No file available
To read the file of this research,
you can request a copy directly from the authors.
The Inflation Technique for Causal Inference with Latent Variables
Preprints and early-stage research may not have been peer reviewed yet.
Abstract
The problem of causal inference is to determine if a given probability distribution on observed variables is compatible with some causal structure. The difficult case is when the causal structure includes latent variables. We here introduce the for tackling this problem. An inflation of a causal structure is a new causal structure that can contain multiple copies of each of the original variables, but where the ancestry of each copy mirrors that of the original. To every distribution of the observed variables that is compatible with the original causal structure, we assign a family of marginal distributions on certain subsets of the copies that are compatible with the inflated causal structure. It follows that compatibility constraints for the inflation can be translated into compatibility constraints for the original causal structure. Even if the constraints at the level of inflation are weak, such as observable statistical independences implied by disjoint causal ancestry, the translated constraints can be strong. We apply this method to derive new inequalities whose violation by a distribution witnesses that distribution's incompatibility with the causal structure (of which Bell inequalities and Pearl's instrumental inequality are prominent examples). We describe an algorithm for deriving all such inequalities for the original causal structure that follow from ancestral independences in the inflation. For three observed binary variables with pairwise common causes, it yields inequalities that are stronger in at least some aspects than those obtainable by existing methods. We also describe an algorithm that derives a weaker set of inequalities but is more efficient. Finally, we discuss which inflations are such that the inequalities one obtains from them remain valid even for quantum (and post-quantum) generalizations of the notion of a causal model.
ResearchGate has not been able to resolve any citations for this publication.
Causality is a seminal concept in science: Any research discipline, from sociology and medicine to physics and chemistry, aims at understanding the causes that could explain the correlations observed among some measured variables. While several methods exist to characterize classical causal models, no general construction is known for the quantum case. In this work, we present quantum inflation, a systematic technique to falsify if a given quantum causal model is compatible with some observed correlations. We demonstrate the power of the technique by reproducing known results and solving open problems for some paradigmatic examples of causal networks. Our results may find applications in many fields: from the characterization of correlations in quantum networks to the study of quantum effects in thermodynamic and biological processes.
The causal compatibility question asks whether a given causal structure graph — possibly involving latent variables — constitutes a genuinely plausible causal explanation for a given probability distribution over the graph’s observed categorical variables. Algorithms predicated on merely necessary constraints for causal compatibility typically suffer from false negatives, i.e. they admit incompatible distributions as apparently compatible with the given graph. In 10.1515/jci-2017-0020, one of us introduced the inflation technique for formulating useful relaxations of the causal compatibility problem in terms of linear programming. In this work, we develop a formal hierarchy of such causal compatibility relaxations. We prove that inflation is asymptotically tight, i.e., that the hierarchy converges to a zero-error test for causal compatibility. In this sense, the inflation technique fulfills a longstanding desideratum in the field of causal inference. We quantify the rate of convergence by showing that any distribution which passes the nth-order inflation test must be On−1/2-close in Euclidean norm to some distribution genuinely compatible with the given causal structure. Furthermore, we show that for many causal structures, the (unrelaxed) causal compatibility problem is faithfully formulated already by either the first or second order inflation test.
Marginal problems naturally arise in a variety of different fields: basically, the question is whether some marginal/partial information is compatible with a joint probability distribution. To this aim, the characterization of marginal sets via quantifier elimination and polyhedral projection algorithms is of primal importance. In this work, before considering specific problems, we review polyhedral projection algorithms with focus on applications in information theory, and, alongside known algorithms, we also present a newly developed geometric algorithm which walks along the face lattice of the polyhedron in the projection space. One important application of this is in the field of quantum non-locality, where marginal problems arise in the computation of Bell inequalities. We apply the discussed algorithms to discover many tight entropic Bell inequalities of the tripartite Bell scenario as well as more complex networks arising in the field of causal inference. Finally, we analyze the usefulness of these inequalities as nonlocality witnesses by searching for violating quantum states.
We present an algebraic description of the sets of local correlations in arbitrary networks, when the parties have finite inputs and outputs. We consider networks generalizing the usual Bell scenarios by the presence of multiple uncorrelated sources. We prove a finite upper bound on the cardinality of the value sets of the local hidden variables. Consequently, we find that the sets of local correlations are connected, closed and semialgebraic, and bounded by tight polynomial Bell-like inequalities.
