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The algebraization of classical propositional logic is Boolean algebra.

Bayesian networks are a generalization of classical propositional logic with probability truth-values.

What is the corresponding algebraic form of Bayesian networks? Or at least outline how such an algebra would look like? I seem to be at a loss as to how to begin...

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A bayesian network corresponds to an independence (algebraic) variety and hence to a polynomial ring over $\mathbb{R}$. You can start from

Garcia, Luis David, Michael Stillman, and Bernd Sturmfels. "Algebraic geometry of Bayesian networks." Journal of Symbolic Computation 39.3-4 (2005): 331-355.

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