Probabilistic Approaches to Recommendations

The importance of accurate recommender systems has been widely recognized by academia and industry, and recommendation is rapidly becoming one of the most successful applications of data mining and machine learning. Understanding and predicting the choices and preferences of users is a challenging task: real-world scenarios involve users behaving in complex situations, where prior beliefs, specific tendencies, and reciprocal influences jointly contribute to determining the preferences of users toward huge amounts of information, services, and products. Probabilistic modeling represents a robust formal mathematical framework to model these assumptions and study their effects in the recommendation process. This book starts with a brief summary of the recommendation problem and its challenges and a review of some widely used techniques Next, we introduce and discuss probabilistic approaches for modeling preference data. We focus our attention on methods based on latent factors, such as mixture models, probabilistic matrix factorization, and topic models, for explicit and implicit preference data. These methods represent a significant advance in the research and technology of recommendation. The resulting models allow us to identify complex patterns in preference data, which can be exploited to predict future purchases effectively. The extreme sparsity of preference data poses serious challenges to the modeling of user preferences, especially in the cases where few observations are available. Bayesian inference techniques elegantly address the need for regularization, and their integration with latent factor modeling helps to boost the performances of the basic techniques. We summarize the strengths and weakness of several approaches by considering two different but related evaluation perspectives, namely, rating prediction and recommendation accuracy. Furthermore, we describe how probabilistic methods based on latent factors enable the exploitation of preference patterns in novel applications beyond rating prediction or recommendation accuracy. We finally discuss the application of probabilistic techniques in two additional scenarios, characterized by the availability of side information besides preference data. In summary, the book categorizes the myriad probabilistic approaches to recommendations and provides guidelines for their adoption in real-world situations.

[1]  Arun Yadav,et al.  MapReduce implementation of Variational Bayesian Probabilistic Matrix Factorization algorithm , 2013, 2013 IEEE International Conference on Big Data.

[2]  G. Manco,et al.  Probabilistic topic models for sequence data , 2013, Machine Learning.

[3]  Saul Vargas,et al.  Exploiting the diversity of user preferences for recommendation , 2013, OAIR.

[4]  Dan Cosley,et al.  Do social explanations work?: studying and modeling the effects of social explanations in recommender systems , 2013, WWW.

[5]  Nicola Barbieri,et al.  2012 IEEE 12th International Conference on Data Mining Topic-aware Social Influence Propagation Models , 2022 .

[6]  Jun Zhu,et al.  Online Nonparametric Max-Margin Matrix Factorization for Collaborative Prediction , 2012, 2014 IEEE International Conference on Data Mining.

[7]  Michael J. Freedman,et al.  Scalable Inference of Overlapping Communities , 2012, NIPS.

[8]  Guillermo Sapiro,et al.  Kernelized Probabilistic Matrix Factorization: Exploiting Graphs and Side Information , 2012, SDM.

[9]  Ricardo Giglio,et al.  Case study on the business value impact of personalized recommendations on a large online retailer , 2012, RecSys.

[10]  Suh-Yin Lee,et al.  Efficient algorithms for influence maximization in social networks , 2012, Knowledge and Information Systems.

[11]  Laks V. S. Lakshmanan,et al.  RecMax: exploiting recommender systems for fun and profit , 2012, KDD.

[12]  Mao Ye,et al.  Exploring social influence for recommendation: a generative model approach , 2012, SIGIR '12.

[13]  Junghoo Cho,et al.  Social-network analysis using topic models , 2012, SIGIR '12.

[14]  Xiaoqian Jiang,et al.  Predicting accurate probabilities with a ranking loss , 2012, ICML.

[15]  Haiyi Zhu,et al.  To Switch or Not To Switch , 2012, CHI.

[16]  Rossano Schifanella,et al.  Friendship prediction and homophily in social media , 2012, TWEB.

[17]  Alexander J. Smola,et al.  Discovering geographical topics in the twitter stream , 2012, WWW.

[18]  Jordan L. Boyd-Graber,et al.  Mr. LDA: a flexible large scale topic modeling package using variational inference in MapReduce , 2012, WWW.

[19]  Kazumi Saito,et al.  Efficient discovery of influential nodes for SIS models in social networks , 2012, Knowledge and Information Systems.

[20]  Vikas Sindhwani,et al.  Learning evolving and emerging topics in social media: a dynamic nmf approach with temporal regularization , 2012, WSDM '12.

[21]  Alexander J. Smola,et al.  Scalable inference in latent variable models , 2012, WSDM '12.

[22]  Zhang Xiong,et al.  Improving neighborhood based Collaborative Filtering via integrated folksonomy information , 2012, Pattern Recognit. Lett..

[23]  Sanjeev R. Kulkarni,et al.  Wisdom of the Crowd: Incorporating Social Influence in Recommendation Models , 2011, 2011 IEEE 17th International Conference on Parallel and Distributed Systems.

[24]  Martin Ester,et al.  A generalized stochastic block model for recommendation in social rating networks , 2011, RecSys '11.

[25]  Saul Vargas,et al.  Rank and relevance in novelty and diversity metrics for recommender systems , 2011, RecSys '11.

[26]  Nicola Barbieri,et al.  Modeling item selection and relevance for accurate recommendations: a bayesian approach , 2011, RecSys '11.

[27]  Nicola Barbieri,et al.  An Analysis of Probabilistic Methods for Top-N Recommendation in Collaborative Filtering , 2011, ECML/PKDD.

[28]  Charles Elkan,et al.  Link Prediction via Matrix Factorization , 2011, ECML/PKDD.

[29]  Laks V. S. Lakshmanan,et al.  A Data-Based Approach to Social Influence Maximization , 2011, Proc. VLDB Endow..

[30]  F. Bonchi Influence Propagation in Social Networks: A Data Mining Perspective , 2011, 2011 IEEE/WIC/ACM International Conferences on Web Intelligence and Intelligent Agent Technology.

[31]  David B. Dunson,et al.  Probabilistic topic models , 2012, Commun. ACM.

[32]  Nicola Barbieri,et al.  Regularized Gibbs Sampling for User Profiling with Soft Constraints , 2011, 2011 International Conference on Advances in Social Networks Analysis and Mining.

[33]  Samuel J. Gershman,et al.  A Tutorial on Bayesian Nonparametric Models , 2011, 1106.2697.

[34]  Chao Liu,et al.  Recommender systems with social regularization , 2011, WSDM '11.

[35]  Charles Elkan,et al.  Fast Algorithms for Approximating the Singular Value Decomposition , 2011, TKDD.

[36]  Arindam Banerjee,et al.  Generalized Probabilistic Matrix Factorizations for Collaborative Filtering , 2010, 2010 IEEE International Conference on Data Mining.

[37]  Charles Elkan,et al.  A Log-Linear Model with Latent Features for Dyadic Prediction , 2010, 2010 IEEE International Conference on Data Mining.

[38]  Huidong Jin,et al.  Sequential Latent Dirichlet Allocation: Discover Underlying Topic Structures within a Document , 2010, 2010 IEEE International Conference on Data Mining.

[39]  Gholamreza Haffari,et al.  Modeling the Temporal Dynamics of Social Rating Networks Using Bidirectional Effects of Social Relations and Rating Patterns , 2010, 2010 IEEE International Conference on Data Mining Workshops.

[40]  Francis R. Bach,et al.  Online Learning for Latent Dirichlet Allocation , 2010, NIPS.

[41]  Arindam Banerjee,et al.  Residual Bayesian Co-clustering for Matrix Approximation , 2010, SDM.

[42]  Jiawei Han,et al.  Mining topic-level influence in heterogeneous networks , 2010, CIKM.

[43]  Mouzhi Ge,et al.  Beyond accuracy: evaluating recommender systems by coverage and serendipity , 2010, RecSys '10.

[44]  Roberto Turrin,et al.  Performance of recommender algorithms on top-n recommendation tasks , 2010, RecSys '10.

[45]  Yasushi Sakurai,et al.  Online multiscale dynamic topic models , 2010, KDD.

[46]  Wei Chen,et al.  Scalable influence maximization for prevalent viral marketing in large-scale social networks , 2010, KDD.

[47]  Licia Capra,et al.  Temporal diversity in recommender systems , 2010, SIGIR.

[48]  Krishna P. Gummadi,et al.  Measuring User Influence in Twitter: The Million Follower Fallacy , 2010, ICWSM.

[49]  Craig MacDonald,et al.  Exploiting query reformulations for web search result diversification , 2010, WWW '10.

[50]  Ryan P. Adams,et al.  Incorporating Side Information in Probabilistic Matrix Factorization with Gaussian Processes , 2010, UAI.

[51]  Cho-Jui Hsieh,et al.  Iterative Scaling and Coordinate Descent Methods for Maximum Entropy Models , 2010, J. Mach. Learn. Res..

[52]  Deepak Agarwal,et al.  fLDA: matrix factorization through latent dirichlet allocation , 2010, WSDM '10.

[53]  Laks V. S. Lakshmanan,et al.  Learning influence probabilities in social networks , 2010, WSDM '10.

[54]  Max Welling,et al.  Distributed Algorithms for Topic Models , 2009, J. Mach. Learn. Res..

[55]  Neil J. Hurley,et al.  Novel Item Recommendation by User Profile Partitioning , 2009, 2009 IEEE/WIC/ACM International Joint Conference on Web Intelligence and Intelligent Agent Technology.

[56]  Kathryn B. Laskey,et al.  Latent Dirichlet Bayesian Co-Clustering , 2009, ECML/PKDD.

[57]  Yehuda Koren,et al.  Matrix Factorization Techniques for Recommender Systems , 2009, Computer.

[58]  Yihong Gong,et al.  Fast nonparametric matrix factorization for large-scale collaborative filtering , 2009, SIGIR.

[59]  Martin Ester,et al.  TrustWalker: a random walk model for combining trust-based and item-based recommendation , 2009, KDD.

[60]  Deepak Agarwal,et al.  Regression-based latent factor models , 2009, KDD.

[61]  Andrew McCallum,et al.  Efficient methods for topic model inference on streaming document collections , 2009, KDD.

[62]  Lars Schmidt-Thieme,et al.  BPR: Bayesian Personalized Ranking from Implicit Feedback , 2009, UAI.

[63]  Yee Whye Teh,et al.  On Smoothing and Inference for Topic Models , 2009, UAI.

[64]  Thore Graepel,et al.  Matchbox: large scale online bayesian recommendations , 2009, WWW '09.

[65]  Nathan Sakunkoo,et al.  Analysis of Social Influence in Online Book Reviews , 2009, ICWSM.

[66]  Daniel Barbará,et al.  On-line LDA: Adaptive Topic Models for Mining Text Streams with Applications to Topic Detection and Tracking , 2008, 2008 Eighth IEEE International Conference on Data Mining.

[67]  Arindam Banerjee,et al.  Bayesian Co-clustering , 2008, 2008 Eighth IEEE International Conference on Data Mining.

[68]  Michael R. Lyu,et al.  SoRec: social recommendation using probabilistic matrix factorization , 2008, CIKM '08.

[69]  Mi Zhang,et al.  Avoiding monotony: improving the diversity of recommendation lists , 2008, RecSys '08.

[70]  Masahiro Kimura,et al.  Prediction of Information Diffusion Probabilities for Independent Cascade Model , 2008, KES.

[71]  Alexander J. Smola,et al.  Improving maximum margin matrix factorization , 2008, Machine Learning.

[72]  Ravi Kumar,et al.  Influence and correlation in social networks , 2008, KDD.

[73]  Jon M. Kleinberg,et al.  Feedback effects between similarity and social influence in online communities , 2008, KDD.

[74]  Max Welling,et al.  Fast collapsed gibbs sampling for latent dirichlet allocation , 2008, KDD.

[75]  Yehuda Koren,et al.  Factorization meets the neighborhood: a multifaceted collaborative filtering model , 2008, KDD.

[76]  Max Welling,et al.  Multi-HDP: A Non Parametric Bayesian Model for Tensor Factorization , 2008, AAAI.

[77]  Ruslan Salakhutdinov,et al.  Bayesian probabilistic matrix factorization using Markov chain Monte Carlo , 2008, ICML '08.

[78]  Deng Cai,et al.  Topic modeling with network regularization , 2008, WWW.

[79]  David M. Blei,et al.  Supervised Topic Models , 2007, NIPS.

[80]  Ruslan Salakhutdinov,et al.  Probabilistic Matrix Factorization , 2007, NIPS.

[81]  Robert M. Bell,et al.  Scalable Collaborative Filtering with Jointly Derived Neighborhood Interpolation Weights , 2007, Seventh IEEE International Conference on Data Mining (ICDM 2007).

[82]  A. McCallum,et al.  Topical N-Grams: Phrase and Topic Discovery, with an Application to Information Retrieval , 2007, Seventh IEEE International Conference on Data Mining (ICDM 2007).

[83]  William W. Cohen,et al.  Parallelized Variational EM for Latent Dirichlet Allocation: An Experimental Evaluation of Speed and Scalability , 2007, Seventh IEEE International Conference on Data Mining Workshops (ICDMW 2007).

[84]  Bamshad Mobasher,et al.  Robustness of collaborative recommendation based on association rule mining , 2007, RecSys '07.

[85]  R. Burke,et al.  Toward trustworthy recommender systems: An analysis of attack models and algorithm robustness , 2007, TOIT.

[86]  Bamshad Mobasher,et al.  Defending recommender systems: detection of profile injection attacks , 2007, Service Oriented Computing and Applications.

[87]  Yehuda Koren,et al.  Modeling relationships at multiple scales to improve accuracy of large recommender systems , 2007, KDD '07.

[88]  Andreas Krause,et al.  Cost-effective outbreak detection in networks , 2007, KDD '07.

[89]  Deepak Agarwal,et al.  Predictive discrete latent factor models for large scale dyadic data , 2007, KDD '07.

[90]  ChengXiang Zhai,et al.  Automatic labeling of multinomial topic models , 2007, KDD '07.

[91]  Ryohei Orihara,et al.  Metrics for Evaluating the Serendipity of Recommendation Lists , 2007, JSAI.

[92]  Edoardo M. Airoldi,et al.  Mixed Membership Stochastic Blockmodels , 2007, NIPS.

[93]  Abhinandan Das,et al.  Google news personalization: scalable online collaborative filtering , 2007, WWW '07.

[94]  Neil J. Hurley,et al.  Attacking Recommender Systems: A Cost-Benefit Analysis , 2007, IEEE Intelligent Systems.

[95]  Evangelos E. Milios,et al.  Latent Dirichlet Co-Clustering , 2006, Sixth International Conference on Data Mining (ICDM'06).

[96]  Masahiro Kimura,et al.  Tractable Models for Information Diffusion in Social Networks , 2006, PKDD.

[97]  Hanna M. Wallach Topic modeling: beyond bag-of-words , 2006, ICML.

[98]  Tom Fawcett,et al.  An introduction to ROC analysis , 2006, Pattern Recognit. Lett..

[99]  Luo Si,et al.  A study of mixture models for collaborative filtering , 2006, Information Retrieval.

[100]  Sean M. McNee,et al.  Being accurate is not enough: how accuracy metrics have hurt recommender systems , 2006, CHI Extended Abstracts.

[101]  Srujana Merugu,et al.  A scalable collaborative filtering framework based on co-clustering , 2005, Fifth IEEE International Conference on Data Mining (ICDM'05).

[102]  Nathan Srebro,et al.  Fast maximum margin matrix factorization for collaborative prediction , 2005, ICML.

[103]  Dimitris Plexousakis,et al.  Alleviating the Sparsity Problem of Collaborative Filtering Using Trust Inferences , 2005, iTrust.

[104]  J. Konstan,et al.  Improving recommendation lists through topic diversification , 2005, WWW '05.

[105]  Gérard Govaert,et al.  An EM algorithm for the block mixture model , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[106]  Tommi S. Jaakkola,et al.  Maximum-Margin Matrix Factorization , 2004, NIPS.

[107]  Neil J. Hurley,et al.  Collaborative recommendation: A robustness analysis , 2004, TOIT.

[108]  David M. Pennock,et al.  Collaborative filtering with maximum entropy , 2004, IEEE Intelligent Systems.

[109]  Patrik O. Hoyer,et al.  Non-negative Matrix Factorization with Sparseness Constraints , 2004, J. Mach. Learn. Res..

[110]  Xin Jin,et al.  Web usage mining based on probabilistic latent semantic analysis , 2004, KDD.

[111]  Takeo Kanade,et al.  Maximum Entropy for Collaborative Filtering , 2004, UAI.

[112]  John Riedl,et al.  Shilling recommender systems for fun and profit , 2004, WWW '04.

[113]  T. Griffiths,et al.  Finding scientific topics , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[114]  J. Lafferty,et al.  Mixed-membership models of scientific publications , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[115]  Benjamin M. Marlin,et al.  Modeling User Rating Profiles For Collaborative Filtering , 2003, NIPS.

[116]  Wenliang Du,et al.  Privacy-preserving collaborative filtering using randomized perturbation techniques , 2003, Third IEEE International Conference on Data Mining.

[117]  Jon Kleinberg,et al.  Maximizing the spread of influence through a social network , 2003, KDD '03.

[118]  Rong Yan,et al.  A Faster Iterative Scaling Algorithm for Conditional Exponential Model , 2003, ICML.

[119]  Tommi S. Jaakkola,et al.  Weighted Low-Rank Approximations , 2003, ICML.

[120]  Thomas Hofmann,et al.  Collaborative filtering via gaussian probabilistic latent semantic analysis , 2003, SIGIR.

[121]  Ata Kabán,et al.  On an equivalence between PLSI and LDA , 2003, SIGIR.

[122]  Gérard Govaert,et al.  Clustering with block mixture models , 2003, Pattern Recognit..

[123]  John Riedl,et al.  An Empirical Analysis of Design Choices in Neighborhood-Based Collaborative Filtering Algorithms , 2002, Information Retrieval.

[124]  Rob Malouf,et al.  A Comparison of Algorithms for Maximum Entropy Parameter Estimation , 2002, CoNLL.

[125]  John F. Canny,et al.  Collaborative filtering with privacy via factor analysis , 2002, SIGIR '02.

[126]  Tom Minka,et al.  Expectation-Propogation for the Generative Aspect Model , 2002, UAI.

[127]  Matthew Richardson,et al.  Mining knowledge-sharing sites for viral marketing , 2002, KDD.

[128]  John F. Canny,et al.  Collaborative filtering with privacy , 2002, Proceedings 2002 IEEE Symposium on Security and Privacy.

[129]  Anil K. Jain,et al.  Unsupervised Learning of Finite Mixture Models , 2002, IEEE Trans. Pattern Anal. Mach. Intell..

[130]  Sean M. McNee,et al.  Getting to know you: learning new user preferences in recommender systems , 2002, IUI '02.

[131]  Benjamin J. Keller,et al.  Privacy Risks in Recommender Systems , 2001, IEEE Internet Comput..

[132]  George Karypis,et al.  Evaluation of Item-Based Top-N Recommendation Algorithms , 2001, CIKM '01.

[133]  Thomas Hofmann,et al.  Learning What People (Don't) Want , 2001, ECML.

[134]  Matthew Richardson,et al.  Mining the network value of customers , 2001, KDD '01.

[135]  David M. Pennock,et al.  Probabilistic Models for Unified Collaborative and Content-Based Recommendation in Sparse-Data Environments , 2001, UAI.

[136]  M. McPherson,et al.  Birds of a Feather: Homophily in Social Networks , 2001 .

[137]  John Riedl,et al.  Item-based collaborative filtering recommendation algorithms , 2001, WWW '01.

[138]  Padhraic Smyth,et al.  Visualization of navigation patterns on a Web site using model-based clustering , 2000, KDD '00.

[139]  Udi Manber,et al.  Experience with personalization of Yahoo! , 2000, CACM.

[140]  John Riedl,et al.  Application of Dimensionality Reduction in Recommender System - A Case Study , 2000 .

[141]  Joydeep Ghosh,et al.  Value-based customer grouping from large retail data sets , 2000, SPIE Defense + Commercial Sensing.

[142]  Michael J. Pazzani,et al.  A Framework for Collaborative, Content-Based and Demographic Filtering , 1999, Artificial Intelligence Review.

[143]  A. Walker,et al.  Multiple stellar populations in the globular cluster ω Centauri as tracers of a merger event , 1999, Nature.

[144]  Anil K. Jain,et al.  Data clustering: a review , 1999, CSUR.

[145]  Philip S. Yu,et al.  Horting hatches an egg: a new graph-theoretic approach to collaborative filtering , 1999, KDD '99.

[146]  Thomas Hofmann,et al.  Latent Class Models for Collaborative Filtering , 1999, IJCAI.

[147]  Michael I. Jordan,et al.  An Introduction to Variational Methods for Graphical Models , 1999, Machine Learning.

[148]  Michael J. Pazzani,et al.  Learning Collaborative Information Filters , 1998, ICML.