We provide a scheme for inferring causal relations from uncontrolled statistical data based on tools from computational algebraic geometry, in particular, the computation of Groebner bases. We focus on causal structures containing just two observed variables, each of which is binary. We consider the consequences of imposing different restrictions on the number and cardinality of latent variables and of assuming different functional dependences of the observed variables on the latent ones (in particular, the noise need not be additive). We provide an inductive scheme for classifying functional causal structures into distinct observational equivalence classes. For each observational equivalence class, we provide a procedure for deriving constraints on the joint distribution that are necessary and sufficient conditions for it to arise from a model in that class. We also demonstrate how this sort of approach provides a means of determining which causal parameters are identifiable and how to solve for these. Prospects for expanding the scope of our scheme, in particular to the problem of quantum causal inference, are also discussed.
Acausal scenario is a graph that describes the cause and effect relationships between all relevant variables in an experiment. A scenario is deemed not interesting if there is no device-independent way to distinguish the predictions of classical physics from any generalised probabilistic theory (including quantum mechanics). Conversely, an interesting scenario is one in which there exists a gap between the predictions of different operational probabilistic theories, as occurs for example in Belltype experiments. Henson, Lal and Pusey (HLP) recently proposed a sufficient condition for a causal scenario to not be interesting. In this paper we supplement their analysis with some new techniques and results.Wefirst show that existing graphical techniques due to Evans can be used to confirm by inspection that many graphs are interesting without having to explicitly search for inequality violations. For three exceptional cases-the graphs numbered #15, 16, 20 in HLP-we show that there exist non-Shannon type entropic inequalities that imply these graphs are interesting. In doing so, we find that existing methods of entropic inequalities can be greatly enhanced by conditioning on the specific values of certain variables.
Reichenbach's principle asserts that if two observed variables are found to be correlated, then there should be a causal explanation of these correlations. Furthermore, if the explanation is in terms of a common cause, then the conditional probability distribution over the variables given the complete common cause should factorize. The principle is generalized by the formalism of causal models, in which the causal relationships among variables constrain the form of their joint probability distribution. In the quantum case, however, the observed correlations in Bell experiments cannot be explained in the manner Reichenbach's principle would seem to demand. Motivated by this, we introduce a quantum counterpart to the principle. We demonstrate that under the assumption that quantum dynamics is fundamentally unitary, if a quantum channel with input A and outputs B and C is compatible with A being a complete common cause of B and C, then it must factorize in a particular way. Finally, we show how to generalize our quantum version of Reichenbach's principle to a formalism for quantum causal models, and provide examples of how the formalism works.
A causal structure is a relationship between observed variables that in general restricts the set of possible correlations between them. This relationship can be mediated by unobserved systems, which correspond to random variables in the classical case and shared density operators in the quantum case. Here we investigate the use of entropy vectors - vectors whose components correspond to the entropies of each subset of the observed variables - as a means to differentiate between the classical and quantum cases of particular causal structures. We use non-Shannon entropic inequalities to give additional constraints on the sets of possible entropy vectors in the classical case and hence obtain tighter outer approximations of the set of realisable entropy vectors. We also discuss how to obtain inner approximations. We provide several examples and analyse the triangle causal structure in detail. In most causal structures where it is unknown whether or not there is a classical-quantum separation in terms of entropy vectors, non-Shannon inequalities give a tighter characterisation in the classical case. The question of whether these improved characterisations also hold in the quantum case remains open, i.e., we remain without an example of an entropy vector that is attainable in a quantum causal structure, but not in the classical analogue. These techniques may also shed light on the question of whether non-Shannon inequalities also hold for the von Neumann entropy.
Causal modelling provides a powerful set of tools for identifying causal
structure from observed correlations. It is well known that such techniques
fail for quantum systems, unless one introduces `spooky' hidden mechanisms.
Whether one can produce a genuinely quantum framework in order to discover
causal structure remains an open question. Here we introduce a new framework
for quantum causal modelling that allows for the discovery of causal structure.
We define quantum analogues for many of the core features of classical causal
modelling techniques, including the Causal Markov Condition and Faithfulness.
