Piotr Zwiernik >



  • Total positivity in multivariate extremes (with Frank Röttger and Sebastian Engelke)
  • Maximum Likelihood Estimation for Brownian Motion Tree Models Based on One Sample (with Michael Truell, Jan-Christian Hütter, Chandler Squires, and Caroline Uhler)
  • Robust Estimation of Tree Structured Models (with Marta Casanellas and Marina Garrote-López).
  • Likelihood geometry of correlation models (with Carlos Améndola). Le Matematiche (special issue on Linear Spaces of Symmetric Matrices), 76(2), 2021, 559-583.
  • Locally associated graphical models and mixed convex exponential families (with Steffen Lauritzen).
  • Dependence in elliptical partial correlation graphs (with David Rossell). Electronic Journal of Statistics, Vol. 15, No. 2, 2021, 4236-4263.
  • Estimating linear covariance models with numerical nonlinear algebra (with Bernd Sturmfels and Sascha Timme). Algebraic Statistics, 11(1), 2020, 31-52.
  • Secant varieties of toric varieties arising from simplicial complexes (with M. Azeem Khadam and Mateusz Michałek). Linear Algebra and its Applications, 588, 2020, 428-457.
  • The identification problem for linear rational expectations models (with Majid Al-Sadoon).
  • Learning partial correlation graphs and graphical models by covariance queries (with Gábor Lugosi, Jakub Truszkowski, and Vasiliki Velona). Journal of Machine Learning Research, 22(203):1−41, 2021.
  • Total positivity in exponential families with application to binary variables (with Steffen Lauritzen and Caroline Uhler). Annals of Statistics 2021, Vol. 49, No. 3, 1436-1459.
  • Brownian motion tree models are toric (with Bernd Sturmfels and Caroline Uhler). Kybernetika, Vol. 56 (special volume in memory of František Matúš), No. 6, 2020, 1154-1175.
  • Maximum likelihood estimation in Gaussian models under total positivity (with Steffen Lauritzen and Caroline Uhler). Annals of Statistics, Vol. 47, No. 4, 2019, 1835--1863.
  • Latent tree models, in Handbook of Graphical Models (Editors: Mathias Drton, Steffen Lauritzen, Marloes Maathuis, Martin Wainwright), CRC Press, 2019.
  • Maximum likelihood estimation of the Latent Class Model through model boundary decomposition (with Elizabeth Allman, Hector Baños Cervantes, Robin Evans, Serkan Hoşten, Kaie Kubjas, Daniel Lemke, and John Rhodes). Journal of Algebraic Statistics: Issue in honor of S. E. Fienberg, 10(1), 2019, 51-84.
  • Maximum likelihood estimation for linear Gaussian covariance models (with Caroline Uhler, Donald Richards), Journal of the Royal Statistical Society Series B 79(4), 2017, 1269–1292.
  • Total positivity in Markov structures (with Shaun Fallat, Steffen Lauritzen, Kayvan Sadeghi, Caroline Uhler, Nanny Wermuth), Annals of Statistics 45(3), 2017, 1152-1184.
  • The correlation space of Gaussian latent tree models and model selection without fitting (with John Aston, Nat Shiers, and Jim Q. Smith), Biometrika (2016) 103 (3): 531-545.
  • Exponential varieties (with Mateusz Michałek, Bernd Sturmfels, and Caroline Uhler), Proceedings of the London Mathematical Society (3) 112 (2016), no. 1, 27–56.
  • Marginal likelihood and model selection for Gaussian latent tree and forest models (with Mathias Drton, Shaowei Lin, and Luca Weihs), Bernoulli 23(2), 2017, 1202-1232.
  • Automorphism groups of Gaussian Bayesian networks (with Jan Draisma), Bernoulli 23(2), 2017, 1102–1129.
  • Tensors of nonnegative rank two (with Elizabeth Allman, John A. Rhodes, and Bernd Sturmfels), Linear Algebra and its Applications: Special issue on Statistics, Volume 473 (2015), pp 37-53.
  • Secant cumulants and toric geometry (with Mateusz Michałek, Luke Oeding), International Mathematics Research Notices, Issue 12 (2015), pp 4019-4063.
  • Cremona linearizations of some classical varieties (with Ciro Ciliberto, Angie Cueto, Massimiliano Mella, and Kristian Ranestad), In: Casnati G., Conte A., Gatto L., Giacardi L., Marchisio M., Verra A. (eds) From Classical to Modern Algebraic Geometry. Trends in the History of Science. Birkhäuser (2016).
  • Binary distributions of concentric rings (with Giovanni Marchetti, Nanny Wermuth), Journal of Multivariate Analysis, 130 (2014), pp 252–260.
  • Groups acting on Gaussian graphical models (with Jan Draisma, Sonja Kuhnt), Annals of Statistics, Volume 41, Number 4 (2013), pp 1944-1969.
  • Binary cumulant varieties (with Bernd Sturmfels), Annals of Combinatorics, Volume 17, Issue 1 (2013), pp 229-250.
  • L-cumulants, L-cumulant embeddings and algebraic statistics, Journal of Algebraic Statistics, Volume 3 (2012), pp 11-43.
  • The dependence of routine Bayesian model selection methods on irrelevant alternatives (with Jim Q. Smith).
  • Tree-cumulants and the geometry of binary tree models (with Jim Q. Smith), Bernoulli Volume 18, Number 1 (2012), 290-321.
  • Asymptotic Behaviour of the Marginal Likelihood for General Markov Models, J. of Machine Learning Research. Volume 12 (2011), pp 3283−3310.
  • Implicit inequality constraints in a binary tree model (with Jim Q. Smith), Electron. J. Statist. Volume 5 (2011), pp 1276-1312.


