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.

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.

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.

Finding a directed acyclic graph (DAG) that best encodes the conditional inde-pendence statements observable from data is a central question within causality. Algorithms that greedily transform one candidate DAG into another given a fixed set of moves have been particularly successful, for example, the greedy equivalence search, greedy interventional equivalence search, and max-min hill climbing algorithms. In 2010, Studenty, Hemmecke, and Lindner introduced the char-acteristic imset (CIM) polytope, CIMp, whose vertices correspond to Markov equivalence classes, as a way of transforming causal discovery into a linear optimization problem. We show that the moves of the aforementioned algorithms are included within classes of edges of CIMp and that restrictions placed on the skeleton of the candidate DAGs correspond to faces of CIMp. Thus, we observe that greedy equivalence search, greedy interventional equivalence search, and max-min hill climbing all have geometric realizations as greedy edge-walks along CIMp. Furthermore, the identified edges of CIMp strictly generalize the moves of these algorithms. Exploiting this generalization, we introduce a greedy simplex-type algorithm called greedy CIM, and a hybrid variant, skeletal greedy CIM, that outperforms current competitors among hybrid and constraint-based algorithms.

We consider the problem of characterizing Bayesian networks up to unconditional equivalence, i.e., when directed acyclic graphs (DAGs) have the same set of unconditional $d$-separation statements. Each unconditional equivalence class (UEC) is uniquely represented with an undirected graph whose clique structure encodes the members of the class. Via this structure, we provide a transformational characterization of unconditional equivalence; i.e., we show that two DAGs are in the same UEC if and only if one can be transformed into the other via a finite sequence of specified moves. We also extend this characterization to the essential graphs representing the Markov equivalence classes (MECs) in the UEC. UECs form a partition coarsening of the space of MECs and are easily estimable from marginal independence tests. Thus, a characterization of unconditional equivalence has applications in methods that involve searching the space of MECs of Bayesian networks.

In geometric, algebraic, and topological combinatorics, the unimodality of combinatorial generating polynomials is frequently studied. Unimodality follows when the polynomial is (real) stable, a property often deduced via the theory of interlacing polynomials. Many of the open questions on stability and unimodality of polynomials pertain to the enumeration of faces of cell complexes. In this paper, we relate the theory of interlacing polynomials to the shellability of cell complexes. We first derive a sufficient condition for stability of the h-polynomial of a subdivision of a shellable complex. To apply it, we generalize the notion of reciprocal domains for convex embeddings of polytopes to abstract polytopes and use this generalization to define the family of stable shellings of a polytopal complex. We characterize the stable shellings of cubical and simplicial complexes, and apply this theory to answer a question of Brenti and Welker on barycentric subdivisions for the well-known cubical polytopes. We also give a positive solution to a problem of Mohammadi and Welker on edgewise subdivisions of cell complexes. We end by relating the family of stable line shellings to the combinatorics of hyperplane arrangements. We pose related questions, answers to which would resolve some long-standing problems while strengthening ties between the theory of interlacing polynomials and the combinatorics of hyperplane arrangements.

Directed acyclic graphical models are widely used to represent complex causal systems. Since the basic task of learning such a model from data is NP-hard, a standard approach is greedy search over the space of directed acyclic graphs or Markov equivalence classes of directed acyclic graphs. As the space of directed acyclic graphs on p nodes and the associated space of Markov equivalence classes are both much larger than the space of permutations, it is desirable to consider permutation-based greedy searches. Here, we provide the first consistency guarantees, both uniform and high dimensional, of a greedy permutation-based search. This search corresponds to a simplex-like algorithm operating over the edge-graph of a subpolytope of the permutohedron, called a directed acyclic graph associahedron. Every vertex in this polytope is associated with a directed acyclic graph, and hence with a collection of permutations that are consistent with the directed acyclic graph ordering. A walk is performed on the edges of the polytope maximizing the sparsity of the associated directed acyclic graphs. We show via simulated and real data that this permutation search is competitive with current approaches.

This extended abstract is a summary of a recent paper which studies the enumeration of faces of subdivisions of cell complexes. Motivated by a conjecture of Brenti and Welker on the real-rootedness of the h-polynomial of the barycentric subdivision of the boundary complex of a convex polytope, we introduce a framework for proving real-rootedness of h-polynomials for subdivisions of polytopal complexes by relating interlacing polynomials to shellability via the existence of so-called stable shellings. We show that any shellable cubical, or simplicial, complex admitting a stable shelling has barycentric and edgewise subdivisions with real-rooted h-polynomials. Such shellings are shown to exist for well-studied families of cubical polytopes, giving a positive answer to the conjecture of Brenti and Welker in these cases. The framework of stable shellings is also applied to answer to a conjecture of Mohammadi and Welker on edgewise subdivisions in the case of shellable simplicial complexes.

In algebraic, topological, and geometric combinatorics, inequalities among the coefficients of combinatorial polynomials are frequently studied. Recently, a notion called the alternatingly increasing property, which is stronger than unimodality, was introduced. In this paper, we relate the alternatingly increasing property to real-rootedness of the symmetric decomposition of a polynomial to develop a systematic approach for proving the alternatingly increasing property for several classes of polynomials. We apply our results to strengthen and generalize real-rootedness, unimodality, and alternatingly increasing results pertaining to colored Eulerian and derangement polynomials, Ehrhart h*-polynomials for lattice zonotopes, h-polynomials of barycentric subdivisions of doubly Cohen-Macaulay level simplicial complexes, and certain local h-polynomials for subdivisions of simplices. In particular, we prove two conjectures of Athanasiadis.