Books

  1. Andrew Gelman, and Aki Vehtari (2024). Active Statistics. Cambridge University Press.
    • Publisher’s webpage for the book
    • Author’s webpage for the book.
    • This book provides statistics instructors and students with complete classroom material for a one- or two-semester course on applied regression and causal inference. It is built around 52 stories, 52 class-participation activities, 52 hands-on computer demonstrations, and 52 discussion problems that allow instructors and students to explore in a fun way the real-world complexity of the subject. The book fosters an engaging ‘flipped classroom’ environment with a focus on visualization and understanding. The book provides instructors with frameworks for self-study or for structuring the course, along with tips for maintaining student engagement at all levels, and practice exam questions to help guide learning. Designed to accompany the authors’ previous textbook Regression and Other Stories, its modular nature and wealth of material allow this book to be adapted to different courses and texts or be used by learners as a hands-on workbook.

  2. Andrew Gelman, Jennifer Hill, and Aki Vehtari (2020). Regression and other stories. Cambridge University Press.
    • Publisher’s webpage for the book.
    • Author’s webpage for the book.
    • Back cover text: Many textbooks on regression focus on theory and the simplest of examples. Real statistical problems, however, are complex and subtle. This is not a book about the theory of regression. It is a book about how to use regression to solve real problems of comparison, estimation, prediction, and causal inference. It focuses on practical issues such as sample size and missing data and a wide range of goals and techniques. It jumps right in to methods and computer code you can use fresh out of the box.

  3. Andrew Gelman, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari and Donald B. Rubin (2013). Bayesian Data Analysis, Third Edition. Chapman and Hall/CRC.

Pre-prints

  1. Marvin Schmitt, Chengkun Li, Aki Vehtari, Luigi Acerbi, Paul-Christian Bürkner, and Stefan T. Radev (2024). Amortized Bayesian Workflow (Extended Abstract). arXiv preprint arXiv:2409.04332.

  2. Kunal Ghosh, Milica Todorović, Aki Vehtari, and Patrick Rinke (2024). Active Learning of Molecular Data for Task-Specific Objectives. arXiv preprint arXiv:2408.11191.

  3. Måns Magnusson, Jakob Torgander, Paul-Christian Bürkner, Lu Zhang, Bob Carpenter, and Aki Vehtari (2024). posteriordb: Testing, benchmarking and developing Bayesian inference algorithms. arXiv preprint arXiv:2407.04967.

  4. David Kohns, Noa Kallionen, Yann McLatchie, and Aki Vehtari (2024). The ARR2 prior: flexible predictive prior definition for Bayesian auto-regressions. arXiv preprint arXiv:2405.19920.

  5. Anna Elisabeth Riha, Nikolas Siccha, Antti Oulasvirta, and Aki Vehtari (2024). Supporting Bayesian modelling workflows with iterative filtering for multiverse analysis. arXiv preprint arXiv:2404.01688.

  6. Guangzhao Cheng, Aki Vehtari, and Lu Cheng (2024). Raw signal segmentation for estimating RNA modifications and structures from Nanopore direct RNA sequencing data. bioRxiv preprint.

  7. Lauren Kennedy, Aki Vehtari, and Andrew Gelman (2024). Model validation for aggregate inferences in out-of-sample prediction. arXiv preprint arXiv:2312.06334.

  8. Tuomas Sivula, Måns Magnusson, Asael Alonzo Matamoros, and Aki Vehtari (2023). Uncertainty in Bayesian leave-one-out cross-validation based model comparison. arXiv preprint arXiv:2008.10296.
    Video 30min.

  9. Yann McLatchie, Asael Alonzo Matamoros, David Kohns, and Aki Vehtari (2022). Bayesian order identification of ARMA models with projection predictive inference. arXiv preprint arXiv:2208.14824.

  10. Alejandro Catalina, Paul Bürkner, and Aki Vehtari (2021). Latent space projection predictive inference. arXiv preprint arXiv:2109.04702.

  11. Shabnam Salimi, Aki Vehtari, Marcel Salive, Luigi Ferrucci (2021). Body Clock: Matching Personalized Multimorbidity and Fast Aging Using Information Entropy. medRxiv preprint, doi:10.1101/2021.03.29.21254372.

  12. Andrew Gelman, Aki Vehtari, Daniel Simpson, Charles C. Margossian, Bob Carpenter, Yuling Yao, Lauren Kennedy, Jonah Gabry, Paul-Christian Bürkner, and Martin Modrák (2020). Bayesian workflow. arXiv preprint arXiv:2011.01808.
    Video 65min
    Video 44min

  13. Yuling Yao, Collin Cademartori, Aki Vehtari, and Andrew Gelman (2020). Adaptive path sampling in metastable posterior distributions. arXiv preprint arXiv:2009.00471.

  14. Oscar Oelrich, Shutong Ding, Måns Magnusson, Aki Vehtari, and Mattias Villani (2020). When are Bayesian model probabilities overconfident? arXiv preprint arXiv:2003.04026.

Peer-reviewed

  1. Christopher Tosh, Philip Greengard, Ben Goodrich, Andrew Gelman, Aki Vehtari, and Daniel Hsu (2025). The piranha problem: Large effects swimming in a small pond. Notices of the American Mathematical Society, accepted for publication. arXiv preprint arXiv:2105.13445. preprint

  2. Yann McLatchie, Sölvi Rögnvaldsson, Frank Weber, and Aki Vehtari (2024). Advances in projection predictive inference. Statistical Science, Online.
    arXiv preprint arXiv:2306.15581.

  3. Charles C. Margossian, Matthew D. Hoffman, Pavel Sountsov, Lionel Riou-Durand, Aki Vehtari, and Andrew Gelman (2024). Nested \(\widehat{R}\): Assessing the convergence of Markov chain Monte Carlo when running many short chains. Bayesian Analysis, doi:10.1214/24-BA1453.
    arXiv preprint arXiv:2110.13017.

  4. Yann McLatchie and Aki Vehtari (2024). Efficient estimation and correction of selection-induced bias with order statistics. Statistics and Computing, 34(132). doi:10.1007/s11222-024-10442-4.
    arXiv preprint arXiv:2309.03742.

  5. Frank Weber, Änne Glass, and Aki Vehtari (2024). Projection predictive variable selection for discrete response families with finite support. Computational Statistics, doi:10.1007/s00180-024-01506-0.
    arXiv preprint arXiv:2301.01660.

  6. Aki Vehtari, Daniel Simpson, Andrew Gelman, Yuling Yao, and Jonah Gabry (2024). Pareto smoothed importance sampling. Journal of Machine Learning Research, 25(72):1-58. Online
    arXiv preprint arXiv:1507.02646. Python code. R code.
    Video of StanCon 2024 talk

  7. Manushi Welandawe, Michael Riis Andersen, Aki Vehtari, and Jonathan H. Huggins (2024). A framework for improving the reliability of black-box variational inference. Journal of Machine Learning Research, 25(219):1-71. Online.
    arXiv preprint arXiv:2203.15945.

  8. Noa Kallioinen, Topi Paananen, Paul-Christian Bürkner, and Aki Vehtari (2024). Detecting and diagnosing prior and likelihood sensitivity with power-scaling. Statistics and Computing, 34(57). Online.
    arXiv preprint arXiv:2107.14054.
    Supplementary code.
    priorsense: R package

  9. Erik Štrumbelj, Alexandre Bouchard-Côté, Jukka Corander, Andrew Gelman, Håvard Rue, Lawrence Murray, Henri Pesonen, Martyn Plummer, and Aki Vehtari (2024). Past, Present, and Future of Software for Bayesian Inference. Statistical Science, 39(1):46-61. Online.
    preprint

  10. Alex Cooper, Dan Simpson, Lauren Kennedy, Catherine Forbes, and Aki Vehtari (2024). Cross-validatory model selection for Bayesian autoregressions with exogenous regressors. Bayesian Analysis, doi:10.1214/23-BA1409.
    arXiv preprint arXiv:2301.08276.

  11. Marta Kołczyńska, Paul-Christian Bürkner, Lauren Kennedy, and Aki Vehtari (2024). Trust in state institutions in Europe, 1989–2019. Survey Research Methods, 18(1). doi:10.18148/srm/2024.v18i1.8119
    SocArXiv preprint doi:10.31235/osf.io/3v5g7.

  12. Alex Cooper, Aki Vehtari, Catherine Forbes, Lauren Kennedy, and Dan Simpson (2024). Bayesian cross-validation by parallel Markov chain Monte Carlo. Statistics and Computing, 34:119. doi:10.1007/s11222-024-10404-w.
    arXiv preprint arXiv:2310.07002.

