23-BA1364","url":null,"abstract":"<p><p>The exponential random graph model (ERGM) is a popular model for social networks, which is known to have an intractable likelihood function. Sampling from the posterior for such a model is a long-standing problem in statistical research. We analyze the performance of the stochastic gradient Langevin dynamics (SGLD) algorithm (also known as noisy Longevin Monte Carlo) in tackling this problem, where the stochastic gradient is calculated via running a short Markov chain (the so-called inner Markov chain in this paper) at each iteration. We show that if the model size grows with the network size slowly enough, then SGLD converges to the true posterior in 2-Wasserstein distance as the network size and iteration number become large regardless of the length of the inner Markov chain performed at each iteration. Our study provides a scalable algorithm for analyzing large-scale social networks with possibly high-dimensional ERGMs.</p>","PeriodicalId":55398,"journal":{"name":"Bayesian Analysis","volume":" ","pages":"595-621"},"PeriodicalIF":4.9,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11756892/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46968695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bayesian model selection is premised on the assumption that the data are generated from one of the postulated models. However, in many applications, all of these models are incorrect (that is, there is misspecification). When the models are misspecified, two or more models can provide a nearly equally good fit to the data, in which case Bayesian model selection can be highly unstable, potentially leading to self-contradictory findings. To remedy this instability, we propose to use bagging on the posterior distribution ("BayesBag") - that is, to average the posterior model probabilities over many bootstrapped datasets. We provide theoretical results characterizing the asymptotic behavior of the posterior and the bagged posterior in the (misspecified) model selection setting. We empirically assess the BayesBag approach on synthetic and real-world data in (i) feature selection for linear regression and (ii) phylogenetic tree reconstruction. Our theory and experiments show that, when all models are misspecified, BayesBag (a) provides greater reproducibility and (b) places posterior mass on optimal models more reliably, compared to the usual Bayesian posterior; on the other hand, under correct specification, BayesBag is slightly more conservative than the usual posterior, in the sense that BayesBag posterior probabilities tend to be slightly farther from the extremes of zero and one. Overall, our results demonstrate that BayesBag provides an easy-to-use and widely applicable approach that improves upon Bayesian model selection by making it more stable and reproducible.
{"title":"Reproducible Model Selection Using Bagged Posteriors.","authors":"Jonathan H Huggins, Jeffrey W Miller","doi":"10.1214/21-ba1301","DOIUrl":"https://doi.org/10.1214/21-ba1301","url":null,"abstract":"<p><p>Bayesian model selection is premised on the assumption that the data are generated from one of the postulated models. However, in many applications, all of these models are incorrect (that is, there is misspecification). When the models are misspecified, two or more models can provide a nearly equally good fit to the data, in which case Bayesian model selection can be highly unstable, potentially leading to self-contradictory findings. To remedy this instability, we propose to use bagging on the posterior distribution (\"BayesBag\") - that is, to average the posterior model probabilities over many bootstrapped datasets. We provide theoretical results characterizing the asymptotic behavior of the posterior and the bagged posterior in the (misspecified) model selection setting. We empirically assess the BayesBag approach on synthetic and real-world data in (i) feature selection for linear regression and (ii) phylogenetic tree reconstruction. Our theory and experiments show that, when all models are misspecified, BayesBag (a) provides greater reproducibility and (b) places posterior mass on optimal models more reliably, compared to the usual Bayesian posterior; on the other hand, under correct specification, BayesBag is slightly more conservative than the usual posterior, in the sense that BayesBag posterior probabilities tend to be slightly farther from the extremes of zero and one. Overall, our results demonstrate that BayesBag provides an easy-to-use and widely applicable approach that improves upon Bayesian model selection by making it more stable and reproducible.</p>","PeriodicalId":55398,"journal":{"name":"Bayesian Analysis","volume":"18 1","pages":"79-104"},"PeriodicalIF":4.4,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9838736/pdf/nihms-1796997.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9229540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The evolution of communities in dynamic (time-varying) network data is a prominent topic of interest. A popular approach to understanding these dynamic networks is to embed the dyadic relations into a latent metric space. While methods for clustering with this approach exist for dynamic networks, they all assume a static community structure. This paper presents a Bayesian nonparametric model for dynamic networks that can model networks with evolving community structures. Our model extends existing latent space approaches by explicitly modeling the additions, deletions, splits, and mergers of groups with a hierarchical Dirichlet process hidden Markov model. Our proposed approach, the hierarchical Dirichlet process latent position cluster model (HDP-LPCM), incorporates transitivity, models both individual and group level aspects of the data, and avoids the computationally expensive selection of the number of groups required by most popular methods. We provide a Markov chain Monte Carlo estimation algorithm and demonstrate its ability to detect evolving community structure in a network of military alliances during the Cold War and a narrative network constructed from the Game of Thrones television series.
