Quantum analogues of Bayes' theorem, sufficient statistics and the pooling problem

          @misc{ pirsa_PIRSA:09060021,
            doi = {10.48660/09060021},
            url = {https://pirsa.org/09060021},
            author = {Spekkens, Robert},
            keywords = {Quantum Information, Quantum Foundations},
            language = {en},
            title = {Quantum analogues of Bayes{\textquoteright} theorem, sufficient statistics and the pooling problem},
            publisher = {Perimeter Institute},
            year = {2009},
            month = {jun},
            note = {PIRSA:09060021 see, \url{https://pirsa.org}}
          }
          

Abstract

The notion of a conditional probability is critical for Bayesian reasoning. Bayes’ theorem, the engine of inference, concerns the inversion of conditional probabilities. Also critical are the concepts of conditional independence and sufficient statistics. The conditional density operator introduced by Leifer is a natural generalization of conditional probability to quantum theory. This talk will pursue this generalization to define quantum analogues of Bayes' theorem, conditional independence and sufficient statistics. These can be used to provide simple proofs of certain well-known results in quantum information theory, such as the isomorphism between POVMs and convex decompositions of a mixed state and the remote collapse postulate, and to prove some novel results on how to pool quantum states. This is joint work with Matt Leifer. I will also briefly discuss the possibility of a diagrammatic calculus for classical and quantum Bayesian inference (joint work with Bob Coecke).