  13. Ryoko Noda, Michael Francis Mechenich, Juha Saarinen, Aki Vehtari, Indrė Žliobaitė (2024). Predicting habitat suitability for Asian elephants in non-analog ecosystems with Bayesian models. Ecological Informatics, 82:102658. doi:10.1016/j.ecoinf.2024.102658.

  14. Petrus Mikkola, Osvaldo A. Martin, Suyog Chandramouli, Marcelo Hartmann, Oriol Abril Pla, Owen Thomas, Henri Pesonen, Jukka Corander, Aki Vehtari, Samuel Kaski, Paul-Christian Bürkner, Arto Klami (2024). Prior knowledge elicitation: The past, present, and future. Bayesian Analysis, 19(49):1129-1161. doi:10.1214/23-BA1381.
    arXiv preprint arXiv:2112.01380.

  15. Martin Modrák, Angie H. Moon, Shinyoung Kim, Paul Bürkner, Niko Huurre, Kateřina Faltejsková, Andrew Gelman, and Aki Vehtari (2023). Simulation-based calibration checking for Bayesian computation: The choice of test quantities shapes sensitivity. Bayesian Analysis, doi:10.1214/23-BA1404.
    arXiv preprint arXiv:2211.02383.
    Code
    SBC R package

  16. Juho Timonen, Nikolas Siccha, Ben Bales, Harri Lähdesmäki, and Aki Vehtari (2023). An importance sampling approach for reliable and efficient inference in Bayesian ordinary differential equation models. Stat, 12(1):e614. doi:10.1002/sta4.614.
    arXiv preprint arXiv:2205.09059.

  17. Peter Mikula, Oldřich Tomášek, Dušan Romportl, Timothy K. Aikins, Jorge E. Avendaño, Bukola D. A. Braimoh-Azaki, Adams Chaskda, Will Cresswell, Susan J. Cunningham, Svein Dale, Gabriela R. Favoretto, Kelvin S. Floyd, Hayley Glover, Tomáš Grim, Dominic A. W. Henry, Tomas Holmern, Martin Hromada, Soladoye B. Iwajomo, Amanda Lilleyman, Flora J. Magige, Rowan O. Martin, Marina F. de A. Maximiano, Eric D. Nana, Emmanuel Ncube, Henry Ndaimani, Emma Nelson, Johann H. van Niekerk, Carina Pienaar, Augusto J. Piratelli, Penny Pistorius, Anna Radkovic, Chevonne Reynolds, Eivin Røskaft, Griffin K. Shanungu, Paulo R. Siqueira, Tawanda Tarakini, Nattaly Tejeiro-Mahecha, Michelle L. Thompson, Wanyoike Wamiti, Mark Wilson, Donovan R. C. Tye, Nicholas D. Tye, Aki Vehtari, Piotr Tryjanowski, Michael A. Weston, Daniel T. Blumstein, and Tomáš Albrecht (2023). Bird tolerance to humans in open tropical ecosystems. Nature Communications, 14:2146. doi:10.1038/s41467-023-37936-5.

  18. Gabriel Riutort-Mayol, Paul-Christian Bürkner, Michael R. Andersen, Arno Solin, and Aki Vehtari (2023). Practical Hilbert space approximate Bayesian Gaussian processes for probabilistic programming. Statistics and Computing, 33(17):1-28. doi:10.1007/s11222-022-10167-2.
    arXiv preprint arXiv:2004.11408.

  19. Philip Greengard, Jeremy Hoskins, Charles C. Margossian, Andrew Gelman, and Aki Vehtari (2023). Fast methods for posterior inference of two-group normal-normal models. Bayesian Analysis, 18(3):889-907. doi:10.1214/22-BA1329.
    Code.
    arXiv preprint arXiv:2110.03055.

  20. Lu Zhang, Bob Carpenter, Andrew Gelman, and Aki Vehtari (2022). Pathfinder: Parallel quasi-Newton variational inference. Journal of Machine Learning Research, 23(306):1-49. Online
    arXiv preprint arXiv:2108.03782. Supplementary code.

  21. Federico Pavone, Juho Piironen, Paul-Christian Bürkner, and Aki Vehtari (2022). Using reference models in variable selection. Computational Statistics, 38:349-371. doi:10.1007/s00180-022-01231-6. arXiv preprint arXiv:2004.13118.
    Video 80min

  22. Teemu Säilynoja, Paul-Christian Bürkner, and Aki Vehtari (2022). Graphical test for discrete uniformity and its applications in goodness of fit evaluation and multiple sample comparison. Statistics and Computing, 32(32). doi:10.1007/s11222-022-10090-6. arXiv preprint arXiv:2103.10522.
    Code.
    Video

  23. Alejandro Catalina, Paul-Christian Bürkner, and Aki Vehtari (2022). Projection predictive inference for generalized linear and additive multilevel models. Proceedings of the 24th International Conference on Artificial Intelligence and Statistics (AISTATS), PMLR 151:4446–4461. Online. arXiv preprint arXiv:2010.06994.

  24. Isaac Sebenius, Topi Paananen, Aki Vehtari (2022). Feature collapsing for Gaussian Process variable ranking. Proceedings of the 24th International Conference on Artificial Intelligence and Statistics (AISTATS), PMLR 151:11341-11355. Online

  25. Yuling Yao, Aki Vehtari, and Andrew Gelman (2022). Stacking for non-mixing Bayesian computations: The curse and blessing of multimodal posteriors. Journal of Machine Learning Research, 23(79):1-45. Online. arXiv preprint arXiv:2006.12335.

  26. Ben Lambert and Aki Vehtari (2022). \(R^*\): A robust MCMC convergence diagnostic with uncertainty using decision tree classifiers. Bayesian Analysis, 17(2):353-379. doi:10.1214/20-BA1252. arXiv preprint arXiv:2003.07900.

  27. Tuulia Malén, Tomi Karjalainen, Janne Isojärvi, Aki Vehtari, Paul-Christian Bürkner, Vesa Putkinen, Valtteri Kaasinen, Jarmo Hietala, Pirjo Nuutila, Juha Rinne, and Lauri Nummenmaa (2022). Atlas of type 2 dopamine receptors in the human brain: Age and sex dependent variability in a large PET cohort. NeuroImage, 255:119149. doi:10.1016/j.neuroimage.2022.119149.
    bioRxiv preprint 10.1101/2021.08.10.455776.

  28. Tuomas Sivula, Måns Magnusson, and Aki Vehtari (2022). Unbiased estimator for the variance of the leave-one-out cross-validation estimator for a Bayesian normal model with fixed variance. Communications in Statistics – Theory and Methods, doi:10.1080/03610926.2021.2021240.
    arXiv preprint arXiv:2008.10859.

  29. Philip Greengard, Andrew Gelman, and Aki Vehtari (2022). A fast linear regression via SVD and marginalization. Computational Statistics , 37:701–720.
    Online.
    arXiv preprint arXiv:2011.04829.

  30. Yuling Yao, Gregor Pirš, Aki Vehtari, and Andrew Gelman (2022). Bayesian hierarchical stacking: Some models are (somewhere) useful. Bayesian Analysis, 17(4), 1043-1071. doi:10.1214/21-BA1287.
    arXiv preprint arXiv:2101.08954.

  31. Akash Kumar Dhaka, Alejandro Catalina, Manushi Welandawe, Michael Riis Andersen, Jonathan Huggins, and Aki Vehtari (2021). Challenges and opportunities in high-dimensional variational inference. NeurIPS 2021, 34:7787–7798. Online.
    arXiv preprint arXiv:2103.01085.

  32. Eero Siivola, Akash Kumar Dhaka, Michael Riis Andersen, Javier González, Pablo Garcia Moreno, and Aki Vehtari (2021). Preferential batch Bayesian optimization. 2021 IEEE 31st International Workshop on Machine Learning for Signal Processing (MLSP), doi:10.1109/MLSP52302.2021.9596494.
    arXiv preprint arXiv:2003.11435.

  33. Andrew Gelman and Aki Vehtari (2021). What are the most important statistical ideas of the past 50 years? Journal of American Statistical Association, 116(536):2087–2097. doi:10.1080/01621459.2021.1938081. arXiv preprint arXiv:2012.00174.

  34. Topi Paananen, Michael Riis Andersen, Aki Vehtari (2021). Uncertainty-aware sensitivity analysis using Rényi divergences,. Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence, PMLR 161:1185-1194.
    Online.
    arXiv preprint arXiv:1910.07942.

  35. Eero Siivola, Javier González, Andrei Paleyes, and Aki Vehtari (2021). Good practices for Bayesian optimization of high dimensional structured spaces. Applied AI Letters, 2(2):e24, doi:10.1002/ail2.24.
    arXiv preprint arXiv:2012.15471.