Based on the process matrix formalism, this framework naturally extends to
generalised structures with indefinite causal order.
The known Fourier-Chernikov algorithm of linear inequality system convolution is complemented with an original procedure of all dependent (redundant) inequalities deletion. The concept of "almost dependent" inequalities is defined and an algorithm for further reducing the system by deletion of these is considered. The concluding algorithm makes it possible to hold actual-time convolution of a general inequality system containing up to 50 variables with the rigorous method of dependent inequalities deletion and up to 100 variables with the approximate method of one. The main ap-plication of such an approach consists in solving linear inequality system in an explicit form. These results are illus-trated with a series of computer experiments.
Bellʼs inequality fundamentally changed our understanding of quantum mechanics. Bellʼs insight that non-local correlations between quantum systems cannot be explained classically can be verified experimentally, and has numerous applications in modern quantum information. Today, the Clauser–Horne–Shimony–Holt (CHSH) inequality is probably the most well-known Bell inequality and it has given us a wealth of understanding in what differentiates the classical from the quantum world. Yet, there are certainly other means of quantifying ‘Bell non-locality without inequalities’ such as the famous Hardyʼs paradox. As such, one may wonder whether these are entirely different approaches to non-locality. For this anniversary issue, we unify the perspective of the CHSH inequality and Hardy’s paradox into one family of non-local games which include both as special cases.
This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell’s theorem’.
The problem of using observed correlations to infer causal relations is
relevant to a wide variety of scientific disciplines. Yet given correlations
between just two classical variables, it is impossible to determine whether
they arose from a causal influence of one on the other or a common cause
influencing both, unless one can implement a randomized intervention. We here
consider the problem of causal inference for quantum variables. We introduce
causal tomography, which unifies and generalizes conventional quantum
tomography schemes to provide a complete solution to the causal inference
problem using a quantum analogue of a randomized trial. We furthermore show
that, in contrast to the classical case, observed quantum correlations alone
can sometimes provide a solution. We implement a quantum-optical experiment
that allows us to control the causal relation between two optical modes, and
two measurement schemes -- one with and one without randomization -- that
extract this relation from the observed correlations. Our results show that
entanglement and coherence, known to be central to quantum information
processing, also provide a quantum advantage for causal inference.
Bayesian networks provide a powerful tool for reasoning about probabilistic
causation, used in many areas of science. They are, however, intrinsically
classical. In particular, Bayesian networks naturally yield the Bell
inequalities. Inspired by this connection, we generalise the formalism of
classical Bayesian networks in order to investigate non-classical correlations
in arbitrary causal structures. Our framework of `generalised Bayesian
networks' replaces latent variables with the resources of any generalised
probabilistic theory, most importantly quantum theory, but also, for example,
Popescu-Rohrlich boxes. We obtain three main sets of results. Firstly, we prove
that all of the observable conditional independences required by the classical
theory also hold in our generalisation; to obtain this, we extend the classical
d-separation theorem to our setting. Secondly, we find that the
theory-independent constraints on probabilities can go beyond these conditional
independences. For example we find that no probabilistic theory predicts
perfect correlation between three parties using only bipartite common causes.
Finally, we begin a classification of those causal structures, such as the Bell
scenario, that may yield a separation between classical, quantum and
general-probabilistic correlations.
It has recently been found that Bell scenarios are only a small subclass of
interesting setups for studying the non-classical features of quantum theory
within spacetime. We find that it is possible to talk about classical
correlations, quantum correlations and other kinds of correlations on any
directed acyclic graph, and this captures various extensions of Bell scenarios
which have been considered in the literature. From a conceptual point of view,
the main feature of our approach is its high level of unification: while the
notions of source, choice of setting and measurement play all seemingly
different roles in a Bell scenario, our formalism shows that they are all
instances of the same concept of "event".
Our work can also be understood as a contribution to the subject of causal
inference with latent variables. Among other things, we introduce hidden
Bayesian networks as a generalization of hidden Markov models.
