Picturing classical and quantum Bayesian inference
- 370 Downloads
We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropic quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer’s calculus of ‘conditional density operators’. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category.
KeywordsBayesian inference Graphical calculus Conditional density operator Symmetric monoidal category Frobenius algebra
Unable to display preview. Download preview PDF.
- Abramsky, S., & Coecke, B. (2004). A categorical semantics of quantum protocols. In Proceedings of 19th IEEE conference on logic in computer science (pp. 415–425). IEEE Press. arXiv:quant-ph/0402130.Google Scholar
- Baez, J. C. (2006). Quantum quandaries: A category-theoretic perspective. In D. Rickles, S. French, & J. T. Saatsi (Eds.), The structural foundations of quantum gravity (pp. 240–266). Oxford: Oxford University Press. arXiv:quant-ph/0404040.Google Scholar
- Barnum, H., & Wilce, A. (2009). Ordered linear spaces and categories as frameworks for information-processing characterizations of quantum and classical theory. arXiv:0908.2354.Google Scholar
- Barnum, H., Barrett, J., Leifer, M., & Wilce, A. (2006). Cloning and broadcasting in generic probabilistic theories. arXiv:quant-ph/0611295.Google Scholar
- Coecke B., Paquette E.O. (2006) POVMs and Naimark’s theorem without sums. Electronic Notes in Theoretical Computer Science 210: 131–152 arXiv:quant-ph/0608072Google Scholar
- Coecke, B., & Paquette, E. O. (2011). Categories for the practising physicist. In B. Coecke (Ed.), New structures for physics. Lecture notes in physics (pp. 173–286). New York: Springer-Verlag. arXiv:0905.3010.Google Scholar
- Coecke B., & Pavlovic D. (2007). Quantum measurements without sums. In G. Chen, L. Kauffman, & S. Lamonaco (Eds.), Mathematics of quantum computing and technology (pp. 567–604). Abington: Taylor and Francis. arXiv:quant-ph/0608035.Google Scholar
- Coecke, B., Paquette, E. O., & Pavlovic, D. (2009). Classical and quantum structuralism. In I. Mackie & S. Gay (Eds.), Semantic techniques for quantum computation (pp. 29–69). Cambridge: Cambridge University Press. arXiv:0904.1997 (to appear).Google Scholar
- Coecke, B., Pavlovic, D., Vicary J. (2008b). A new description of orthogonal bases. Mathematical Structures in Computer Science. To appear. arXiv:0810.0812.Google Scholar
- Dixon, L., Duncan, R., Kissinger, A., & Merry, A. (2010). quantomatic software tool. http://dream.inf.ed.ac.uk/projects/quantomatic/.
- Fuchs, C. A., & Schack, R. (2009). Quantum-Bayesian coherence. arXiv:0906.2187v1.Google Scholar
- Hasegawa, M., Hofmann, M., & Plotkin, G. (2008). Finite dimensional vector spaces are complete for traced symmetric monoidal categories. Lecture notes in computer science Vol. 4800 (pp. 367–385). Heidelberg: Springer-Verlag.Google Scholar
- Kelly, G. M. (1972). Many-variable functorial calculus. In G. M. Kelly, M. L. Laplaza, G. Lewis, & S. Mac Lane (Eds.), Coherence in categories. Lecture notes in mathematics Vol. 281 (pp. 66–105). Berlin: Springer-Verlag.Google Scholar
- Lack S. (2004) Composing PROPs. Theory and Applications of Categories 13: 147–163Google Scholar
- Lauda A.D. (2006) Frobenius algebras and ambidextrous adjunctions. Theory and Applications of Categories 16: 84–122 arXiv:math.CT/0502550Google Scholar
- Leifer, M. S., & Spekkens, R. W. (2008). Quantum analogues of Bayes’ theorem, sufficient statistics and the pooling problem (in preparation, 2009).Google Scholar
- Melliés, P.-A. (2009). Categorical semantics of linear logic. http://www.pps.jussieu.fr/~mellies/papers/panorama.pdf.
- Pearl J. (1988) Probabilistic reasoning in intelligent systems. Morgan Kaufmann, San FranciscoGoogle Scholar
- Pearl J. (2000) Causality: Models, reasoning and inference. Cambridge University Press, CambridgeGoogle Scholar
- Penrose R. (1971) Applications of negative dimensional tensors. In: Welsh D. (eds) Combinatorial mathematics and its applications. Academic Press, New York, pp 221–244Google Scholar
- Selinger P. (2011) A survey of graphical languages for monoidal categories. In: Coecke B. (eds) New structures for physics. Lecture notes in physics. Springer-Verlag, Heidelberg, pp 275–337 arXiv:0908.3347Google Scholar
- Turaev V. (1994) Quantum invariants of knots and 3-manifolds. de Gruyter, BerlinGoogle Scholar