  36. Eero Siivola, Sebastian Weber, and Aki Vehtari (2021). Qualifying drug dosing regimens in pediatrics using Gaussian processes. Statistics in Medicine, 40(10):2355-2372. doi:10.1002/sim.8907.

  37. Stefano Mangiola, Evan A. Thomas, Martin Modrák, Aki Vehtari, and Anthony T. Papenfuss (2021). Probabilistic outlier identification for RNA sequencing generalised linear model. NAR Genomics and Bioinformatics, 3(1):lqab005. Online.

  38. Juho Timonen, Henrik Mannerström, Aki Vehtari, and Harri Lähdesmäki (2021). lgpr: An interpretable nonparametric method for inferring covariate effects from longitudinal data. Bioinformatics, 38(13):1860-1867. doi:10.1093/bioinformatics/btab021.
    arXiv preprint arXiv:1912.03549.
    lgpr software package.

  39. Marko Järvenpää, Michael Gutmann, Aki Vehtari, and Pekka Marttinen (2021). Parallel Gaussian process surrogate Bayesian inference with noisy likelihood evaluations. Bayesian Analysis, 16(1):147-148. doi:10.1214/20-BA1200.
    arXiv preprint arXiv:1905.01252.

  40. Topi Paananen, Juho Piironen, Paul-Christian Bürkner, and Aki Vehtari (2021). Implicitly adaptive importance sampling. Statistics and Computing, 31, 16. doi:10.1007/s11222-020-09982-2.
    arXiv preprint arXiv:1906.08850.

  41. Aki Vehtari, Andrew Gelman, Daniel Simpson, Bob Carpenter, and Paul-Christian Bürkner (2021). Rank-normalization, folding, and localization: An improved \(\widehat{R}\) for assessing convergence of MCMC. Bayesian Analysis, 16(2):667-718. doi:10.1214/20-BA1221. Online. arXiv preprint arXiv:1903.08008.
    Code.
    Online Appendix.
    Online Appendix 2: Comparison of ESS/MCSE methods.
    Video

  42. Joaquin Cavieres, Cole C. Monnahan, and Aki Vehtari (2021). Accounting for spatial dependence improves relative abundance estimates in a benthic marine species structured as a metapopulation. Fisheries Research, 240:105960, doi:10.1016/j.fishres.2021.105960.

  43. Paul-Christian Bürkner, Jonah Gabry, and Aki Vehtari (2020). Efficient leave-one-out cross-validation for Bayesian non-factorized normal and Student-\(t\) models. Computational Statistics, 36:1243-1261. doi:10.1007/s00180-020-01045-4.
    arXiv preprint arXiv:1810.10559. Code.

  44. Akash Kumar Dhaka, Alejandro Catalina, Michael Riis Andersen, Måns Magnusson, Jonathan H. Huggins, and Aki Vehtari (2020). Robust, accurate stochastic optimization for variational inference. NeurIPS 2020, 33:10961–10973. Online.
    arXiv preprint arXiv:2009.00666.

  45. Charles C. Margossian, Aki Vehtari, Daniel Simpson, and Raj Agrawal (2020). Hamiltonian Monte Carlo using an adjoint-differentiated Laplace approximation: Bayesian inference for latent Gaussian models and beyond. NeurIPS 2020, 33:9086–9097. Online.
    arXiv preprint arXiv:2004.12550.

  46. Homayun Afrabandpey, Tomi Peltola, Juho Piironen, Aki Vehtari, and Samuel Kaski (2020). A decision-theoretic approach for model interpretability in Bayesian framework. Machine Learning, 109:1855–1876. Special Issue of the ECML PKDD 2020 Journal Track. Online.
    arXiv preprint arXiv:1910.09358

  47. Akash Kumar Dhaka, Michael Riis Andersen, Pablo Moreno, and Aki Vehtari (2020). Scalable Gaussian process for extreme classification. 2020 IEEE 30th International Workshop on Machine Learning for Signal Processing (MLSP), doi:10.1109/MLSP49062.2020.9231675.
    preprint.

  48. Paul-Christian Bürkner, Jonah Gabry, and Aki Vehtari (2020). Approximate leave-future-out cross-validation for time series models. Journal of Statistical Computation and Simulation, 90(14):2499-2523. Online.
    arXiv preprint arXiv:1902.06281. Code.

  49. Marko Järvenpää, Aki Vehtari, and Pekka Marttinen (2020). Batch simulations and uncertainty quantification in Gaussian process surrogate-based approximate Bayesian computation. Proceedings of Conference on Uncertainty in Artificial Intelligence (UAI), ID:334. Online.
    arXiv preprint arXiv:1910.06121.

  50. Tatu Kantonen, Tomi Karjalainen, Janne Isojärvi, Pirjo Nuutila, Jouni Tuisku, Juha Rinne, Jarmo Hietala, Valtteri Kaasinen, Kari Kalliokoski, Harry Scheinin, Jussi Hirvonen, Aki Vehtari, and Lauri Nummenmaa (2020). Interindividual variability and lateralization of \(\mu\)-opioid receptors in the human brain. NeuroImage, 217:116922. doi:10.1016/j.neuroimage.2020.116922.
    bioRxiv preprint.

  51. Juho Piironen, Markus Paasiniemi, and Aki Vehtari (2020). Projective inference in high-dimensional problems: prediction and feature selection. Electronic Journal of Statistics, 14(1):2155-2197. Online.
    arXiv preprint arXiv:1810.02406.
    Code.
    Video

  52. Aki Vehtari, Andrew Gelman, Tuomas Sivula, Pasi Jylänki, Dustin Tran, Swupnil Sahai, Paul Blomstedt, John P. Cunningham, David Schiminovich, and Christian Robert (2020). Expectation propagation as a way of life: A framework for Bayesian inference on partitioned data. Journal of Machine Learning Research, 21(17):1-53. Online.
    arXiv preprint arXiv:1412.4869.

  53. Måns Magnusson, Michael Riis Andersen, Johan Jonasson, Aki Vehtari (2020). Leave-one-out cross-validation for Bayesian model comparison in large data. Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), PMLR 108:341-351. Online.
    arXiv preprint arXiv:2001.00980.

  54. Olli-Pekka Koistinen, Vilhjálmur Ásgeirsson, Aki Vehtari, and Hannes Jónsson (2020). Minimum mode saddle point searches using Gaussian process regression with inverse-distance covariance function. Journal of Chemical Theory and Computation, 16(1):499-509. doi:10.1021/acs.jctc.9b01038.
    ChemRxiv preprint 10.26434/chemrxiv.9994868.v1.

  55. Olli-Pekka Koistinen, Vilhjálmur Ásgeirsson, Aki Vehtari, and Hannes Jónsson (2019). Nudged elastic band calculations accelerated with Gaussian process regression based on inverse inter-atomic distances. Journal of Chemical Theory and Computation, 15:6738-6751, doi:10.1021/acs.jctc.9b00692.
    ChemRxiv preprint 10.26434/chemrxiv.8850440.v1.

  56. Måns Magnusson, Michael Riis Andersen, Johan Jonasson, Aki Vehtari (2019). Bayesian leave-one-out cross-validation for large data. Thirty-sixth International Conference on Machine Learning, PMLR 97:4244-4253. Online.
    arXiv preprint arXiv:1904.10679.

  57. Iiris Sundin, Peter Schulam, Eero Siivola, Aki Vehtari, Suchi Saria, and Samuel Kaski (2019). Active learning for decision-making from imbalanced observational data. Thirty-sixth International Conference on Machine Learning, PMLR 97:6046-6055. Online.
    arXiv preprint arXiv:1904.05268.

  58. Lu Cheng, Siddharth Ramchandran, Tommi Vatanen, Niina Lietzen, Riitta Lahesmaa, Aki Vehtari, and Harri Lähdesmäki (2019). LonGP: an additive Gaussian process regression model for longitudinal study designs. Nature Communications, 10:1798. Online.
    bioRxiv preprint.

  59. Topi Paananen, Juho Piironen, Michael Riis Andersen, and Aki Vehtari (2019). Variable selection for Gaussian processes via sensitivity analysis of the posterior predictive distribution. Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS), PMLR 89:1743-1752. Online.
    arXiv preprint arXiv:1712.08048.

  60. Kunal Ghosh, Annika Stuke, Milica Todorović, Peter Bjørn Jørgensen, Mikkel N. Schmidt, Aki Vehtari, and Patrick Rinke (2019). Deep learning spectroscopy: neural networks for molecular excitation spectra. Advanced Science, 6(9):1801367. doi:10.1002/advs.201801367.

  61. Jonah Gabry, Daniel Simpson, Aki Vehtari, Michael Betancourt, and Andrew Gelman (2019). Visualization in Bayesian workflow. Journal of the Royal Statistical Society Series A, 182(2):389-402. Online.
    arXiv preprint arXiv:1709.01449.
    Discussion and rejoinder.