We present a new characterization of quantum theory in terms of simple
physical principles that is different from previous ones in two important
respects: first, it only refers to properties of single systems without any
assumptions on composition of systems; and second, it is closer to experiment
by having absence of higher-order interference as a postulate, which is
currently the subject of experimental investigation. We give three postulates
-- no higher-order interference, classical decomposability of states, and
strong symmetry -- and prove that the only non-classical convex operational
theories satisfying them are real, complex, and quaternionic quantum theory,
together with 3x3 octonionic quantum theory and ball state spaces of arbitrary
dimension. Then we show that adding observability of energy as a fourth
postulate yields complex quantum theory as the unique solution, relating the
emergence of the complex numbers to the possibility of Hamiltonian dynamics. We
also show that there may be interesting non-quantum theories satisfying only
the first two of our postulates, which would allow for higher-order
interference in experiments, while still respecting the contextuality analogue
of the local orthogonality principle.
The framework of generalized probabilistic theories is a powerful tool for
studying the foundations of quantum physics. It provides the basis for a
variety of recent findings that significantly improve our understanding of the
rich physical structure of quantum theory. This review paper tries to present
the framework and recent results to a broader readership in an accessible
manner. To achieve this, we follow a constructive approach. Starting from few
basic physically motivated assumptions we show how a given set of observations
can be manifested in an operational theory. Furthermore, we characterize
consistency conditions limiting the range of possible extensions. In this
framework classical and quantum physics appear as special cases, and the aim is
to understand what distinguishes quantum mechanics as the fundamental theory
realized in nature. It turns out non-classical features of single systems can
equivalently result from higher dimensional classical theories that have been
restricted. Entanglement and non-locality, however, are shown to be genuine
non-classical features.
The fields of quantum non-locality in physics, and causal discovery in
machine learning, both face the problem of deciding whether observed data is
compatible with a presumed causal relationship between the variables (for
example a local hidden variable model). Traditionally, Bell inequalities have
been used to describe the restrictions imposed by causal structures on marginal
distributions. However, some structures give rise to non-convex constraints on
the accessible data, and it has recently been noted that linear inequalities on
the observable entropies capture these situations more naturally. In this
paper, we show the versatility of the entropic approach by greatly expanding
the set of scenarios for which entropic constraints are known. For the first
time, we treat Bell scenarios involving multiple parties and multiple
observables per party. Going beyond the usual Bell setup, we exhibit
inequalities for scenarios with extra conditional independence assumptions, as
well as a limited amount of shared randomness between the parties. Many of our
results are based on a geometric observation: Bell polytopes for two-outcome
measurements can be naturally imbedded into the convex cone of attainable
marginal entropies. Thus, any entropic inequality can be translated into one
valid for probabilities. In some situations the converse also holds, which
provides us with a rich source of candidate entropic inequalities.
In recent years, the use of information principles to understand quantum correlations has been very successful. Unfortunately, all principles considered so far have a bipartite formulation, but intrinsically multipartite principles, yet to be discovered, are necessary for reproducing quantum correlations. Here we introduce local orthogonality, an intrinsically multipartite principle stating that events involving different outcomes of the same local measurement must be exclusive or orthogonal. We prove that it is equivalent to no-signalling in the bipartite scenario but more restrictive for more than two parties. By exploiting this non-equivalence, it is then demonstrated that some bipartite supra-quantum correlations do violate the local orthogonality when distributed among several parties. Finally, we show how its multipartite character allows revealing the non-quantumness of correlations for which any bipartite principle fails. We believe that local orthogonality is a crucial ingredient for understanding no-signalling and quantum correlations.
What assumptions and methods allow us to turn observations into causal knowledge, and how can even incomplete causal knowledge be used in planning and prediction to influence and control our environment? In this book Peter Spirtes, Clark Glymour, and Richard Scheines address these questions using the formalism of Bayes networks, with results that have been applied in diverse areas of research in the social, behavioral, and physical sciences. The authors show that although experimental and observational study designs may not always permit the same inferences, they are subject to uniform principles. They axiomatize the connection between causal structure and probabilistic independence, explore several varieties of causal indistinguishability, formulate a theory of manipulation, and develop asymptotically reliable procedures for searching over equivalence classes of causal models, including models of categorical data and structural equation models with and without latent variables. The authors show that the relationship between causality and probability can also help to clarify such diverse topics in statistics as the comparative power of experimentation versus observation, Simpson's paradox, errors in regression models, retrospective versus prospective sampling, and variable selection. The second edition contains a new introduction and an extensive survey of advances and applications that have appeared since the first edition was published in 1993.
In this paper we introduce a new algorithm called Equality Set Projection (ESP) for computing the orthogonal projection of bounded, convex polytopes. Our solution addresses the case where the input polytope is represented as the intersection of a finite number of halfplanes and its projection is given in an irredundant halfspace form. Unlike many existing approaches, the key advantage oered by ESP is its output sensitivity, i.e., its complexity is a function of the number of facets in the projection of the polytope. This feature makes it particularly suited for many problems of theoretical and practical importance in which the number of vertices far exceeds the number of facets. Further, it is shown that for non-degenerate polytopes of fixed size (dimension and number of facets) the complexity is linear in the number of facets in the projection. Numerical results are presented that demonstrate that high dimensional polytopes can be projected eciently.
An active area of research in the fields of machine learning and statistics
is the development of causal discovery algorithms, the purpose of which is to
infer the causal relations that hold among a set of variables from the
correlations that these exhibit. We apply some of these algorithms to the
correlations that arise for entangled quantum systems. We show that they cannot
distinguish correlations that satisfy Bell inequalities from correlations that
violate Bell inequalities, and consequently that they cannot do justice to the
challenges of explaining certain quantum correlations causally. Nonetheless, by
adapting the conceptual tools of causal inference, we can show that any attempt
to provide a causal explanation of nonsignalling correlations that violate a
Bell inequality must contradict a core principle of these algorithms, namely,
that an observed statistical independence between variables should not be
explained by fine-tuning of the causal parameters. We demonstrate the need for
such fine-tuning for most of the causal mechanisms that have been proposed to
underlie Bell correlations, including superluminal causal influences,
superdeterminism (that is, a denial of freedom of choice of settings), and
retrocausal influences which do not introduce causal cycles.
We present a class of inequality constraints on the set of distributions
induced by local interventions on variables governed by a causal Bayesian
network, in which some of the variables remain unmeasured. We derive bounds on
causal effects that are not directly measured in randomized experiments. We
derive instrumental inequality type of constraints on nonexperimental
distributions. The results have applications in testing causal models with
observational or experimental data.
We use the implicitization procedure to generate polynomial equality
constraints on the set of distributions induced by local interventions on
variables governed by a causal Bayesian network with hidden variables. We show
how we may reduce the complexity of the implicitization problem and make the
problem tractable in certain causal Bayesian networks. We also show some
preliminary results on the algebraic structure of polynomial constraints. The
results have applications in distinguishing between causal models and in
testing causal models with combined observational and experimental data.
Bell's Theorem witnesses that the predictions of quantum theory cannot be
reproduced by theories of local hidden variables in which observers can choose
their measurements independently of the source. Working out an idea of
Branciard, Rosset, Gisin and Pironio, we consider scenarios which feature
several sources, but no choice of measurement for the observers. Every Bell
scenario can be mapped into such a \emph{correlation scenario}, and Bell's
Theorem then discards those local hidden variable theories in which the sources
are independent. However, most correlation scenarios do not arise from Bell
scenarios, and we describe examples of (quantum) nonlocality in some of these
scenarios, while posing many open problems along the way. Some of our scenarios
have been considered before by mathematicians in the context of causal
inference.
Satisfiability Modulo Theories (SMT) refers to the problem of determining whether a first-order formula is satisfiable with respect to some logical theory. Solvers based on SMT are used as back-end engines in model-checking applications such as bounded, interpolation-based, and predicate-abstraction-based model checking. After a brief illustration of these uses, we survey the predominant techniques for solving SMT problems with an emphasis on the lazy approach, in which a propositional satisfiability (SAT) solver is combined with one or more theory solvers. We discuss the architecture of a lazy SMT solver, give examples of theory solvers, show how to combine such solvers modularly, and mention several extensions of the lazy approach. We also briefly describe the eager approach in which the SMT problem is reduced to a SAT problem. Finally, we discuss how the basic framework for determining satisfiability can be extended with additional functionality such as producing models, proofs, unsatisfiable cores, and interpolants.
Current machine learning systems operate, almost exclusively, in a statistical, or model-blind mode, which entails severe theoretical limits on their power and performance. Such systems cannot reason about interventions and retrospection and, therefore, cannot serve as the basis for strong AI. To achieve human level intelligence, learning machines need the guidance of a model of reality, similar to the ones used in causal inference. To demonstrate the essential role of such models, I will present a summary of seven tasks which are beyond reach of current machine learning systems and which have been accomplished using the tools of causal inference.
Information Theory and Network Coding consists of two parts: Components of Information Theory, and Fundamentals of Network Coding Theory. Part I is a rigorous treatment of information theory for discrete and continuous systems. In addition to the classical topics, there are such modern topics as the I-Measure, Shannon-type and non-Shannon-type information inequalities, and a fundamental relation between entropy and group theory. With information theory as the foundation, Part II is a comprehensive treatment of network coding theory with detailed discussions on linear network codes, convolutional network codes, and multi-source network coding.
Other important features include:
• Derivations that are from the first principle
• A large number of examples throughout the book
• Many original exercise problems
• Easy-to-use chapter summaries
• Two parts that can be used separately or together for a comprehensive course
Information Theory and Network Coding is for senior undergraduate and graduate students in electrical engineering, computer science, and applied mathematics. This work can also be used as a reference for professional engineers in the area of communications.
Testing whether a probability distribution is compatible with a given Bayesian network is a fundamental task in the field of causal inference, where Bayesian networks model causal relations. Here we consider the class of causal structures where all correlations between observed quantities are solely due to the influence from latent variables. We show that each model of this type imposes a certain signature on the observable covariance matrix in terms of a particular decomposition into positive semidefinite components. This signature, and thus the underlying hypothetical latent structure, can be tested in a computationally efficient manner via semidefinite programming. This stands in stark contrast with the algebraic geometric tools required if the full observable probability distribution is taken into account. The semidefinite test is compared with tests based on entropic inequalities.
In this paper, we present the software Parallel AdjaceNcy Decomposition Algorithm (PANDA), an efficient implementation of the vertex and facet enumeration problem for polyhedra. The software is based on the double description method. Special features are the possibility to employ the so-called adjacency decomposition, the option of doing computations in exact integer arithmetic, the exploitation of symmetry information and the easy usability of parallel hardware. With computational experiments, we demonstrate the effectiveness of the software.
This text is an introduction to an operational outlook on Bell inequalities, which has been very fruitful in the past few years. It has lead to the recognition that Bell tests have their own place in applied quantum technologies, because they quantify non-classicality in a device-independent way, that is, without any need to describe the degrees of freedom under study and the measurements that are performed. At the more fundamental level, the same device-independent outlook has allowed the falsification of several other alternative models that could hope to reproduce the observed statistics while keeping some classical features that quantum theory denies; and it has shed new light on the long-standing quest for deriving quantum theory from physical principles.
The problem of eliminating unknowns from a system of linear inequalities is considered. A new fast technique for verifying Chernikov rules in Fourier-Motzkin elimination is proposed, which is an adaptation of the “graph” test for adjacency in the double description method. Numerical results are presented that confirm the effectiveness of this technique.
One of Yakir Aharonov's endlessly captivating physics ideas is the conjecture
that two axioms, namely relativistic causality ("no superluminal signalling")
and nonlocality, so nearly contradict each other that a unique theory - quantum
mechanics - reconciles them. But superquantum (or "PR-box") correlations imply
that quantum mechanics is not the most nonlocal theory (in the sense of
nonlocal correlations) consistent with relativistic causality. Let us consider
supplementing these two axioms with a minimal third axiom: there exists a
classical limit in which macroscopic observables commute. That is, just as
quantum mechanics has a classical limit, so must any generalization of quantum
mechanics. In this classical limit, PR-box correlations violate relativistic
causality. Generalized to all stronger-than-quantum bipartite correlations,
this result is a derivation of Tsirelson's bound without assuming quantum
mechanics.
The correlations that can be observed between a set of variables depend on
the causal structure underpinning them. Causal structures can be modeled using
directed acyclic graphs, where nodes represent variables and edges denote
functional dependencies. In this work, we describe a general algorithm for
computing information-theoretic constraints on the correlations that can arise
from a given interaction pattern, where we allow for classical as well as
quantum variables. We apply the general technique to two relevant cases: First,
we show that the principle of information causality appears naturally in our
framework and go on to generalize and strengthen it. Second, we derive bounds
on the correlations that can occur in a networked architecture, where a set of
few-body quantum systems is distributed among a larger number of parties.
