30 citations · 169 across the 28 of their papers we have counts for
40 papers
Bayesian Networks of Density Operators
Steffen Lauritzen, Piotr Zwiernik
We study quantum analogues of Bayesian networks on a directed acyclic graph (DAG), distinguishing two constructions for positive definite density operators on finite-dimensional te…
The Markov Marginal Problem for Density Operators
Steffen Lauritzen, Piotr Zwiernik
We study when local reduced density operators, viewed as quantum marginals, can be assembled into a global quantum state with a prescribed Markov structure. The starting point is a…
Learning latent tree models with small query complexity
Luc Devroye, Gabor Lugosi, Piotr Zwiernik
We consider the problem of structure recovery in a graphical model of a tree where some variables are latent. Specifically, we focus on the Gaussian case, which can be reformulated…
Property testing in graphical models: testing small separation numbers
Luc Devroye, Gábor Lugosi, Piotr Zwiernik
In many statistical applications, the dimension is too large to handle for standard high-dimensional machine learning procedures. This is particularly true for graphical models, wh…
Entropic covariance models
Piotr Zwiernik
In covariance matrix estimation, one of the challenges lies in finding a suitable model and an efficient estimation method. Two commonly used modelling approaches in the literature…
Positivity in Linear Gaussian Structural Equation Models
Asad Lodhia, Jan-Christian Hütter, Caroline Uhler +1
We study a notion of positivity of Gaussian directed acyclic graphical models corresponding to a non-negativity constraint on the coefficients of the associated structural equation…
Graphical model inference with external network data
Jack Jewson, Li Li, Laura Battaglia +3
We consider two applications where we study how dependence structure between many variables is linked to external network data. We first study the interplay between social media co…
Non-Independent Components Analysis
Geert Mesters, Piotr Zwiernik
A seminal result in the ICA literature states that for , if the components of are independent and at most one is Gaussian, then is identified up…
Total positivity in multivariate extremes
Frank Röttger, Sebastian Engelke, Piotr Zwiernik
Positive dependence is present in many real world data sets and has appealing stochastic properties that can be exploited in statistical modeling and in estimation. In particular,…
Maximum Likelihood Estimation for Brownian Motion Tree Models Based on One Sample
Michael Truell, Jan-Christian Hütter, Chandler Squires +2
We study the problem of maximum likelihood estimation given one data sample () over Brownian Motion Tree Models (BMTMs), a class of Gaussian models on trees. BMTMs are often u…
Robust estimation of tree structured models
Marta Casanellas, Marina Garrote-López, Piotr Zwiernik
Consider the problem of learning undirected graphical models on trees from corrupted data. Recently Katiyar et al. showed that it is possible to recover trees from noisy binary dat…
Likelihood Geometry of Correlation Models
Carlos Améndola, Piotr Zwiernik
Correlation matrices are standardized covariance matrices. They form an affine space of symmetric matrices defined by setting the diagonal entries to one. We study the geometry of…
Locally associated graphical models and mixed convex exponential families
Steffen Lauritzen, Piotr Zwiernik
The notion of multivariate total positivity has proved to be useful in finance and psychology but may be too restrictive in other applications. In this paper we propose a concept o…
Dependence in elliptical partial correlation graphs
David Rossell, Piotr Zwiernik
The Gaussian model equips strong properties that facilitate studying and interpreting graphical models. Specifically it reduces conditional independence and the study of positive a…
Estimating linear covariance models with numerical nonlinear algebra
Bernd Sturmfels, Sascha Timme, Piotr Zwiernik
Numerical nonlinear algebra is applied to maximum likelihood estimation for Gaussian models defined by linear constraints on the covariance matrix. We examine the generic case as w…
Secant varieties of toric varieties arising from simplicial complexes
M. Azeem Khadam, Mateusz Michałek, Piotr Zwiernik
Motivated by the study of the secant variety of the Segre-Veronese variety we propose a general framework to analyze properties of the secant varieties of toric embeddings of affin…
The Identification Problem for Linear Rational Expectations Models
Majid M. Al-Sadoon, Piotr Zwiernik
This version corrects a number of mistakes that appeared in the previous draft. In particular, the (EU-LREM) condition is sufficient for existence and uniqueness but not necessary,…
Learning partial correlation graphs and graphical models by covariance queries
Gábor Lugosi, Jakub Truszkowski, Vasiliki Velona +1
We study the problem of recovering the structure underlying large Gaussian graphical models or, more generally, partial correlation graphs. In high-dimensional problems it is often…
Total positivity in exponential families with application to binary variables
Steffen Lauritzen, Caroline Uhler, Piotr Zwiernik
We study exponential families of distributions that are multivariate totally positive of order 2 (MTP2), show that these are convex exponential families, and derive conditions for…
Brownian motion tree models are toric
Bernd Sturmfels, Caroline Uhler, Piotr Zwiernik
Felsenstein's classical model for Gaussian distributions on a phylogenetic tree is shown to be a toric variety in the space of concentration matrices. We present an exact semialgeb…
Maximum likelihood estimation of the Latent Class Model through model boundary decomposition
Elizabeth S. Allman, Hector Baños Cervantes, Robin Evans +5
The Expectation-Maximization (EM) algorithm is routinely used for the maximum likelihood estimation in the latent class analysis. However, the EM algorithm comes with no guarantees…
Latent tree models
Piotr Zwiernik
Latent tree models are graphical models defined on trees, in which only a subset of variables is observed. They were first discussed by Judea Pearl as tree-decomposable distributio…
Maximum likelihood estimation in Gaussian models under total positivity
Steffen Lauritzen, Caroline Uhler, Piotr Zwiernik
We analyze the problem of maximum likelihood estimation for Gaussian distributions that are multivariate totally positive of order two (MTP2). By exploiting connections to phylogen…
Total positivity in Markov structures
Shaun Fallat, Steffen Lauritzen, Kayvan Sadeghi +3
We discuss properties of distributions that are multivariate totally positive of order two (MTP2) related to conditional independence. In particular, we show that any independence…