  62. Andrew Gelman, Ben Goodrich, Jonah Gabry, and Aki Vehtari (2019). R-squared for Bayesian regression models. The American Statistician, 73(3):307-309. Online.
    Preprint.
    Online appendix with code.

  63. Marko Järvenpää, Michael U. Gutmann, Arijus Pleska, Aki Vehtari, and Pekka Marttinen (2019). Efficient acquisition rules for model-based approximate Bayesian computation. Bayesian Analysis, 14(2):595-622. Online.
    arXiv preprint arXiv:1704.00520.

  64. Eero Siivola, Aki Vehtari, Jarno Vanhatalo, and Javier González (2018). Correcting boundary over-exploration deficiencies in Bayesian optimization with virtual derivative sign observations. 2018 IEEE International Workshop on Machine Learning for Signal Processing (MLSP), doi:10.1109/MLSP.2018.8516936. Online.
    arXiv preprint arXiv:1704.00963.

  65. Yuling Yao, Aki Vehtari, Daniel Simpson, and Andrew Gelman (2018). Yes, but Did It Work?: Evaluating Variational Inference. Thirty-fifth International Conference on Machine Learning, PMLR 80:5577-5586. Online.
    arXiv preprint arXiv:1802.02538.

  66. Marko Järvenpää, Michael Gutmann, Aki Vehtari and Pekka Marttinen (2018). Gaussian process modeling in approximate Bayesian computation to estimate horizontal gene transfer in bacteria. The Annals of Applied Statistics, 12(4):2228-2251. Online.
    arXiv preprint arXiv:1610.06462.

  67. Juho Piironen, and Aki Vehtari (2018). Iterative supervised principal components. Proceedings of the 21st International Conference on Artificial Intelligence and Statistics, PMLR 84:106-114. Online.
    arXiv preprint arXiv:1710.06229.

  68. Pedram Daee, Tomi Peltola, Aki Vehtari, and Samuel Kaski (2018). User modelling for avoiding overfitting in interactive knowledge elicitation for prediction. Proceedings IUI’18 23rd International Conference on Intelligent User Interfaces, pp. 305-310. Online.
    arXiv preprint arXiv:1710.04881.

  69. Niina Lietzén, Lu Cheng, Robert Moulder, Heli Siljander, Essi Laajala , Taina Härkönen, Aleksandr Peet, Aki Vehtari, Vallo Tillmann, Mikael Knip, and Harri Lähdesmäki (2018). Characterization and non-parametric modeling of the developing serum proteome during early childhood. Scientific Reports, 8:5883. doi:10.1038/s41598-018-24019-5.

  70. Jarmo Rantonen, Jaro Karppinen, Aki Vehtari, Satu Luoto, Eira Viikari-Juntura, Markku Hupli, Antti Malmivaara, and Simo Taimela (2018). Effectiveness of three interventions for secondary prevention of low back pain in the occupational health setting. A randomised controlled trial with a natural course control. BMC Public Health, 18:598. Online.

  71. Jeff Sperinde, Weidong Huang, Aki Vehtari, Ahmed Chenna, Pirkko-Liisa Lehtinen-Kellokumpu, John Winslow, Petri Bono, Yolanda Lie, Christos Petropoulos, Jodi Weidler, and Heikki Joensuu (2018). p95HER2 methionine 611 carboxy-terminal fragment is predictive of trastuzumab adjuvant treatment benefit in the FinHer trial. Clinical Cancer Research, 24(13):3046-3052. doi:10.1158/1078-0432.CCR-17-3250.

  72. Yuling Yao, Aki Vehtari, Daniel Simpson, and Andrew Gelman (2018). Using stacking to average Bayesian predictive distributions (with discussion). Bayesian Analysis, 13(3):917-1003, doi:10.1214/17-BA1091. Online.
    arXiv preprint arXiv:1704.02030.

  73. Sebastian Weber, Andrew Gelman, Daniel Lee, Michael Betancourt, Aki Vehtari, and Amy Racine-Poon (2018). Bayesian aggregation of average data: An application in drug development. Annals of Applied Statistics, 12(3):1583-1604. Online. Preprint. [*]Youden Award in Interlaboratory Testing from the American Statistical Association.

  74. Jarno Lintusaari, Henri Vuollekoski, Antti Kangasrääsiö, Kusti Skytén, Marko Järvenpää, Michael Gutmann, Aki Vehtari, Jukka Corander, and Samuel Kaski (2018). ELFI: Engine for Likelihood Free Inference. Journal of Machine Learning Research, 19(16):1-7. Online. arXiv preprint arXiv:1708.00707.

  75. Juho Piironen and Aki Vehtari (2017). Sparsity information and regularization in the horseshoe and other shrinkage priors. Electronic Journal of Statistics, 11(2):5018-5051. Online. arXiv preprint arXiv:1707.01694. After the article was published, the regularized horseshoe prior has been implemented in rstanarm and brms (brms is not conditioning on sigma).

  76. Michael Riis Andersen, Aki Vehtari, Ole Winther and Lars Kai Hansen (2017). Bayesian inference for spatio-temporal spike and slab priors. Journal of Machine Learning Research, 18(139):1-58, Online. arXiv preprint arXiv:1509.04752.

  77. Olli-Pekka Koistinen, Freyja B. Dagbjartsdóttir, Vilhjálmur Ásgeirsson, Aki Vehtari and Hannes Jónsson (2017). Nudged elastic band calculations accelerated with Gaussian process regression. Journal of Chemical Physics, 147, 152720, doi:10.1063/1.4986787. arXiv preprint arXiv:1706.04606.

  78. Kristiina Santalahti, Aki Havulinna, Mikael Maksimow, Tanja Zeller, Stefan Blankenberg, Aki Vehtari, Heikki Joensuu, Sirpa Jalkanen, Veikko Salomaa, and Marko Salmi (2017). Plasma levels of HGF and PlGF predict mortality in a general population: a prospective cohort study. Journal of Internal Medicine, 282:340–352. doi:10.1111/joim.12648. Online.

  79. Aki Vehtari, Andrew Gelman and Jonah Gabry (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing, 27(5):1413–1432. doi:10.1007/s11222-016-9696-4. Online. arXiv preprint arXiv:1507.04544. Python code. R code.

  80. Juho Piironen and Aki Vehtari (2017). On the hyperprior choice for the global shrinkage parameter in the horseshoe prior. Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:905-913. Online.

  81. Juho Piironen and Aki Vehtari (2017). Comparison of Bayesian predictive methods for model selection. Statistics and Computing, 27(3):711-735. doi:10.1007/s11222-016-9649-y. First Online 07 April 2016. arXiv preprint arXiv:1503.08650.
    Supplement: Juho Piironen and Aki Vehtari (2015). Projection predictive variable selection using Stan+R. arXiv preprint arXiv:1508.02502. Code

  82. Olli-Pekka Pulkka, Bengt Nilsson, Maarit Sarlomo-Rikala, Peter Reichardt, Mikael Eriksson, Kirsten Sundby Hall, Eva Wardelmann, Aki Vehtari, Heikki Joensuu and Harri Sihto (2017). SLUG transcription factor: A pro-survival and prognostic factor in gastrointestinal stromal tumour. British Journal of Cancer, 116:1195–1202. doi:10.1038/bjc.2017.82, advance online publication.

  83. Heikki Joensuu, Eva Wardelmann, Harri Sihto, Mikael Eriksson, Kirsten Sundby Hall, Annette Reichardt, Jörg T. Hartmann, Daniel Pink, Silke Cameron, Peter Hohenberger, Salah-Eddin Al-Batran, Marcus Schlemmer, Sebastian Bauer, Bengt Nilsson, Raija Kallio, Jouni Junnila, Aki Vehtari and Peter Reichardt (2017). Effect of KIT and PDGFRA Mutations on Survival in Patients With Gastrointestinal Stromal Tumor Treated With Adjuvant Imatinib: An Analysis of a Randomized Trial. JAMA Oncology, 3(5):602–609. doi:10.1001/jamaoncol.2016.5751. Online.

  84. Dmitry Smirnov, Fanny Lachat, Tomi Peltola, Juha M. Lahnakoski, Olli-Pekka Koistinen, Enrico Glerean, Aki Vehtari, Riitta Hari, Mikko Sams and Lauri Nummenmaa (2017). Brain-to-brain hyperclassification reveals action-specific motor mapping of observed actions in humans. PLOS ONE, doi:10.1371/journal.pone.0189508. Online.