[149]  David Heckerman,et al.  Empirical Analysis of Predictive Algorithms for Collaborative Filtering , 1998, UAI.

[150]  Bruce Krulwich,et al.  LIFESTYLE FINDER: Intelligent User Profiling Using Large-Scale Demographic Data , 1997, AI Mag..

[151]  Michael J. Pazzani,et al.  Learning and Revising User Profiles: The Identification of Interesting Web Sites , 1997, Machine Learning.

[152]  Susan T. Dumais,et al.  Using Linear Algebra for Intelligent Information Retrieval , 1995, SIAM Rev..

[153]  Henry Lieberman,et al.  Letizia: An Agent That Assists Web Browsing , 1995, IJCAI.

[154]  Ken Lang,et al.  NewsWeeder: Learning to Filter Netnews , 1995, ICML.

[155]  Jonathan Grudin,et al.  Computer-supported cooperative work: history and focus , 1994, Computer.

[156]  Jorge Nocedal,et al.  On the limited memory BFGS method for large scale optimization , 1989, Math. Program..

[157]  M. L. Fisher,et al.  An analysis of approximations for maximizing submodular set functions—I , 1978, Math. Program..

[158]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[159]  Nicola Barbieri,et al.  Balancing Prediction and Recommendation Accuracy: Hierarchical Latent Factors for Preference Data , 2012, SDM.

[160]  Nicola Barbieri,et al.  Characterizing Relationships through Co-clustering - A Probabilistic Approach , 2011, KDIR.

[161]  Mohammad Al Hasan,et al.  A Survey of Link Prediction in Social Networks , 2011, Social Network Data Analytics.

[162]  Nicola Barbieri,et al.  A Probabilistic Hierarchical Approach for Pattern Discovery in Collaborative Filtering Data (Extended Abstract) , 2011, SEBD.

[163]  Xi Chen,et al.  Temporal Collaborative Filtering with Bayesian Probabilistic Tensor Factorization , 2010, SDM.

[164]  Michael I. Jordan,et al.  Latent Dirichlet Allocation , 2009 .

[165]  Myra Spiliopoulou,et al.  Topic Evolution in a Stream of Documents , 2009, SDM.

[166]  Yehuda Koren,et al.  Improved Neighborhood-based Collaborative Filtering , 2007 .

[167]  J. Tenenbaum,et al.  Topics in Semantic Representation , 2007 .

[168]  Yee Whye Teh,et al.  Variational Bayesian Approach to Movie Rating Prediction , 2007, KDD 2007.

[169]  Arindam Banerjee,et al.  Topic Models over Text Streams: A Study of Batch and Online Unsupervised Learning , 2007, SDM.

[170]  Nando de Freitas,et al.  An Introduction to MCMC for Machine Learning , 2004, Machine Learning.

[171]  John Riedl,et al.  E-Commerce Recommendation Applications , 2004, Data Mining and Knowledge Discovery.

[172]  Inderjit S. Dhillon,et al.  Concept Decompositions for Large Sparse Text Data Using Clustering , 2004, Machine Learning.

[173]  Jonathan L. Herlocker,et al.  Evaluating collaborative filtering recommender systems , 2004, TOIS.

[174]  Thomas Hofmann,et al.  Unsupervised Learning by Probabilistic Latent Semantic Analysis , 2004, Machine Learning.

[175]  Thomas Hofmann,et al.  Latent semantic models for collaborative filtering , 2004, TOIS.

[176]  David M. Pennock,et al.  Categories and Subject Descriptors , 2001 .

[177]  Rashmi R. Sinha,et al.  Comparing Recommendations Made by Online Systems and Friends , 2001, DELOS.

[178]  P. Resnick,et al.  An open architecture for collaborative filtering of netnews , 1994 .

[179]  Tapani Raiko,et al.  Tkk Reports in Information and Computer Science Practical Approaches to Principal Component Analysis in the Presence of Missing Values Tkk Reports in Information and Computer Science Practical Approaches to Principal Component Analysis in the Presence of Missing Values , 2022 .

[180]  THIS IS A DRAFT VERSION. FINAL VERSION TO BE PUBLISHED AT NIPS ’06 A Collapsed Variational Bayesian Inference Algorithm for Latent Dirichlet Allocation , 2022 .