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Gaussoids are two-antecedental approximations of Gaussian conditional independence structures


The gaussoid axioms are conditional independence inference rules which characterize regular Gaussian CI structures over a three-element ground set. It is known that no finite set of inference rules completely describes regular Gaussian CI as the ground set grows. In this article we show that the gaussoid axioms logically imply every inference rule of at most two antecedents which is valid for regular Gaussians over any ground set. The proof is accomplished by exhibiting for each inclusion-minimal gaussoid extension of at most two CI statements a regular Gaussian realization. Moreover we prove that all those gaussoids have rational positive-definite realizations inside every ε-ball around the identity matrix. For the proof we introduce the concept of algebraic Gaussians over arbitrary fields and of positive Gaussians over ordered fields and obtain the same two-antecedental completeness of the gaussoid axioms for algebraic and positive Gaussians over all fields of characteristic zero as a byproduct.


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The author wishes to thank Thomas Kahle, Andreas Kretschmer and Milan Studený for helpful discussions, and the anonymous referees for their clear and thoughtful suggestions for improvement of the earlier manuscript. Thanks to Xiangying Chen for spotting an error in the previous version of Example 7.4.


Open Access funding enabled and organized by Projekt DEAL. This work is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 314838170, GRK 2297 MathCoRe.

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Correspondence to Tobias Boege.

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Boege, T. Gaussoids are two-antecedental approximations of Gaussian conditional independence structures. Ann Math Artif Intell (2021).

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  • Conditional independence
  • Inference
  • Completeness
  • Gaussoid
  • Realizability
  • Rationality

Mathematics Subject Classification (2010)

  • 62B10
  • 62R01
  • 14P10