In this note we give a short overview on symmetry exploiting techniques in
three different branches of polyhedral computations: The representation
conversion problem, integer linear programming and lattice point counting. We
describe some of the future challenges and sketch some directions of potential
developments.
Nonlocality is the most characteristic feature of quantum mechanics, but recent research seems to suggest the possible existence of nonlocal correlations stronger than those predicted by theory. This raises the question of whether nature is in fact more nonlocal than expected from quantum theory or, alternatively, whether there could be an as yet undiscovered principle limiting the strength of nonlocal correlations. Here, I review some of the recent directions in the intensive theoretical effort to answer this question.
There have been a number of attempts to derive the set of quantum non-local
correlations from reasonable physical principles. Here we introduce
, a set of multipartite supra-quantum correlations that has appeared
under different names in fields as diverse as graph theory, quantum gravity and
quantum information science. We argue that may correspond to the
set of correlations of a reasonable physical theory, in which case the research
program to reconstruct quantum theory from device-independent principles is met
with strong obstacles. In support of this conjecture, we prove that
is closed under classical operations and satisfies the physical principles of
Non-Trivial Communication Complexity, No Advantage for Nonlocal Computation,
Macroscopic Locality and Local Orthogonality. We also review numerical evidence
that almost quantum correlations satisfy Information Causality.
A neighborliness property of marginal polytopes of hierarchical models, depending on the cardinality of the smallest non-face of the underlying simplicial complex, is shown. The case of binary variables is studied explicitly, then the general case is reduced to the binary case. A Markov basis for binary hierarchical models whose simplicial complexes is the complement of an interval is given.
In the mid-nineteenth century George Boole formulated his ‘conditions of possible experience’. These are equations and ineqaulities
that the relative frequencies of (logically connected) events must satisfy. Some of Boole's conditions have been rediscovered
in more recent years by physicists, including Bell inequalities, Clauser Horne inequalities, and many others. In this paper,
the nature of Boole's conditions and their relation to propositional logic is explained, and the puzzle associated with their
violation by quantum frequencies is investigated in relation to a variety of approaches to the interpretation of quantum mechanics.
In 1960, the mathematician Ernst Specker described a simple example of
nonclassical correlations which he dramatized using a parable about a seer who
sets an impossible prediction task to his daughter's suitors. We revisit this
example here, using it as an entree to three central concepts in quantum
foundations: contextuality, Bell-nonlocality, and complementarity.
Specifically, we show that Specker's parable offers a narrative thread that
weaves together a large number of results, including: the impossibility of
measurement-noncontextual and outcome-deterministic ontological models of
quantum theory (the Kochen-Specker theorem), in particular the proof of
Klyachko; the impossibility of Bell-local models of quantum theory (Bell's
theorem), especially the proofs by Mermin and Hardy; the impossibility of a
preparation-noncontextual ontological model of quantum theory; and the
existence of triples of positive operator valued measures (POVMs) that can be
measured jointly pairwise but not triplewise. Along the way, several novel
results are presented, including: a generalization of a theorem by Fine
connecting the existence of a joint distribution over outcomes of
counterfactual measurements to the existence of a noncontextual model; a
generalization of Klyachko's proof of the Kochen-Specker theorem; a proof of
the Kochen-Specker theorem in the style of Hardy's proof of Bell's theorem; a
categorization of contextual and Bell-nonlocal correlations in terms of
frustrated networks; a new inequality testing preparation noncontextuality; and
lastly, some results on the joint measurability of POVMs and the question of
whether these can be modeled noncontextually. Finally, we emphasize that
Specker's parable provides a novel type of foil to quantum theory, challenging
us to explain why the particular sort of contextuality and complementarity
embodied therein does not arise in a quantum world.
In general if a linear program has an optimal solution, then a primal and dual optimal solution is a certificate of the solvable status. Furthermore, it is well known that in the solvable case, then the linear program always has an optimal basic solution. Similarly, when a linear program is primal or dual infeasible then by Farkas's Lemma a certificate of the infeasible status exists. However, in the primal or dual infeasible case then there is not an uniform definition of what a suitable basis certificate of the infeasible status is.