  • Semialgebraic Statistics and Latent Tree Models, Monographs on Statistics and Applied Probability 146, Chapman&Hall, 2015.
  • Piotr Białowolski, Maria Drozdowicz-Bieć, Robert Pater, Katarzyna Lada, Piotr Zwiernik, Dawid Żochowski, Forecasting with Composite Coincident and Leading Indexes and the CLIMA Model. Case of Poland, Szkoła Główna Handlowa, Warsaw, 2007.


  • GOLAZO R package for a general assymetric regularization of the log-likelihood in graphical models.
    To install, just call in R: devtools::install_github("pzwiernik/golazo")
  • MTP2binary R package for a fitting MTP2 binary distributions and the Ising model in particular.
    To install, just call in R: devtools::install_github("pzwiernik/MTP2binary")
  • mtp R package for working with Gaussian MTP2 distributions. Now made pretty much obsolete by golazo.
    To install, just call in R: devtools::install_github("pzwiernik/mtp2")
  • Structural EM R package for the structural EM algorithm for Gaussian latent tree models (no documentation yet).
    To install, just call in R: devtools::install_github("pzwiernik/StructuralEM")
  • LatentClass R package with the EM algorithm for discrete latent class models (no documentation yet). It has implemented the exact maximum likelihood for the simple 2x2x2 latent class model.
    To install, just call in R: devtools::install_github("pzwiernik/LatentClass")

secondary publications:

  • Berg, J.B. van den, Castro, R., Draisma, J., Evers, J.H.M., Hendriks, M., Khimshiashvili, G., Krehel, O., Kryven, I., Mora, K.,Szabó, B.T., Zwiernik, P. (2012). Non-imaging optics for LED-lighting. Proceedings of the 84th European Study Group Mathematics with Industry, 30 January - 3 February 2012, Eindhoven, The Netherlands.
  • Piotr Białowolski, Piotr Zwiernik, Dawid Żochowski, Modelling Inflation using Markov Switching Models. Case of Poland, in: M. Drozdowicz-Bieć ed., "Business Surveys, Business Cycles. Polish Contribution to the 30th CIRET Conference", Prace i Materiały IRG SGH, 86, 2011.
  • Piotr Białowolski, Katarzyna Rosiak-Lada, Piotr Zwiernik, Dawid Żochowski, Joint modelling of the GDP, unemployment rate and inflation using composite indicators (in Polish), Prace i Materiały IRG SGH, 80, 2007.
  • Piotr Białowolski, Piotr Zwiernik, Dawid Żochowski, Diagnosing and Predicting Inflation in Poland with Application of Markov Switching Models and Future Inflation Indicator, 26th CIRET Conference, Rome 2006.
  • Katarzyna Maciejowska, Piotr Zwiernik, Nonlinear modelling for business cycle indicators (in Polish), Prace i Materiały IRG SGH, 77, 2006.
  • Katarzyna Maciejowska, Piotr Zwiernik, Switching Markov models in the analysis of business cycles using business cycle indicators (in Polish), Zeszyty koniunktury w gospodarce polskiej, 21, 2005.


  • The Algebra and Geometry of general Markov models (PhD thesis, University of Warwick), 2010. Written under supervision of Diane Maclagan and Jim Q. Smith.
  • Geometria toryczna statystycznych binarnych modeli grafów (Master's thesis, University of Warsaw), 2007. Written under supervision of Jarosław Wiśniewski.
  • Ukryte modele Markowa w analizie danych dotyczacych koniunktury gospodarczej (Master's thesis, Warsaw School of Economics), 2003. Written under supervision of Monika Dędys.

technical reports: