Colored multiset Eulerian polynomials are a common generalization of MacMahon’s multiset Eulerian polynomials and the colored Eulerian polynomials, both of which are known to satisfy well-studied distributional properties including real-rootedness, logconcavity and unimodality. The symmetric colored multiset Eulerian polynomials are characterized and used to prove sufficient conditions for a colored multiset Eulerian polynomial to be self-interlacing. The latter property implies the aforementioned distributional properties as well as others, including the alternatingly increasing property and bi-γ-positivity. To derive these results, multivariate generalizations of an identity due to MacMahon are deduced. The results are applied to a pair of questions, both previously studied in several special cases, that are seen to admit more general answers when framed in the context of colored multiset Eulerian polynomials. The first question pertains to s-Eulerian polynomials, and the second to interpretations of γ-coefficients.

The basic problem of causal discovery is concerned with estimating a directed acyclic graph (DAG)representing the dependence relations in multivariate data. Several successful causal discoveryalgorithms have optimization-based aspects, which operate via a set of rules for searching thespace of DAGs. Recent results have revealed that the edge graph of the so-called characteristicimset polytope, CIM𝑝, can provide a diverse set of such rules. Characterizing the edge graph ofCIM𝑝 is a generally challenging problem. However, many algorithms first estimate the adjacen-cies in the causal DAG, in the form of an undirected graph 𝐺, prior to orienting the edges. In thisregime, knowledge of the subpolytope CIM𝐺 defined for DAGs with adjacencies specified by 𝐺 isvaluable. In this paper, we characterize the edge graph of CIM𝐺 when 𝐺 is an undirected tree,providing the first family of characteristic imset polytopes for which the edge graph is completelyunderstood. These results are applied to give a new causal discovery algorithm that estimates apolytree representing the dependencies in the given multivariate data. Our algorithm is shownto out-perform comparable methods on both real and synthetic data. Our results also reveal con-nections between characteristic imset polytopes and the well-studied stable set polytopes fromcombinatorial optimization.

We study submodels of Gaussian directed acyclic graph (DAG) models defined by partial homogeneity constraints imposed on the model error variances and structural coefficients. We represent these models with coloured DAGs and investigate their properties for use in statistical and causal inference. Local and global Markov properties are provided and shown to characterize the coloured DAG model. Additional properties relevant to causal discovery are studied, including the existence and nonexistence of faithful distributions and structural identifiability. Extending prior work of Peters and B & uuml;hlmann and Wu and Drton, we prove structural identifiability under the assumption of homogeneous structural coefficients, as well as for a family of models with partially homogeneous structural coefficients. The latter models, termed blocked properly edge-coloured DAGS (BPEC-DAGs), capture additional causal insights by clustering the direct causes of each node into communities according to their effect on their common target. An analogue of the greedy equivalence search algorithm for learning BPEC-DAGs is given and evaluated on real and synthetic data. Regarding model geometry, we provide a proof of a conjecture of Sullivant which generalizes to coloured DAG models, coloured undirected graphical models and directed ancestral graph models. The proof yields a tool for identification of Markov properties for any rationally parametrized model with globally, rationally identifiable parameters.

Behaviour modelling has been widely explored using both statistical and machine learning techniques, primarily relying on analyzing correlations to understand passenger responses under different conditions and scenarios. However, correlation alone does not imply causation. This paper introduces a data-driven causal behaviour modelling approach, comprising two phases: causal discovery and causal inference. Causal discovery phase uses Peter-Clark (PC) algorithm to learn a directed acyclic graph that captures the causal relationships among variables. Causal inference phase estimates the corresponding model parameters and infers (conditional) causal effects of interventions designed to influence user behaviour. The method is validated by comparing the results with those from conventional modelling approaches (logistic regression and expert knowledge) using smart card data from a real-world use case on a pre-peak fare discount incentive program in the Hong Kong Mass Transit Railway system. The results highlight that the purely data-driven causal discovery method can produce reasonable causal graph. The method can also quantify the behavioural impacts of the incentive, identify key influencing factors, and estimate the corresponding causal effects. The overall causal effect of the incentive is approximately 0.7 %, with about 3 % of the population changing behaviour from previous statistical analysis. Interestingly, passengers with the highest flexibility exhibit a negative response, while those with medium-to-high flexibility demonstrate 3 times of the general level of responsiveness. The approach initiates the data-driven, causal modelling of human behaviour dynamics to support policy developments and managerial interventions.

