Semidefinite tests for latent causal structures

Abstract

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.

25 pages, 7 figures

References in corpus (10)

• Beyond Bell's Theorem: Correlation Scenarios
• 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
• Experimental non-locality in a quantum network
• Can a quantum state over time resemble a quantum state at a single time?
• Quantifier Elimination for Statistical Problems
• Analysing causal structures with entropy
• Covariance Bell inequalities

Cited by in corpus (13)

• Bell nonlocality in networks
• Semidefinite tests for quantum network topologies
• Semidefinite programming relaxations for quantum correlations
• The Inflation Technique Completely Solves the Causal Compatibility Problem
• Characterizing Quantum Networks: Insights from Coherence Theory
• Experimental nonclassicality in a causal network without assuming freedom of choice
• Inflation: a Python library for classical and quantum causal compatibility
• A convergent inflation hierarchy for quantum causal structures
• Covariance Decomposition as a Universal Limit on Correlations in Networks
• Inferring Quantum Network Topology using Local Measurements
• Witnessing Non-Classicality in a Simple Causal Structure with Three Observable Variables
• Local models and Bell inequalities for the minimal triangle network
• Causal Networks and Freedom of Choice in Bell's Theorem