  85. Olli-Pekka Koistinen, Emile Maras, Aki Vehtari and Hannes Jónsson (2016). Minimum energy path calculations with Gaussian process regression. Nanosystems: Physics, Chemistry, Mathematics, 7(6):925–935. Online.

  86. Juha Salmi, Olli-Pekka Koistinen, Enrico Glerean, Pasi Jylänki, Aki Vehtari, Iiro P. Jääskeläinen, Sasu Mäkelä, Lauri Nummenmaa, Katarina Nummi-Kuisma, Ilari Nummi and Mikko Sams (2016). Distributed neural signatures of natural audiovisual speech and music in the human auditory cortex. NeuroImage, 157:108-117. doi:10.1016/j.neuroimage.2016.12.005. Online. Preprint.

  87. Santosh Tirunagari, Simon C Bull, Aki Vehtari, Christopher Farmer, Simon de Lusignan and Norman Poh (2016). Automatic Detection of Acute Kidney Injury Episodes from Primary Care Data. 2016 IEEE Symposium Series on Computational Intelligence (SSCI), doi:10.1109/SSCI.2016.7849885. Online.

  88. Juho Piironen and Aki Vehtari (2016). Projection predictive input variable selection for Gaussian process models. 2016 IEEE 26th International Workshop on Machine Learning for Signal Processing (MLSP), doi:10.1109/MLSP.2016.7738829. arXiv preprint arXiv:1510.04813. Online.

  89. Aki Vehtari, Tommi Mononen, Ville Tolvanen, Tuomas Sivula and Ole Winther (2016). Bayesian leave-one-out cross-validation approximations for Gaussian latent variable models. Journal of Machine Learning Research, 17(103):1−38. Online. arXiv preprint arXiv:1412.7461.
    Errata: In Equation (34) “-1/2 log cbar_ii” -> “+1/2 log cbar_ii”

  90. Alan Saul, James Hensman, Aki Vehtari, Neil Lawrence (2016). Chained Gaussian processes. Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS) PMLR 51:1431-1440. Online. arXiv preprint arXiv:1604.05263. Code

  91. Jarmo Rantonen, Jaro Karppinen, Aki Vehtari, Satu Luoto, Eira Viikari-Juntura, Markku Hupli, Antti Malmivaara and Simo Taimela (2016). Cost-effectiveness of providing patients with information on managing mild low-back symptoms. A controlled trial in an occupational health setting. BMC Public Health, 16:316, doi:10.1186/s12889-016-2974-4. Online.

  92. Dario Gasbarra, Elja Arjas, Aki Vehtari, Rémy Slama and Niels Keiding (2015). The current duration design for estimating the time to pregnancy distribution: a nonparametric Bayesian perspective. Lifetime Data Analysis, 21(4):594–625. DOI:10.1007/s10985-015-9333-0. Online. Preprint.

  93. Ville Tolvanen, Pasi Jylänki and Aki Vehtari (2014). Expectation propagation for nonstationary heteroscedastic Gaussian process regression. Machine Learning for Signal Processing (MLSP), 2014 IEEE International Workshop on, doi:10.1109/MLSP.2014.6958906. Online. Preprint. Code available in GPstuff toolbox.

  94. Tomi Peltola, Aki S. Havulinna, Veikko Salomaa and Aki Vehtari (2014). Hierarchical Bayesian survival analysis and projective covariate selection in cardiovascular event risk prediction. In Laskey, K. B., Jones, J. and Almond, R. (eds.) Proceedings of Eleventh UAI Bayesian Modeling Applications Workshop (BMAW 2014), CEUR Workshop Proceedings Vol-1218, 79-88. Preprint Online.

  95. Jaakko Riihimäki and Aki Vehtari (2014). Laplace approximation for logistic Gaussian process density estimation and regression. Bayesian Analysis, 9(2):425-448. Online 3 February, 2014. Code available in GPstuff toolbox.

  96. Tomi Peltola, Pasi Jylänki and Aki Vehtari (2014). Expectation propagation for likelihoods depending on an inner product of two multivariate random variables. Proceedings of the 17th International Conference on Artificial Intelligence and Statistics (AISTATS) PMLR 33:769-777. Online.

  97. Heikki Joensuu, Peter Reichardt, Mikael Eriksson, Kirsten Sundby Hall and Aki Vehtari (2014). Gastrointestinal stromal tumor: A method for optimizing the timing of CT scans in the follow-up of cancer patients. Radiology, 271(1):96-106. Online 18 November, 2013. PDF. Appendix PDF. Preprint of the statistical appendix. Related poster presented at The Third Workshop on Bayesian Inference for Latent Gaussian Models with Applications. [*]Highlighted in “This Month in Radiology”.

  98. Pasi Jylänki, Aapo Nummenmaa and Aki Vehtari (2014). Expectation propagation for neural networks with sparsity-promoting priors. Journal of Machine Learning Research, 15(May):1849-1901. Online.

  99. Aki Vehtari, Karita Reijonsaari, Olli-Pekka Kahilakoski, Markus V. Paananen, Willem van Mechelen and Simo Taimela (2014). The influence of selective participation in a physical activity intervention on the generalizability of findings. Journal of Occupational and Environmental Medicine, 56(3):291–297. Online 13 January 2014

  100. Jarmo Rantonen, Aki Vehtari, Jaro Karppinen, Satu Luoto, Eira Viikari-Juntura, Markku Hupli, Antti Malmivaara and Simo Taimela (2014). Face-to-face information in addition to a booklet versus a booklet alone for treating mild back pain, a randomized controlled trial. Scandinavian journal of Work Environment & Health, 40(2):156-166. Online 2 November, 2013. PDF.

  101. Mari Myllymäki, Aila Särkkä and Aki Vehtari (2014). Hierarchical second-order analysis of replicated spatial point patterns with non-spatial covariates. Spatial Statistics, 8:104-121. Online 13 August, 2013. Preprint.

  102. Andrew Gelman, Jessica Hwang and Aki Vehtari (2014). Understanding predictive information criteria for Bayesian models. Statistics and Computing, 24(6):997-1016. Online 20 August, 2013. Preprint.

  103. Jarno Vanhatalo, Jaakko Riihimäki, Jouni Hartikainen, Pasi Jylänki, Ville Tolvanen and Aki Vehtari (2013). GPstuff: Bayesian modeling with Gaussian processes. Journal of Machine Learning Research, 14(Apr):1175-1179. Online. Software homepage.

  104. Jaakko Riihimäki, Pasi Jylänki and Aki Vehtari (2013). Nested expectation propagation for Gaussian process classification with a multinomial probit likelihood. Journal of Machine Learning Research, 14(Jan):75-109. Online. Code available in GPstuff toolbox.

  105. Aki Vehtari and Janne Ojanen (2012). A survey of Bayesian predictive methods for model assessment, selection and comparison. Statistics Surveys, 6:142-228. Online. Errata was published in Statistics Surveys, 8 (2014), 1-1.

  106. Tomi Peltola, Pekka Marttinen and Aki Vehtari (2012). Finite adaptation and multistep moves in the Metropolis-Hastings algorithm for variable selection in genome-wide association analysis. PLoS One, 7(11):e49445. Online.

  107. Karita Reijonsaari, Aki Vehtari, Olli-Pekka Kahilakoski, Willem van Mechelen, Timo Aro and Simo Taimela (2012). The effectiveness of physical activity monitoring and distance counseling in an occupational setting - Results from a randomized controlled trial (CoAct). BMC Public Health, 12:344 (11 May 2012). Online.

  108. Heikki Joensuu, Mikael Eriksson, Kirsten Sundby Hall, Jörg T. Hartmann, Daniel Pink, Jochen Schütte, Giuliano Ramadori, Peter Hohenberger, Justus Duyster, Salah-Eddin Al-Batran, Marcus Schlemmer, Sebastian Bauer, Eva Wardelmann, Maarit Sarlomo-Rikala, Bengt Nilsson, Harri Sihto, Odd R. Monge, Petri Bono, Raija Kallio, Aki Vehtari, Mika Leinonen, Thor Alvegård and Peter Reichardt (2012). One vs three years of adjuvant imatinib for operable gastrointestinal stromal tumor: A randomized trial. The Journal of American Medical Association, 307(12):1265-1272. Online. [*]Featured article. [*]Top 10 Journal Watch Oncology and Hematology story.

  109. Heikki Joensuu, Aki Vehtari, Jaakko Riihimäki, Toshirou Nishida, Sonja E Steigen, Peter Brabec, Lukas Plank, Bengt Nilsson, Claudia Cirilli, Chiara Braconi, Andrea Bordoni, Magnus K Magnusson, Zdenek Linke, Jozef Sufliarsky, Federico Massimo, Jon G Jonasson, Angelo Paolo Dei Tos and Piotr Rutkowski (2012). Risk of gastrointestinal stromal tumour recurrence after surgery: an analysis of pooled population-based cohorts. The Lancet Oncology, 13(3):265-274. Published Online: 07 December 2011. Statistical appendix. [*]Commented in editorial.

