A Combinatorial Solution to Causal Compatibility

Abstract

Within the field of causal inference, it is desirable to learn the structure of causal relationships holding between a system of variables from the correlations that these variables exhibit; a sub-problem of which is to certify whether or not a given causal hypothesis is compatible with the observed correlations. A particularly challenging setting for assessing causal compatibility is in the presence of partial information; i.e. when some of the variables are hidden/latent. This paper introduces the possible worlds framework as a method for deciding causal compatibility in this difficult setting. We define a graphical object called a possible worlds diagram, which compactly depicts the set of all possible observations. From this construction, we demonstrate explicitly, using several examples, how to prove causal incompatibility. In fact, we use these constructions to prove causal incompatibility where no other techniques have been able to. Moreover, we prove that the possible worlds framework can be adapted to provide a complete solution to the possibilistic causal compatibility problem. Even more, we also discuss how to exploit graphical symmetries and cross-world consistency constraints in order to implement a hierarchy of necessary compatibility tests that we prove converges to sufficiency.

arXiv comment44 pages, 21 figures, 3 appendices, originally submitted for partial fulfilment of Perimeter Scholars International masters program, comments welcomed

References in corpus (28)

• Quantum correlations with no causal order
• The Sheaf-Theoretic Structure Of Non-Locality and Contextuality
• The lesson of causal discovery algorithms for quantum correlations: Causal explanations of Bell-inequality violations require fine-tuning
• Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference
• Bilocal versus non-bilocal correlations in entanglement swapping experiments
• Quantum causal modelling
• Beyond Bell's Theorem: Correlation Scenarios
• Learning high-dimensional directed acyclic graphs with latent and selection variables
• Information-Theoretic Implications of Quantum Causal Structures
• Theory-independent limits on correlations from generalised Bayesian networks
• Inferring causal structure: a quantum advantage
• Polynomial Bell inequalities
• Directed Cyclic Graphical Representations of Feedback Models
• Logical Bell Inequalities
• The Inflation Technique for Causal Inference with Latent Variables
• On the Testability of Causal Models with Latent and Instrumental Variables
• Entropic Inequalities and Marginal Problems
• Beyond Bell's Theorem II: Scenarios with arbitrary causal structure
• Looking for symmetric Bell inequalities
• Causal Compatibility Inequalities Admitting Quantum Violations in the Triangle Structure
• Causal structures from entropic information: Geometry and novel scenarios
• Graphs for margins of Bayesian networks
• A graph-separation theorem for quantum causal models
• Nested Markov Properties for Acyclic Directed Mixed Graphs
• Causal inference via algebraic geometry: feasibility tests for functional causal structures with two binary observed variables
• A Constraint Propagation Approach to Probabilistic Reasoning
• Probability distributions with summary graph structure
• Notes about the Caratheodory number

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