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Terry Soo

Publications and source records attributed to Terry Soo.

At least 19 recordsLinked to original sources
arXiv2026

Interpretable Causal Graphical Models for Equilibrium Systems with Confounding

In applications, quantities of interest are often modelled in equilibrium or an equilibrium solution is sought. The presence of confounding makes causal inference in this setting challenging. We provide interpretable graphical models for equilibrium systems with confounding using anterial graphs (Lauritzen and Sadeghi, 2018), a class of graphs containing directed acyclic graphs, ancestral graphs, and chain graphs. In this setting, we provide valid graphical representations of both counterfactual variables and observational variables, which we relate to counterfactual graphs (Shpitser and Pearl, 2007) and single-world intervention graphs (Richardson and Robins,2013). As an application of this graphical representation, we provide an element-wise procedure of selecting adjustment sets that flexibly include and exclude given covariates.

stat.ME↗
arXiv2025

Towards Robust Causal Effect Identification Beyond Markov Equivalence

Causal effect identification typically requires a fully specified causal graph, which can be difficult to obtain in practice. We provide a sufficient criterion for identifying causal effects from a candidate set of Markov equivalence classes with added background knowledge, which represents cases where determining the causal graph up to a single Markov equivalence class is challenging. Such cases can happen, for example, when the untestable assumptions (e.g. faithfulness) that underlie causal discovery algorithms do not hold.

stat.ME↗
arXiv2024

A General Framework on Conditions for Constraint-based Causal Learning

Most constraint-based causal learning algorithms provably return the correct causal graph under certain correctness conditions, such as faithfulness. By representing any constraint-based causal learning algorithm using the notion of a property, we provide a general framework to obtain and study correctness conditions for these algorithms. From the framework, we provide exact correctness conditions for the PC algorithm, which are then related to the correctness conditions of some other existing causal discovery algorithms. The framework also suggests a paradigm for designing causal learning algorithms which allows for the correctness conditions of algorithms to be controlled for before designing the actual algorithm, and has the following implications. We show that the sparsest Markov representation condition is the weakest correctness condition for algorithms that output ancestral graphs or directed acyclic graphs satisfying any existing notions of minimality. We also reason that Pearl-minimality is necessary for meaningful causal learning but not sufficient to relax the faithfulness condition and, as such, has to be strengthened, such as by including background knowledge, for causal learning beyond faithfulness.

cs.AI↗
arXiv2024

Localised Natural Causal Learning Algorithms for Weak Consistency Conditions

By relaxing conditions for natural structure learning algorithms, a family of constraint-based algorithms containing all exact structure learning algorithms under the faithfulness assumption, we define localised natural structure learning algorithms (LoNS). We also provide a set of necessary and sufficient assumptions for consistency of LoNS, which can be thought of as a strict relaxation of the restricted faithfulness assumption. We provide a practical LoNS algorithm that runs in exponential time, which is then compared with related existing structure learning algorithms, namely PC/SGS and the relatively recent sparsest permutation algorithm. Simulation studies are also provided.

stat.ME↗
arXiv2023

Axiomatization of Interventional Probability Distributions

Causal intervention is an essential tool in causal inference. It is axiomatized under the rules of do-calculus in the case of structure causal models. We provide simple axiomatizations for families of probability distributions to be different types of interventional distributions. Our axiomatizations neatly lead to a simple and clear theory of causality that has several advantages: it does not need to make use of any modeling assumptions such as those imposed by structural causal models; it only relies on interventions on single variables; it includes most cases with latent variables and causal cycles; and more importantly, it does not assume the existence of an underlying true causal graph as we do not take it as the primitive object--in fact, a causal graph is derived as a by-product of our theory. We show that, under our axiomatizations, the intervened distributions are Markovian to the defined intervened causal graphs, and an observed joint probability distribution is Markovian to the obtained causal graph; these results are consistent with the case of structural causal models, and as a result, the existing theory of causal inference applies. We also show that a large class of natural structural causal models satisfy the theory presented here. We note that the aim of this paper is axiomatization of interventional families, which is subtly different from "causal modeling."

math.ST↗
arXiv2021

Sinai factors of nonsingular systems: Bernoulli shifts and Anosov flows

We show that a totally dissipative system has all nonsingular systems as factors, but that this is no longer true when the factor maps are required to be finitary. In particular, if a nonsingular Bernoulli shift satisfies the Doeblin condition, and has a limiting marginal distribution p, then it cannot have, as a finitary factor, an independent and identically distributed (iid) system of entropy larger than H(p); on the other hand, we show that iid systems with entropy strictly lower than H(p) can be obtained as finitary factors of these Bernoulli shifts, extending Keane and Smorodinsky's finitary version of Sinai's factor theorem to the nonsingular setting. As a consequence of our results we also obtain that every transitive twice continuously differentiable Anosov diffeomorphism on a compact manifold, endowed with volume measure, has iid factors, and if the factor is required to be finitary, then the iid factor cannot have Kolmogorov-Sinai entropy greater than the measure-theoretic entropy of a Sinai-Ruelle-Bowen measure associated with the Anosov diffeomorphism.