We address the problem of representing context-specific causal models based on both observational and experimental data collected under general (e.g. hard or soft) interventions by introducing a new family of context-specific conditional independence models called CStrees. This family is defined via a novel factorization criterion that allows for a generalization of the factorization property defining general interventional directed acyclic graph (DAG) models. We derive a graphical characterization of model equivalence for observational CStrees that extends the Verma and Pearl criterion for DAGs. This characterization is then extended to CStree models under general, context-specific interventions. To obtain these results, we formalize a notion of context-specific intervention that can be incorporated into concise graphical representations of CStree models. We relate CStrees to other context-specific models, showing that the families of DAGs, CStrees, labelled DAGs, and staged trees form a strict chain of inclusions. We then present an algorithm for learning CStrees from a combination of observational and interventional data where the intervention targets are assumed to be unknown with hard or soft and possibly context-specific effects. The algorithm, evaluated on simulated and real data, performs well in the recovery of context-specific dependence structure as well as context-specific interventional perturbations.

A cosmological polytope is defined for a given Feynman diagram, and its canonical form may be used to compute the contribution of the Feynman diagram to the wavefunction of certain cosmological models. Given a subdivision of a polytope, its canonical form is obtained as a sum of the canonical forms of the facets of the subdivision. In this paper, we identify such formulas for the canonical form via algebraic techniques. It is shown that the toric ideal of every cosmological polytope admits a Gröbner basis with a squarefree initial ideal, yielding a regular unimodular triangulation of the polytope. In specific instances, including trees and cycles, we recover graphical characterizations of the facets of such triangulations that may be used to compute the desired canonical form. For paths and cycles, these characterizations admit simple enumeration. Hence, we obtain formulas for the normalized volume of these polytopes, extending previous observations of Kühne and Monin.

Conventional modelling related to travel behaviours is facing challenges from rapidly changing demand dynamics, especially in a post-pandemic context with more volatile and elastic demand. In response to this challenge and to capitalize on the potential of ubiquitous travel trajectory data (such as smart card or mobile phone data), this study proposes data-driven causal behaviour modelling. The approach includes causal discovery and causal inference methods. For causal discovery, we use classic algorithms, like PC and GES, to learn a directed acyclic graph representing the causal relationships among the variables. Using this causal graph, we define a structural equation model and estimate its corresponding parameters, allowing causal inference of the (conditional) average treatment effect. The approach is validated with expert knowledge using a case study for behaviour response of passengers to a pre-peak fare discount incentive in the mass transit systems in Hong Kong. It also presents a comparative analysis of the causal inference results with those from the traditional transport model approach, e.g., logistic regression. We demonstrate the use of causal inference on a learned causal structure, which allows identification of the important factors, estimation of their causal effects, and quantification of the unique contribution of the intervention policy. Importantly, we will illustrate how to derive policy insights not only for future scenarios (what if it is…), but also from counterfactual analysis of historical actions (what if it had been…). Our approach advances data-driven modelling of casual human behaviour dynamics to support policy developments and managerial interventions.

We construct Bayesian and frequentist finite-sample goodness-of-fit tests for three different variants of the stochastic blockmodel for network data. Since all of the stochastic blockmodel variants are log-linear in form when block assignments are known, the tests for the latent block model versions combine a block membership estimator with the algebraic statistics machinery for testing goodness-of-fit in log-linear models. We describe Markov bases and marginal polytopes of the variants of the stochastic blockmodel and discuss how both facilitate the development of goodness-of-fit tests and understanding of model behaviour. The general testing methodology developed here extends to any finite mixture of log-linear models on discrete data, and as such is the first application of the algebraic statistics machinery for latent-variable models.

Reflexive lattice polytopes play a key role in combinatorics, algebraic geometry, physics, and other areas. One important class of lattice polytopes are lattice sim-plices defining weighted projective spaces. We investigate the question of when a reflexive weighted projective space simplex has the integer decomposition prop-erty. We provide a complete classification of reflexive weighted projective space simplices having the integer decomposition property for the case when there are at most three distinct non-unit weights, and conjecture a general classification for an arbitrary number of distinct non-unit weights. Further, for any weighted projective space simplex and m ≥ 1, we define the m-th reflexive stabilization, a reflexive weighted projective space simplex. We prove that when m is 2 or greater, reflexive stabilizations do not have the integer decomposition property. We also prove that as long as one weight is at least three, the Ehrhart h*-polynomial of any sufficiently large reflexive stabilization is not unimodal and has only 1 and 2 as coefficients. We use this construction to generate interesting examples of reflexive weighted projective space simplices that are near the boundary of both h*-unimodality and the integer decomposition property.

Decomposable graphical models, also known as perfect directed acyclic graph (DAG) models, play a fundamental role in standard approaches to probabilistic inference via graph representations in modern machine learning and statistics. However, such models are limited by the assumption that the data-generating distribution does not entail strictly context-specific conditional independence relations. The family of staged tree models generalizes DAG models so as to accommodate context-specific knowledge. We provide a new characterization of perfect discrete DAG models in terms of their staged tree representations. This characterization identifies the family of balanced staged trees as the natural generalization of discrete decomposable models to the context-specific setting.