  110. Simo Särkkä, Arno Solin, Aapo Nummenmaa, Aki Vehtari, Toni Auranen, Simo Vanni and Fa-Hsuan Lin (2012). Dynamic Retrospective Filtering of Physiological Noise in BOLD fMRI: DRIFTER. NeuroImage, 60(2):1517-1527. Online.

  111. Tomi Peltola, Pekka Marttinen, Antti Jula, Veikko Salomaa, Markus Perola and Aki Vehtari (2012). Bayesian variable selection in searching for additive and dominant effects in genome-wide data. PLoS ONE, 7(1):e29115. Online.

  112. Jarmo Rantonen, Satu Luoto, Aki Vehtari, Markku Hupli, Jaro Karppinen, Antti Malmivaara and Simo Taimela (2012). The effectiveness of two active interventions compared to self-care advice in employees with non-acute low back symptoms. A randomised, controlled trial with a 4-year follow-up in the occupational health setting. Occupational and Environmental Medicine, 69(1):12-20. Online. PDF. Statistical Appendix.. [*]Editor’s choice.

  113. Pasi Jylänki, Jarno Vanhatalo and Aki Vehtari (2011). Robust Gaussian process regression with a Student-\(t\) likelihood. Journal of Machine Learning Research, 12(Nov):3227-3257. Online. Code available in GPstuff toolbox.

  114. Jarno Vanhatalo, Pia Mäkelä and Aki Vehtari (2010). Alkoholikuolleisuuden alueelliset erot Suomessa 2000-luvun alussa. Yhteiskuntapolitiikka, 75(3):265-273. Online in Finnish. English translation: Regional differences in alcohol mortality in Finland in the early 2000s.

  115. Jaakko Riihimäki and Aki Vehtari (2010). Gaussian processes with monotonicity information. Proceedings of the 13th International Conference on Artificial Intelligence and Statistics (AISTATS), PMLR 9:645-652. Online. Code available in GPstuff toolbox.

  116. Jarno Vanhatalo, Ville Pietiläinen and Aki Vehtari (2010). Approximate inference for disease mapping with sparse Gaussian processes. Statistics in Medicine, 29(15):1580-1607. Online.

  117. Jarno Vanhatalo and Aki Vehtari (2010). Speeding up the binary Gaussian process classification. In P. Grünwald and P. Spirtes, editors, Proceedings of the 26th Conference on Uncertainty in Artificial Intelligence (UAI 2010), pp. 623-632, AUAI Press. PDF.

  118. Elina Parviainen and Aki Vehtari (2010). Explaining classification by finding response-related subgroups in data. In Ma, J. et al, eds., Proceedings of the 11th ACIS International Conference on Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed Computing, SNPD2010, pp. 69-75, IEEE Computer Society.

  119. Karita Reijonsaari, Aki Vehtari, Willem Van Mechelen, Timo Aro and Simo Taimela (2009). The effectiveness of physical activity monitoring and distance counselling in an occupational health setting - a research protocol for a randomised controlled trial (CoAct). BMC Public Health, 9:494. Online.

  120. Jaakko Riihimäki, Reijo Sund and Aki Vehtari (2009). Analysing the length of care episode after hip fracture: a nonparametric and a parametric Bayesian approach. Health Care Management Science, 10.1007/s10729-009-9121-z. Online 13 November 2009.

  121. Jarno Vanhatalo, Pasi Jylänki and Aki Vehtari (2009). Gaussian process regression with Student-\(t\) likelihood. In Y. Bengio et al, editors, Advances in Neural Information Processing Systems 22, pp. 1910-1918, NIPS Foundation. Online.

  122. Petri Korhonen, Terhi Husa, Teijo Konttila, Ilkka Tierala, Markku Mäkijärvi, Heikki Väänänen, Janne Ojanen, Aki Vehtari and Lauri Toivonen (2009). Fragmented QRS in prediction of cardiac deaths and heart failure hospitalizations after myocardial infarction. Annals of Noninvasive Electrocardiology, 15(2):130–137.

  123. Reijo Sund, Jaakko Riihimäki, Matti Mäkelä, Aki Vehtari, Peter Lüthje, Tiina Huusko and Unto Häkkinen (2009). Modeling the length of care episode after hip fracture: does the type of fracture matter? Scandinavian Journal of Surgery, 98(3):169-174.

  124. Elina Parviainen and Aki Vehtari (2009). Features and metric from a classifier improve visualizations with dimension reduction. In Alippi et al, eds. Artificial Neural Networks - ICANN 2009, part II, pp. 225–234, Springer.

  125. Toni Auranen, Aapo Nummenmaa, Simo Vanni, Aki Vehtari, Matti S. Hämäläinen, Jouko Lampinen and Iiro P. Jääskeläinen (2009). Automatic fMRI-guided MEG multidipole localization for visual responses. Human Brain Mapping, 30(4):1087-1099. Online 8 May 2008.

  126. Jarno Vanhatalo and Aki Vehtari (2008). Modelling local and global phenomena with sparse Gaussian processes. In David McAllester and Petri Myllymäki, editors, Proceedings of the 24th Conference on Uncertainty in Artificial Intelligence (UAI 2008), pp. 571-578, AUAI Press. Online.

  127. Taru Tukiainen, Tuulia Tynkkynen, Ville-Petteri Mäkinen, Pasi Jylänki, Antti Kangas, Johanna Hokkanen, Aki Vehtari, Olli Gröhn, Merja Hallikainen, Hilkka Soininen, Miia Kivipelto, Per-Henrik Groop, Kimmo Kaski, Reino Laatikainen, Pasi Soininen, Tuula Pirttilä and Mika Ala-Korpela (2008). A multi-metabolite analysis of serum by 1H NMR spectroscopy: early systemic signs of Alzheimer’s disease. Biochemical and Biophysical Research Communications, 375(3):356-61.

  128. Aki Vehtari, Ville-Petteri Mäkinen, Pasi Soininen, Petri Ingman, Sanna M. Mäkelä, Markku J. Savolainen, Minna L. Hannuksela, Kimmo Kaski and Mika Ala-Korpela (2007). A novel Bayesian approach to quantify clinical variables and to determine their spectroscopic counterparts in 1H NMR metabonomic data. BMC Bioinformatics, 8(Suppl 2):S8. Online.

  129. Jarno Vanhatalo and Aki Vehtari (2007). Sparse log Gaussian processes via MCMC for spatial epidemiology. Gaussian Processes in Practice, PMLR 1:73-89. Abstract, PDF.

  130. Marko Tapani Sysi-Aho, Aki Vehtari, Vidya Velagapudi, Jukka Westerbacka, Laxman Yetukuri, Robert Bergholm, Marja-Riitta Taskinen, Hannele Yki-Järvinen and Matej Oresic (2007). Exploring the lipoprotein composition using Bayesian regression on serum lipidomic profiles. Bioinformatics, 23(13):i519-i528. Online.

  131. Aapo Nummenmaa, Toni Auranen, Matti S Hämäläinen, Iiro P Jääskeläinen, Mikko Sams, Aki Vehtari and Jouko Lampinen (2007). Automatic relevance-determination based hierarchical Bayesian MEG inversion in practice. NeuroImage, 37(3):876-889. Online.

  132. Toni Auranen, Aapo Nummenmaa, Matti S. Hämäläinen, Iiro P. Jääskeläinen, Jouko Lampinen, Aki Vehtari and Mikko Sams (2007). Bayesian inverse analysis of neuromagnetic data using cortically constrained multiple dipoles. Human Brain Mapping, 28(10):979-994. Online.

  133. Aapo Nummenmaa, Toni Auranen, Matti S. Hämäläinen, Iiro P. Jääskeläinen, Jouko Lampinen, Mikko Sams and Aki Vehtari (2007). Hierarchical Bayesian estimates of distributed MEG sources: theoretical aspects and comparison of variational and MCMC methods. NeuroImage, 35(2):669-685. Online.

  134. Simo Särkkä, Aki Vehtari and Jouko Lampinen (2007). CATS benchmark time series prediction by Kalman smoother with cross-validated noise density. Neurocomputing, 70(13-15):2331-2341. Online.

  135. Simo Särkkä, Aki Vehtari and Jouko Lampinen (2007). Rao-Blackwellized Particle Filter for Multiple Target Tracking. Information Fusion, 8(1):2-15. Online.

  136. Simo Särkkä, Aki Vehtari and Jouko Lampinen (2007). Prediction of ESTSP Competition Time Series by Unscented Kalman Filter and RTS Smoother. In Amaury Lendasse, editor, Proceedings of European Symposium on Time Series Prediction (ESTSP’07), pp.1-10.