In this work we present a definition of a basis certificate and develop a strongly polynomial algorithm which given a Farkas type certificate of infeasibility computes a basis certificate of infeasibility. This result is relevant for the recently developed interior-point methods because they do not compute a basis certificate of infeasibility in general. However, our result demonstrates that a basis certificate can be obtained at a moderate computational cost.
Let be a family of Borel fields of subsets of a set S and probabilistic measures on measurable spaces , where . The family of measures , is denoted by .
The measures and are said to be consistent if for any . If any pair of measures of the family is consistent, the family itself is referred to as consistent.
The consistent family is said to be extendable if there is a measure on the measurable space consistent with each measure of ( is the smallest Borel field containing all ).
For the purposes of the theory of games the following special case of extendability is important. Let be a finite complete complex and M the set of its vertices. Let a finite set correspond to each vertex a of and the set to each subset . Let is a measure on and is the family of all such measures. The extendability of the family is closely related with the combinatorial properties of the complex .
Any maximal face of the complex is said to be an extreme face if it has proper vertices (i.e. such vertices which do not belong to any other maximal face of ). If T is an extreme face of the complex obtained by removing from all proper vertices of T with their stars is said to be a normal subcomplex of . A complex is said to be regular if there is a sequence of subcomplexes of where is a normal subcomplex of , and the last member vanishes.
The main results of the paper consists in the following statement.
Theorem. The regularity of the complexis a necessary and sufficient condition of extendability of any consistent family ofof measures.
Certain causal models involving unmeasured variables induce no independence
constraints among the observed variables but imply, nevertheless, inequality
contraints on the observed distribution. This paper derives a general formula
for such instrumental variables, that is, exogenous variables that directly
affect some variables but not all. With the help of this formula, it is
possible to test whether a model involving instrumental variables may account
for the data, or, conversely, whether a given variables can be deemed
instrumental.
We provide a classification of graphical models according to their
representation as subfamilies of exponential families. Undirected graphical
models with no hidden variables are linear exponential families (LEFs),
directed acyclic graphical models and chain graphs with no hidden variables,
including Bayesian networks with several families of local distributions, are
curved exponential families (CEFs) and graphical models with hidden variables
are stratified exponential families (SEFs). An SEF is a finite union of CEFs
satisfying a frontier condition. In addition, we illustrate how one can
automatically generate independence and non-independence constraints on the
distributions over the observable variables implied by a Bayesian network with
hidden variables. The relevance of these results for model selection is
examined.
Recent improvement on Tarski's procedure for quantifier elimination in the
first order theory of real numbers makes it feasible to solve small instances
of the following problems completely automatically: 1. listing all equality and
inequality constraints implied by a graphical model with hidden variables. 2.
Comparing graphyical models with hidden variables (i.e., model equivalence,
inclusion, and overlap). 3. Answering questions about the identification of a
model or portion of a model, and about bounds on quantities derived from a
model. 4. Determing whether a given set of independence assertions. We discuss
the foundation of quantifier elimination and demonstrate its application to
these problems.
The validity OF a causal model can be tested ONLY IF the model imposes
constraints ON the probability distribution that governs the generated data. IN
the presence OF unmeasured variables, causal models may impose two types OF
constraints : conditional independencies, AS READ through the d - separation
criterion, AND functional constraints, FOR which no general criterion IS
available.This paper offers a systematic way OF identifying functional
constraints AND, thus, facilitates the task OF testing causal models AS well AS
inferring such models FROM data.
We develop the necessary theory in computational algebraic geometry to place
Bayesian networks into the realm of algebraic statistics. We present an
algebra{statistics dictionary focused on statistical modeling. In particular,
we link the notion of effiective dimension of a Bayesian network with the
notion of algebraic dimension of a variety. We also obtain the independence and
non{independence constraints on the distributions over the observable variables
implied by a Bayesian network with hidden variables, via a generating set of an
ideal of polynomials associated to the network. These results extend previous
work on the subject. Finally, the relevance of these results for model
selection is discussed.
Even though quantum correlations violate Bell's inequality, they satisfy weaker inequalities of a similar type. Some particular inequalities of this kind are proved here. The more general case of instruments located in different space-time regions is also discussed in some detail.