{"title":"A Bayesian Nonparametric Latent Space Approach to Modeling Evolving Communities in Dynamic Networks","authors":"Joshua Daniel Loyal, Yuguo Chen","doi":"10.1214/21-ba1300","DOIUrl":"https://doi.org/10.1214/21-ba1300","url":null,"abstract":"The evolution of communities in dynamic (time-varying) network data is a prominent topic of interest. A popular approach to understanding these dynamic networks is to embed the dyadic relations into a latent metric space. While methods for clustering with this approach exist for dynamic networks, they all assume a static community structure. This paper presents a Bayesian nonparametric model for dynamic networks that can model networks with evolving community structures. Our model extends existing latent space approaches by explicitly modeling the additions, deletions, splits, and mergers of groups with a hierarchical Dirichlet process hidden Markov model. Our proposed approach, the hierarchical Dirichlet process latent position cluster model (HDP-LPCM), incorporates transitivity, models both individual and group level aspects of the data, and avoids the computationally expensive selection of the number of groups required by most popular methods. We provide a Markov chain Monte Carlo estimation algorithm and demonstrate its ability to detect evolving community structure in a network of military alliances during the Cold War and a narrative network constructed from the Game of Thrones television series.","PeriodicalId":55398,"journal":{"name":"Bayesian Analysis","volume":"285 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135643392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Interactions between actors are frequently represented using a network. The latent position model is widely used for analysing network data, whereby each actor is positioned in a latent space. Inferring the dimension of this space is challenging. Often, for simplicity, two dimensions are used or model selection criteria are employed to select the dimension, but this requires choosing a criterion and the computational expense of fitting multiple models. Here the latent shrinkage position model (LSPM) is proposed which intrinsically infers the effective dimension of the latent space. The LSPM employs a Bayesian nonparametric multiplicative truncated gamma process prior that ensures shrinkage of the variance of the latent positions across higher dimensions. Dimensions with non-negligible variance are deemed most useful to describe the observed network, inducing automatic inference on the latent space dimension. While the LSPM is applicable to many network types, logistic and Poisson LSPMs are developed here for binary and count networks respectively. Inference proceeds via a Markov chain Monte Carlo algorithm, where novel surrogate proposal distributions reduce the computational burden. The LSPM’s properties are assessed through simulation studies, and its utility is illustrated through application to real network datasets. Open source software assists wider implementation of the LSPM.
{"title":"A Latent Shrinkage Position Model for Binary and Count Network Data","authors":"Xian Yao Gwee, Isobel Claire Gormley, Michael Fop","doi":"10.1214/23-ba1403","DOIUrl":"https://doi.org/10.1214/23-ba1403","url":null,"abstract":"Interactions between actors are frequently represented using a network. The latent position model is widely used for analysing network data, whereby each actor is positioned in a latent space. Inferring the dimension of this space is challenging. Often, for simplicity, two dimensions are used or model selection criteria are employed to select the dimension, but this requires choosing a criterion and the computational expense of fitting multiple models. Here the latent shrinkage position model (LSPM) is proposed which intrinsically infers the effective dimension of the latent space. The LSPM employs a Bayesian nonparametric multiplicative truncated gamma process prior that ensures shrinkage of the variance of the latent positions across higher dimensions. Dimensions with non-negligible variance are deemed most useful to describe the observed network, inducing automatic inference on the latent space dimension. While the LSPM is applicable to many network types, logistic and Poisson LSPMs are developed here for binary and count networks respectively. Inference proceeds via a Markov chain Monte Carlo algorithm, where novel surrogate proposal distributions reduce the computational burden. The LSPM’s properties are assessed through simulation studies, and its utility is illustrated through application to real network datasets. Open source software assists wider implementation of the LSPM.","PeriodicalId":55398,"journal":{"name":"Bayesian Analysis","volume":"124 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135051558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Diagnosing convergence of Markov chain Monte Carlo is crucial and remains an essentially unsolved problem. Among the most popular methods, the potential scale reduction factor, commonly named ˆ R , is an indicator that monitors the convergence of output chains to a target distribution, based on a comparison of the between- and within-variances. Several improvements have been suggested since its introduction in the 90s. Here, we aim at better understanding the ˆ R behavior by proposing a localized version that focuses on quantiles of the target distribution. This new version relies on key theoretical properties of the associated population value. It naturally leads to proposing a new indicator ˆ R ∞ , which is shown to allow both for localizing the Markov chain Monte Carlo convergence in different quantiles of the target distribution, and at the same time for handling some convergence issues not detected by other ˆ R versions.