  137. Toni Auranen, Aapo Nummenmaa, Matti S. Hämäläinen, Iiro P. Jääskeläinen, Jouko Lampinen, Aki Vehtari and Mikko Sams (2005). Bayesian analysis of the neuromagnetic inverse problem with \(L^p\)-norm priors. NeuroImage, 26(3):870-884.

  138. Ilkka Kalliomäki, Aki Vehtari and Jouko Lampinen (2005). Shape analysis of concrete aggregates for statistical quality modeling. Machine Vision and Applications, 16(3):197-201. PDF.

  139. Simo Särkkä, Aki Vehtari and Jouko Lampinen (2004). Time series prediction by Kalman smoother with cross-validated noise density. IJCNN’2004: Proceedings of the 2004 International Joint Conference on Neural Networks, Budabest, July 2004. [*] The Winner of Time Series Prediction Competition - The CATS Benchmark.

  140. Simo Särkkä, Aki Vehtari and Jouko Lampinen (2004). Rao-Blackwellized Monte Carlo data association for multiple target tracking. In Per Svensson and Johan Schubert, editors, Proceedings of the Seventh International Conference on Information Fusion, volume I, pp. 583-590.

  141. Aki Vehtari and Jouko Lampinen (2003). Expected utility estimation via cross-validation. In J. M. Bernardo, et al., editors, Bayesian Statistics 7, pp. 701-710. Oxford University Press. PDF.

  142. Aki Vehtari and Jouko Lampinen (2002). Bayesian model assessment and comparison using cross-validation predictive densities. Neural Computation, 14(10):2439-2468. Online. Preprint.

  143. Jouko Lampinen and Aki Vehtari (2001). Bayesian approach for neural networks - review and case studies. Neural Networks, 14(3):7-24. (Invited article. Note that unfortunately the paper version has some printer’s errors). Online. Preprint (without printer’s errors).

  144. Jouko Lampinen and Aki Vehtari (2001). Bayesian techniques for neural networks - review and case studies. In K. Wang, J Grundespenkis and A. Yerofeyev, editors, Applied Computational Intelligence to Engineering and Business, pp. 7-15.

  145. Aki Vehtari and Jouko Lampinen (2000). Bayesian MLP neural networks for image analysis. Pattern Recognition Letters, 21(13-14):1183-1191. (Special Issue - Selected Papers from The 11th Scandinavian Conference on Image Analysis). PDF.

  146. Jouko Lampinen and Aki Vehtari (2000) Bayesian techniques for neural networks - review and case studies. In M. Gabbouj and P. Kuosmanen, editors, Eusipco’2000: Proceedings of the X European Signal Processing Conference, volume 2, pp. 713-720. PDF.

  147. Aki Vehtari, Simo Särkkä and Jouko Lampinen (2000). On MCMC sampling in Bayesian MLP neural networks. In Shun-Ichi Amari, C. Lee Giles, Marco Gori and Vincenzo Piuri, editors, IJCNN’2000: Proceedings of the 2000 International Joint Conference on Neural Networks, volume I, pp. 317-322. IEEE. PDF.

  148. Aki Vehtari and Jouko Lampinen (2000). Bayesian MLP neural networks - review and case studies. In Leena Yliniemi and Esko Juuso, editors, TOOLMET2000: Proceedings of the Tool Environments and Development Methods for Intelligent Systems, pp. 120-133. Oulun Yliopistopaino.

  149. Aki Vehtari and Jouko Lampinen (2000). Bayesian neural networks: Case studies in industrial applications. In Y. Suzuki, R. Roy, S. J. Ovaska, T. Furuhashi and Y. Dote, editors, Soft Computing in Industrial Applications, pp. 411-420. Springer-Verlag.

  150. Aki Vehtari and Jouko Lampinen (1999). Bayesian neural networks with correlating residuals. IJCNN’99: Proceedings of the 1999 International Joint Conference on Neural Networks [CD-ROM], paper number 2061. IEEE. PDF.

  151. Jouko Lampinen, Aki Vehtari and Kimmo Leinonen (1999). Application of Bayesian neural network in electrical impedance tomography. IJCNN’99: Proceedings of the 1999 International Joint Conference on Neural Networks [CD-ROM], paper number 375. PDF.

  152. Aki Vehtari and Jouko Lampinen (1999). Bayesian neural networks for industrial applications. SMCIA/99: Proceedings of the 1999 IEEE Midnight-Sun Workshop on Soft Computing Methods in Industrial Applications, pp. 63-68. PDF

  153. Aki Vehtari and Jouko Lampinen (1999). Bayesian neural networks for image analysis. In B. K. Ersboll and P. Johansen, editors, SCIA’99: Proceedings of The 11th Scandinavian Conference on Image Analysis, volume 1, pages 95-102. The Pattern Recognition Society of Denmark. PDF.

  154. Jouko Lampinen, Aki Vehtari and Kimmo Leinonen (1999). Using Bayesian neural network to solve the inverse problem in electrical impedance tomography. In B. K. Ersboll and P. Johansen, editors, SCIA’99: Proceedings of the 11th Scandinavian Conference on Image Analysis, volume 1, pages 87-93. The Pattern Recognition Society of Denmark. PDF.

  155. Jukka Heikkonen, Jari Varjo and Aki Vehtari (1999). Forest change detection via Landsat TM difference features. SCIA’99: Proceedings of the 11th Scandinavian Conference on Image Analysis, volume 1, pages 157-164. The Pattern Recognition Society of Denmark.

  156. Aki Vehtari, Jukka Heikkonen, Jouko Lampinen and Jouni Juujärvi (1998). Using Bayesian neural networks to classify forest scenes. In David P. Casasent, editor, Intelligent Robots and Computer Vision XVII: Algorithms, Techniques and Active Vision, volume 3522 of Proceedings of SPIE, pp. 66-73. SPIE.

  157. Aki Vehtari, Jouni Juujärvi, Jukka Heikkonen and Jouko Lampinen (1998). Forest scene classification: Comparison of classifiers. Proceedings of STeP’98, pp. 152-160.

Non-refereed articles

  1. Michael Riis Andersen, Eero Siivola, Gabriel Riutort-Mayol, and Aki Vehtari (2018). A non-parametric probabilistic model for monotonic functions. BNP@NeurIPS 2018 workshop: “All of Bayesian Nonparametrics (Especially the Useful Bits)”. Online.

  2. Måns Magnusson, Michael Riis Andersen, and Aki Vehtari (2018). Bayesian leave-one-out cross-validation for large data sets. Symposium on Advances in Approximate Bayesian Inference, Montréal, Canada. Online.

  3. Aki Vehtari, Daniel Simpson, Yuling Yao, and Andrew Gelman (2018). Limitations of “Limitations of Bayesian leave-one-out cross-validation for model selection” (invited discussion). Computational Brain & Behavior, 2(1):22–27. Online. arXiv preprint.

  4. Juho Piironen, Michael Betancourt, Daniel Simpson, and Aki Vehtari (2017). Contributed comment on ‘Uncertainty Quantification for the Horseshoe’ by Stéphanie van der Pas, Botond Szabó, and Aad van der Vaart. Bayesian Analysis, 12(4):1264. Online. Preprint.

  5. Andrew Gelman and Aki Vehtari (2017). Consensus Monte Carlo using expectation propagation (Discussion to ‘Comparing Consensus Monte Carlo Strategies for Distributed Bayesian Computation’ by Steve Scott.) Brazilian Journal of Probability and Statistics, 31(4):692-696. Online. Preprint.

  6. Rob Trangucci, Michael Betancourt, and Aki Vehtari (2016). Prior Formulation for Gaussian Process Hyperparameters. In Practical Bayesian Nonparameterics Workshop - NeurIPS. Online.

  7. Andrew Gelman and Aki Vehtari (2014). Discussion to ‘Estimation and Accuracy after Model Selection’ by Bradley Efron. Journal of the American Statistical Association, 109(507):1015-1016. Online. Preprint.

  8. Aki Vehtari and Janne Ojanen (2012). Discussion to ‘Catching up faster by switching sooner: a predictive approach to adaptive estimation with an application to the AIC-BIC dilemma’ by Tim van Erven , Peter Grünwald and Steven de Rooij. Journal of the Royal Statistical Society, Series B (Statistical Methodology), 74(3):411-412. Available online 12 April 2012)

  9. Aki Vehtari and Jarno Vanhatalo (2011). Discussion to ‘Riemann manifold Langevin and Hamiltonian Monte Carlo methods’ by Mark Girolami and Ben Calderhead. Journal of the Royal Statistical Society, Series B (Statistical Methodology), 73(2):201. Available online 3 March 2011.

  10. Jarno Vanhatalo and Aki Vehtari (2009). Discussion to ‘Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations’ by Håvard Rue, Sara Martino and Nicolas Chopin. Journal of the Royal Statistical Society, Series B (Statistical Methodology)., 71(2):383. Available online 6 April 2009.

  11. Aki Vehtari (2007). Discussion to `Some Aspects of Bayesian Model Selection for Prediction’ by Chakrabarti, A. and Ghosh, J. K.. In J. M. Bernardo, et al., editors, Bayesian Statistics 8, p. 83-84. Oxford University Press.

  12. Aki Vehtari (2003). Discussion to `Hierarchical multivariate CAR models for spatio-temporally correlated survival data’ by Carlin B. P. and Banerjee, S. In J. M. Bernardo, et al., editors, Bayesian Statistics 7, p. 61. Oxford University Press. PDF.

  13. Aki Vehtari (2003). Discussion to `Bayesian Treed Generalized Linear Models’ by Chipman, H. A., George, E. I. and McCulloch R. E. In J. M. Bernardo, et al., editors, Bayesian Statistics 7, p. 101. Oxford University Press. PDF.

  14. Aki Vehtari (2002). Discussion to `Bayesian measures of model complexity and fit’ by Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. Journal of the Royal Statistical Society, Series B (Statistical Methodology), 64(4):620. PDF.

  15. Jouko Lampinen and Aki Vehtari (2002). Bayesilaiset menetelmät hahmontunnistuksessa (in Finnish). In J. Iivarinen, S. Kaski and E. Oja, editors, Neljännesvuosisata Hatutusta: Hahmontunnistustutkimus Suomessa 1977-2002, pp. 86-96. Suomen hahmontunnistustutkimuksen seura ry.

Software

  1. Stan Development Team (2024). Stan. mc-stan.org.

  2. ArviZ Team (2024). ArviZ. www.arviz.org.

  3. Aki Vehtari, Jonah Gabry, Måns Magnusson, Yuling Yao, Paul-Christian Bürkner, Topi Paananen, and Andrew Gelman (2024). loo: Efficient Leave-One-Out Cross-Validation and WAIC for Bayesian Models. mc-stan.org/loo/. doi:10.32614/CRAN.package.loo.

  4. Juho Piironen, Markus Paasiniemi, Alejandro Catalina, Frank Weber, and Aki Vehtari (2024). projpred: Projection Predictive Feature Selection. mc-stan.org/projpred/. doi:10.32614/CRAN.package.projpred

  5. Paul-Christian Bürkner, Jonah Gabry, Matthew Kay, and Aki Vehtari (2024). posterior: Tools for Working with Posterior Distributions. mc-stan.org/posterior/. doi:10.32614/CRAN.package.posterior

  6. Måns Magnusson, Paul-Christian Bürkner, and Aki Vehtari (2024). posteriordb: a set of posteriors for Bayesian inference and probabilistic programming. github.com/stan-dev/posteriordb.

  7. Noa Kallioinen, Paul-Christian Bürkner, Topi Paananen, Frank Weber, and Aki Vehtari (2024). priorsense: Detecting and diagnosing prior and likelihood sensitivity with power-scaling. github.com/n-kall/priorsense.

Reports

  1. Aki Vehtari (2021). On Reparameterization Invariant Bayesian Point Estimates and Credible Regions. arXiv preprint arXiv:2109.10843.

  2. Topi Paananen, Alejandro Catalina, Paul-Christian Bürkner, and Aki Vehtari (2020). Group heterogeneity assessment for multilevel models. arXiv preprint arXiv:2005.02773.

  3. Sean Talts, Michael Betancourt, Daniel Simpson, Aki Vehtari, and Andrew Gelman (2020). Validating Bayesian inference algorithms with simulation-based calibration. arXiv:1804.06788.

  4. Gabriel Riutort-Mayol, Michael Riis Andersen, Aki Vehtari, and José Luis Lerma (2019). Gaussian process with derivative information for the analysis of the sunlight adverse effects on color of rock art paintings. arXiv:1911.03454.

  5. Ben Bales, Arya Pourzanjani, Aki Vehtari, and Linda Petzold (2019). Selecting the Metric in Hamiltonian Monte Carlo. arXiv:1905.11916.

  6. Donald R. Williams, Juho Piironen, Aki Vehtari, and Philippe Rast (2018). Bayesian estimation of Gaussian graphical models with predictive covariance selection. arXiv:1801.05725.

  7. Eero Siivola, Juho Piironen, and Aki Vehtari (2016). Automatic monotonicity detection for Gaussian Processes. arXiv:1610.05440.

  8. Sebastian Weber, Andrew Gelman, Bob Carpenter, Daniel Lee, Michael Betancourt, Aki Vehtari, Amy Racine (2016). Hierarchical expectation propagation for Bayesian aggregation of average data. arXiv:1602.02055.

  9. Seth Flaxman, Andrew Gelman, Daniel Neill, Alex Smola, Aki Vehtari and Andrew Gordon Wilson (2015), Fast hierarchical Gaussian processes. Working paper.

  10. Andrew Gelman, Aki Vehtari, Pasi Jylänki, Christian Robert, Nicolas Chopin and John P. Cunningham (2014). Expectation propagation as a way of life. arXiv:1412.4869v1.

  11. Pasi Jylänki, Aapo Nummenmaa, Aki Vehtari (2013). Expectation Propagation for Neural Networks with Sparsity-promoting Priors. arXiv:1303.6938.

  12. Jaakko Riihimäki and Aki Vehtari (2012). Laplace approximation for logistic Gaussian process density estimation. arXiv:1211.0174.

  13. Jarno Vanhatalo, Jaakko Riihimäki, Jouni Hartikainen, Pasi Jylänki, Ville Tolvanen, Aki Vehtari (2012-2013). Bayesian Modeling with Gaussian Processes using the GPstuff Toolbox. arXiv:1206.5754.

  14. Jarno Vanhatalo, Pia Mäkelä, ja Aki Vehtari (2010). Regional differences in alcohol mortality in Finland in the early 2000s. Report A20, Department of Biomedical Engineering and Computational Science Publications, Helsinki University of Technology. PDF.

  15. Aki Vehtari and Jouko Lampinen (2004). Model Selection via Predictive Explanatory Power. Report B38, Laboratory of Computational Engineering, Helsinki University of Technology. PDF.

  16. Simo Särkkä, Toni Tamminen, Aki Vehtari and Jouko Lampinen (2004). Probabilistic methods in multiple target tracking - Review and bibliography. Report B36, Laboratory of Computational Engineering, Helsinki University of Technology. PDF.

  17. Aki Vehtari and Jouko Lampinen (2002). Bayesian input variable selection using posterior probabilities and expected utilities. Report B31, Laboratory of Computational Engineering, Helsinki University of Technology. (Revised version of Report B28). PDF.

  18. Aki Vehtari and Jouko Lampinen (2001). Bayesian input variable selection using cross-validation predictive densities and reversible jump MCMC. Report B28, Laboratory of Computational Engineering, Helsinki University of Technology. (Superseded by Aki Vehtari and Jouko Lampinen (2002). Bayesian Input Variable Selection Using Posterior Probabilities and Expected Utilities. Report B31, Laboratory of Computational Engineering, Helsinki University of Technology.)

  19. Aki Vehtari and Jouko Lampinen (2001). Bayesian model assesment and comparison using cross-validation predictive densities. Report B27, Laboratory of Computational Engineering, Helsinki University of Technology. (Revised version of Report B23). (Superseded by Aki Vehtari and Jouko Lampinen (2002). Bayesian model assessment and comparison using cross-validation predictive densities. Neural Computation, 14(10):2439-2468.)

  20. Aki Vehtari and Jouko Lampinen (2001). On Bayesian model assesment and choice using cross-validation predictive densities. Report B23, Laboratory of Computational Engineering, Helsinki University of Technology. (Superseded by Aki Vehtari and Jouko Lampinen (2002). Bayesian model assessment and comparison using cross-validation predictive densities. Neural Computation, 14(10):2439-2468.) Appendix in PDF.

Theses

  1. Aki Vehtari (2001). Bayesian model assessment and selection using expected utilities. Dissertation for the degree of Doctor of Science in Technology, Helsinki University of Technology. Abstract, PDF, Väitöstiedote.
    [*] Dissertation was awarded: The best doctoral dissertation award in the field of pattern recognition in 2000-2001 in Finland issued by the Pattern Recognition Society of Finland.

  2. Aki Vehtari (1997). Pumppausprosessin neuroverkkomallinnus (Neural network modelling of pumping process). Master’s thesis, Helsinki University of Technology.