math.DS↗
arXiv2021

Conditions and Assumptions for Constraint-based Causal Structure Learning

We formalize constraint-based structure learning of the "true" causal graph from observed data when unobserved variables are also existent. We provide conditions for a "natural" family of constraint-based structure-learning algorithms that output graphs that are Markov equivalent to the causal graph. Under the faithfulness assumption, this natural family contains all exact structure-learning algorithms. We also provide a set of assumptions, under which any natural structure-learning algorithm outputs Markov equivalent graphs to the causal graph. These assumptions can be thought of as a relaxation of faithfulness, and most of them can be directly tested from (the underlying distribution) of the data, particularly when one focuses on structural causal models. We specialize the definitions and results for structural causal models.

math.ST↗
arXiv2020

Some factors of nonsingular Bernoulli shifts

We give elementary constructions of factors of nonsingular Bernoulli shifts. In particular, we show that all nonsingular Bernoulli shifts on a finite number of symbols which satisfy the Doeblin condition have a factor that is equivalent to an independent and identically distributed system. We also prove that there are type-III:1 Bernoulli shifts of every possible ergodic index, answering a question of Danilenko and Lemanczyk (Ergodic Theory Dynam. Systems, 39(12):3292-3321, 2019).

math.DS↗
arXiv2019

Finitary isomorphisms of Brownian motions

Ornstein and Shields (Advances in Math., 10:143-146, 1973) proved that Brownian motion reflected on a bounded region is an infinite entropy Bernoulli flow and thus Ornstein theory yielded the existence of a measure-preserving isomorphism between any two such Brownian motions. For fixed h >0, we construct by elementary methods, isomorphisms with almost surely finite coding windows between Brownian motions reflected on the intervals [0, qh] for all positive rationals q.

math.DS↗
arXiv2018

Finitary isomorphisms of some infinite entropy Bernoulli flows

A consequence of Ornstein theory is that the infinite entropy flows associated with Poisson processes and continuous-time irreducible Markov chains on a finite number of states are isomorphic as measure-preserving systems. We give an elementary construction of such an isomorphism that has an additional finitariness property, subject to the additional conditions that the Markov chain has a uniform holding rate and a mixing skeleton.

math.DS↗
arXiv2018

Finitary isomorphisms of Poisson point processes

As part of a general theory for the isomorphism problem for actions of amenable groups, Ornstein and Weiss (J. Anal. Math. 48:1-141,1987) proved that any two Poisson point processes are isomorphic as measure-preserving actions. We give an elementary construction of an isomorphism between Poisson point processes that is finitary.

math.PR↗
arXiv2015

A monotone isomorphism theorem

In the simple case of a Bernoulli shift on two symbols, zero and one, by permuting the symbols, it is obvious that any two equal entropy shifts are isomorphic. We show that the isomorphism can be realized by a factor that maps a binary sequence to another that is coordinatewise smaller than or equal to the original sequence.

math.DS↗
arXiv2014

Ergodic universality of some topological dynamical systems

The Krieger generator theorem says that every invertible ergodic measure-preserving system with finite measure-theoretic entropy can be embedded into a full shift with strictly greater topological entropy. We extend Krieger's theorem to include toral automorphisms, and more generally, any topological dynamical system on a compact metric space that satisfies almost weak specification, asymptotic entropy expansiveness, and the small boundary property. As a corollary, one obtains a complete solution to a natural generalization of an open problem in Halmos's 1956 book regarding an isomorphism invariant that he proposed.

math.DS↗
arXiv2013

A monotone Sinai theorem

Sinai proved that a nonatomic ergodic measure-preserving system has any Bernoulli shift of no greater entropy as a factor. Given a Bernoulli shift, we show that any other Bernoulli shift that is of strictly less entropy and is stochastically dominated by the original measure can be obtained as a monotone factor; that is, the factor map has the property that for each point in the domain, its image under the factor map is coordinatewise smaller than or equal to the original point.

math.DS↗
arXiv2013

Insertion and Deletion Tolerance of Point Processes

We develop a theory of insertion and deletion tolerance for point processes. A process is insertion-tolerant if adding a suitably chosen random point results in a point process that is absolutely continuous in law with respect to the original process. This condition and the related notion of deletion-tolerance are extensions of the so-called finite energy condition for discrete random processes. We prove several equivalent formulations of each condition, including versions involving Palm processes. Certain other seemingly natural variants of the conditions turn out not to be equivalent. We illustrate the concepts in the context of a number of examples, including Gaussian zero processes and randomly perturbed lattices, and we provide applications to continuum percolation and stable matching.

math.PR↗
arXiv2012

Weak mixing suspension flows over shifts of finite type are universal

Let S be an ergodic measure-preserving automorphism on a non-atomic probability space, and let T be the time-one map of a topologically weak mixing suspension flow over an irreducible subshift of finite type under a Holder ceiling function. We show that if the measure-theoretic entropy of S is strictly less than the topological entropy of T, then there exists an embedding from the measure-preserving automorphism into the suspension flow. As a corollary of this result and the symbolic dynamics for geodesic flows on compact surfaces of negative curvature developed by Bowen and Ratner, we also obtain an embedding from the measure-preserving automorphism into a geodesic flow, whenever the measure-theoretic entropy of S is strictly less than the topological entropy of the time-one map of the geodesic flow.

math.DS↗