The Inflation Technique for Causal Inference with Latent Variables

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 $\text{inflation technique}$ 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.

Minor final corrections, updated to match the published version as closely as possible

References in corpus (20)

• A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations
• Beyond Bell's Theorem: Correlation Scenarios
• Almost quantum correlations
• Quantum common causes and quantum causal models
• Information-Theoretic Implications of Quantum Causal Structures
• Nonlocal correlations in the star-network configuration
• Theory-independent limits on correlations from generalised Bayesian networks
• Inferring causal structure: a quantum advantage
• Higher-order interference and single-system postulates characterizing quantum theory
• Causal Compatibility Inequalities Admitting Quantum Violations in the Triangle Structure
• Proofs of nonlocality without inequalities revisited
• Simple explanation of the quantum limits of genuine $n$-body nonlocality
• Non-Shannon inequalities in the entropy vector approach to causal structures
• Which causal structures might support a quantum-classical gap?
• Quantum Bell Inequalities from Macroscopic Locality
• Indistinguishability of causal relations from limited marginals
• Hardy's Test versus the CHSH Test of Quantum Non-Locality: Fundamental and Practical Aspects
• A unified view on Hardy's paradox and the CHSH inequality
• PR-box correlations have no classical limit
• Computational tools for solving a marginal problem with applications in Bell non-locality and causal modeling

Cited by in corpus (84)

• Bell nonlocality in networks
• Genuine Network Multipartite Entanglement
• Entanglement 25 years after Quantum Teleportation: testing joint measurements in quantum networks
• Full network nonlocality
• Bilocal Bell inequalities violated by the quantum Elegant Joint Measurement
• Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes
• Constraints on nonlocality in networks from no-signaling and independence
• Semidefinite tests for quantum network topologies
• A neural network oracle for quantum nonlocality problems in networks
• Quantum Inflation: A General Approach to Quantum Causal Compatibility
• Nested Markov Properties for Acyclic Directed Mixed Graphs
• Network Quantum Steering
• Quantum entanglement in the triangle network
• Semidefinite programming relaxations for quantum correlations
• Bounding the sets of classical and quantum correlations in networks
• Symmetries in quantum networks lead to no-go theorems for entanglement distribution and to verification techniques
• Single-photon nonlocality in quantum networks
• The Inflation Technique Completely Solves the Causal Compatibility Problem
• Nonlocality for Generic Networks
• Proofs of network quantum nonlocality in continuous families of distributions
• Quantifying causal influences in the presence of a quantum common cause
• Generalized $n$-locality inequalities in star-network configuration and their optimal quantum violations
• Contextuality without incompatibility
• Test of Genuine Multipartite Nonlocality
• Understanding the interplay of entanglement and nonlocality: motivating and developing a new branch of entanglement theory
• Certification of non-classicality in all links of a photonic star network without assuming quantum mechanics
• Any Physical Theory of Nature Must Be Boundlessly Multipartite Nonlocal
• Experimental full network nonlocality with independent sources and strict locality constraints
• Partial self-testing and randomness certification in the triangle network
• Towards a minimal example of quantum nonlocality without inputs
• Experimental nonclassicality in a causal network without assuming freedom of choice
• Network nonlocality sharing via weak measurements in the extended bilocal scenario
• Non-Local Boxes for Networks
• Quantum Correlation Sharing: A Review On Recent Progress From Nonlocality To Other Non-Classical Correlations
• Experimental Demonstration that No Tripartite-Nonlocal Causal Theory Explains Nature's Correlations
• Network Nonlocality via Rigidity of Token-Counting and Color-Matching
• Quantum violation of local causality in urban network with hybrid photonic technologies
• Experimental demonstration of genuine tripartite nonlocality under strict locality conditions
• Inflation: a Python library for classical and quantum causal compatibility
• A mathematical framework for operational fine tunings
• Self-testing nonlocality without entanglement
• A convergent inflation hierarchy for quantum causal structures
• Covariance Decomposition as a Universal Limit on Correlations in Networks
• Analysing causal structures in generalised probabilistic theories
• The inflation hierarchy and the polarization hierarchy are complete for the quantum bilocal scenario
• No graph state is preparable in quantum networks with bipartite sources and no classical communication
• Post-quantum nonlocality in the minimal triangle scenario
• Quantum LOSR networks cannot generate graph states with high fidelity
• Numerically assisted determination of local models in network scenarios
• Experimentally adjudicating between different causal accounts of Bell inequality violations via statistical model selection
• Witnessing Non-Classicality in a Simple Causal Structure with Three Observable Variables
• Connector tensor networks: a renormalization-type approach to quantum certification
• Topologically Robust Quantum Network Nonlocality
• Detection loophole in quantum causality and its countermeasures
• Generalized $n$-locality inequalities in linear-chain network for arbitrary inputs scenario and their quantum violations
• A Hierarchy of Multipartite Nonlocality and Device-Independent Effect Witnesses
• Detection of Network and Genuine Network Quantum Steering
• Analysing causal structures using Tsallis entropies
• No nonlocal advantage of quantum coherence beyond quantum instrumentality
• A Combinatorial Solution to Causal Compatibility
• Relating Wigner's Friend Scenarios to Nonclassical Causal Compatibility, Monogamy Relations, and Fine Tuning
• Noise-robust proofs of quantum network nonlocality
• Quantum network-entanglement measures
• Classifying Causal Structures: Ascertaining when Classical Correlations are Constrained by Inequalities
• Two convergent NPA-like hierarchies for the quantum bilocal scenario
• Bell nonlocality in quantum networks with unreliable sources: Loophole-free postelection via self-testing
• Memory attacks in network nonlocality and self-testing
• Violation of the Finner inequality in the four-output triangle network
• Latent-free equivalent mDAGs
• Experimental test of quantum causal influences
• Local models and Bell inequalities for the minimal triangle network
• Causal Networks and Freedom of Choice in Bell's Theorem
• Symmetric observations without symmetric causal explanations
• Experimental genuine quantum nonlocality in the triangle network
• Guarantees on the structure of experimental quantum networks
• The genuinely multipartite nonlocality of graph states is model-dependent
• Quantum non-classicality in the simplest causal network
• Bipartite quantum states admitting a causal explanation
• No Bipartite-Nonlocal Causal Theory Can Explain Nature's Correlations
• Consistency and Causality of Interconnected Nonsignaling Resources
• Features of preparable entangled states in Gaussian quantum networks
• Strength of statistical evidence for genuine tripartite nonlocality
• Fully quantum inflation: quantum marginal problem constraints in the service of causal inference
• Testing Genuine Multipartite Nonlocality via an Inflated Network with Multi-copy Entangled States