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The theory of graphical causal models is a major entry point to the budding field of algebraic statistics, where algebraic geometry meets statistical modelling, and this talk will give an introduction to it from the perspective of an algebraic geometer. I'll introduce some conceptual tools and methods that are peculiar to algebraic statistics, and work through an example such causal inference computation using the commutative algebra software Macaulay2. At the end I'll review some of my research on hidden Markov models, and their close connection to matrix product state models of quantum-entangled qubits.