{"title":"On the Use of a Local Rˆ to Improve MCMC Convergence Diagnostic","authors":"Théo Moins, Julyan Arbel, A. Dutfoy, S. Girard","doi":"10.1214/23-ba1399","DOIUrl":"https://doi.org/10.1214/23-ba1399","url":null,"abstract":"Diagnosing convergence of Markov chain Monte Carlo is crucial and remains an essentially unsolved problem. Among the most popular methods, the potential scale reduction factor, commonly named ˆ R , is an indicator that monitors the convergence of output chains to a target distribution, based on a comparison of the between- and within-variances. Several improvements have been suggested since its introduction in the 90s. Here, we aim at better understanding the ˆ R behavior by proposing a localized version that focuses on quantiles of the target distribution. This new version relies on key theoretical properties of the associated population value. It naturally leads to proposing a new indicator ˆ R ∞ , which is shown to allow both for localizing the Markov chain Monte Carlo convergence in different quantiles of the target distribution, and at the same time for handling some convergence issues not detected by other ˆ R versions.","PeriodicalId":55398,"journal":{"name":"Bayesian Analysis","volume":" ","pages":""},"PeriodicalIF":4.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44085501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Objective Bayesian Model Selection for Spatial Hierarchical Models with Intrinsic Conditional Autoregressive Priors","authors":"Erica M. Porter, C. Franck, Marco A. R. Ferreira","doi":"10.1214/23-ba1375","DOIUrl":"https://doi.org/10.1214/23-ba1375","url":null,"abstract":"","PeriodicalId":55398,"journal":{"name":"Bayesian Analysis","volume":" ","pages":""},"PeriodicalIF":4.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42465458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Coarsened Mixtures of Hierarchical Skew Normal Kernels for Flow and Mass Cytometry Analyses","authors":"S. Gorsky, Cliburn Chan, Li Ma","doi":"10.1214/22-ba1356","DOIUrl":"https://doi.org/10.1214/22-ba1356","url":null,"abstract":"","PeriodicalId":55398,"journal":{"name":"Bayesian Analysis","volume":" ","pages":""},"PeriodicalIF":4.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47319315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Markov chain Monte Carlo methods for exponential family models with intractable normalizing constant, such as the exchange algorithm, require simulations of the sufficient statistics at every iteration of the Markov chain, which often result in expensive computations. Surrogate models for the likelihood function have been developed to accelerate inference algorithms in this context. However, these surrogate models tend to be relatively inflexible, and often provide a poor approximation to the true likelihood function. In this article, we propose the use of a warped, gradient-enhanced, Gaussian process surrogate model for the likelihood function, which jointly models the sample means and variances of the sufficient statistics, and uses warping functions to capture covariance nonstationarity in the input parameter space. We show that both the consideration of nonstationarity and the inclusion of gradient information can be leveraged to obtain a surrogate model that outperforms the conventional stationary Gaussian process surrogate model when making inference, particularly in regions where the likelihood function exhibits a phase transition. We also show that the proposed surrogate model can be used to improve the effective sample size per unit time when embedded in exact inferential algorithms. The utility of our approach in speeding up inferential algorithms is demonstrated on simulated and real-world data.
{"title":"Warped Gradient-Enhanced Gaussian Process Surrogate Models for Exponential Family Likelihoods with Intractable Normalizing Constants","authors":"Quan Vu, Matthew T. Moores, Andrew Zammit-Mangion","doi":"10.1214/23-ba1400","DOIUrl":"https://doi.org/10.1214/23-ba1400","url":null,"abstract":"Markov chain Monte Carlo methods for exponential family models with intractable normalizing constant, such as the exchange algorithm, require simulations of the sufficient statistics at every iteration of the Markov chain, which often result in expensive computations. Surrogate models for the likelihood function have been developed to accelerate inference algorithms in this context. However, these surrogate models tend to be relatively inflexible, and often provide a poor approximation to the true likelihood function. In this article, we propose the use of a warped, gradient-enhanced, Gaussian process surrogate model for the likelihood function, which jointly models the sample means and variances of the sufficient statistics, and uses warping functions to capture covariance nonstationarity in the input parameter space. We show that both the consideration of nonstationarity and the inclusion of gradient information can be leveraged to obtain a surrogate model that outperforms the conventional stationary Gaussian process surrogate model when making inference, particularly in regions where the likelihood function exhibits a phase transition. We also show that the proposed surrogate model can be used to improve the effective sample size per unit time when embedded in exact inferential algorithms. The utility of our approach in speeding up inferential algorithms is demonstrated on simulated and real-world data.","PeriodicalId":55398,"journal":{"name":"Bayesian Analysis","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135103369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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