Built on

R. T. Cox (1946) Probability, frequency, and reasonable expectation. American Journal of Physics
1946
Earlier work this paper cites.
M. S. Leifer and D. Poulin (2008) Quantum graphical models and belief propagation. Annals of Physics
Original
1946
Earlier work this paper cites.
R. Penrose (1971) Applications of negative dimensional tensors. In: Combinatorial Mathematics and its Applications, pages 221–244. Academic Press
1971
Earlier work this paper cites.
G. M. Kelly (1972) Many-variable functorial calculus. In: Coherence in Categories, G. M. Kelly, M. L. Laplaza, G. Lewis and S. Mac Lane (eds), pages 66–105. Lecture notes in Mathematics
1972
Earlier work this paper cites.
A. Uhlmann (1977) Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory. Communications in Mathematical Physics
1977
Earlier work this paper cites.
G. M. Kelly and M. L. Laplaza (1980) Coherence for compact closed categories. Journal of Pure and Applied Algebra
1980
Earlier work this paper cites.
A. Carboni and R. F. C. Walters (1987) Cartesian bicategories I. Journal of Pure and Applied Algebra
1987
Earlier work this paper cites.
J. Pearl. Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, San Francisco, 1988
1988
Earlier work this paper cites.
A. Joyal and R. Street (1991) The Geometry of tensor calculus I. Advances in Mathematics
1991
Earlier work this paper cites.
V. Turaev (1994) Quantum Invariants of Knots and 3-Manifolds, de Gruyter
1994
Earlier work this paper cites.
J. C. Baez and J. Dolan (1995) Higher-dimensional algebra and topological quantum field theory. Journal of Mathematical Physics
1995
Earlier work this paper cites.
H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher (1996) Noncommuting mixed states cannot be broadcast. Physical Review Letters
1996
Earlier work this paper cites.
B. Coecke, E. O. Paquette and D. Pavlovic (2009) Classical and quantum structuralism. To appear in Semantic Techniques for Quantum Computation, I. Mackie and S. Gay (eds), Cambridge University Press. arXiv:0904.1997
Original
1997
Earlier work this paper cites.
J. Pearl. Causality: Models, Reasoning and Inference. Cambridge University Press, 2000
2000
Earlier work this paper cites.
D. N. Yetter (2001) Functorial Knot Theory. Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants. World Scientific
Original
2001
Earlier work this paper cites.

Similar

J. Kock (2003) Frobenius Algebras and 2D Topological Quantum Field Theories. Cambridge University Press
2003
Cited alongside, same era.
S. Abramsky and B. Coecke (2004) A categorical semantics of quantum protocols. In: Proceedings of 19th IEEE conference on Logic in Computer Science, pages 415–425. IEEE Press. arXiv:quant-ph/0402130
2004
Cited alongside, same era.
P. Hayden, R. Jozsa, D. Petz and A. Winter (2004) Structure of states which satisfy strong subadditivity of quantum entropy with equality. Communications in Mathematical Physics
2004
Cited alongside, same era.
S. Lack (2004) Composing PROPs. Theory and Applications of Categories
2004
Cited alongside, same era.
J. C. Baez (2006) Quantum quandaries: a category-theoretic perspective. In: The Structural Foundations of Quantum Gravity, D. Rickles, S. French and J. T. Saatsi (Eds), pages 240–266. Oxford University Press. arXiv:quant-ph/0404040

Then

B. Coecke, D. Pavlovic, and J. Vicary (2008) A new description of orthogonal bases. Mathematical Structures in Computer Science. To appear. arXiv:0810.0812
Original
2008
Later among the works it cites.
M. Hasegawa, M. Hofmann and G. Plotkin (2008) Finite dimensional vector spaces are complete for traced symmetric monoidal categories. Lecture Notes in Computer Science
2008
Later among the works it cites.
J. Vicary (2008) Categorical Formulation of Finite-dimensional C*-algebras. Electronic Notes in Theoretical Computer Science
Original
2008
Later among the works it cites.
H. Barnum and A. Wilce (2009) Ordered linear spaces and categories as frameworks for information-processing characterizations of quantum and classical theory. arXiv:0908.2354
Original
2009
Later among the works it cites.
B. Coecke and E. O. Paquette (2009) Categories for the practising physicist. In: New Structures for Physics, B. Coecke (ed), pages 173–286. Lecture Notes in Physics, Springer-Verlag. arXiv:0905.3010

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Reading the bibliography…

2011

Picturing classical and quantum Bayesian inference

Coecke, Bob, Spekkens, Robert W.

Understand

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'.

Built on

R. T. Cox (1946) Probability, frequency, and reasonable expectation. American Journal of Physics
1946
Earlier work this paper cites.
M. S. Leifer and D. Poulin (2008) Quantum graphical models and belief propagation. Annals of Physics
Original
1946
Earlier work this paper cites.
R. Penrose (1971) Applications of negative dimensional tensors. In: Combinatorial Mathematics and its Applications, pages 221–244. Academic Press
1971
Earlier work this paper cites.
G. M. Kelly (1972) Many-variable functorial calculus. In: Coherence in Categories, G. M. Kelly, M. L. Laplaza, G. Lewis and S. Mac Lane (eds), pages 66–105. Lecture notes in Mathematics
1972
Earlier work this paper cites.
A. Uhlmann (1977) Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory. Communications in Mathematical Physics
1977
Earlier work this paper cites.
G. M. Kelly and M. L. Laplaza (1980) Coherence for compact closed categories. Journal of Pure and Applied Algebra
1980
Earlier work this paper cites.
A. Carboni and R. F. C. Walters (1987) Cartesian bicategories I. Journal of Pure and Applied Algebra
1987
Earlier work this paper cites.
J. Pearl. Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, San Francisco, 1988
1988
Earlier work this paper cites.
A. Joyal and R. Street (1991) The Geometry of tensor calculus I. Advances in Mathematics
1991
Earlier work this paper cites.
V. Turaev (1994) Quantum Invariants of Knots and 3-Manifolds, de Gruyter
1994
Earlier work this paper cites.
J. C. Baez and J. Dolan (1995) Higher-dimensional algebra and topological quantum field theory. Journal of Mathematical Physics
1995
Earlier work this paper cites.
H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher (1996) Noncommuting mixed states cannot be broadcast. Physical Review Letters
1996
Earlier work this paper cites.
B. Coecke, E. O. Paquette and D. Pavlovic (2009) Classical and quantum structuralism. To appear in Semantic Techniques for Quantum Computation, I. Mackie and S. Gay (eds), Cambridge University Press. arXiv:0904.1997
Original
1997
Earlier work this paper cites.
J. Pearl. Causality: Models, Reasoning and Inference. Cambridge University Press, 2000
2000
Earlier work this paper cites.
D. N. Yetter (2001) Functorial Knot Theory. Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants. World Scientific
Original
2001
Earlier work this paper cites.

Similar

J. Kock (2003) Frobenius Algebras and 2D Topological Quantum Field Theories. Cambridge University Press
2003
Cited alongside, same era.
S. Abramsky and B. Coecke (2004) A categorical semantics of quantum protocols. In: Proceedings of 19th IEEE conference on Logic in Computer Science, pages 415–425. IEEE Press. arXiv:quant-ph/0402130
2004
Cited alongside, same era.
P. Hayden, R. Jozsa, D. Petz and A. Winter (2004) Structure of states which satisfy strong subadditivity of quantum entropy with equality. Communications in Mathematical Physics
2004
Cited alongside, same era.
S. Lack (2004) Composing PROPs. Theory and Applications of Categories
2004
Cited alongside, same era.
J. C. Baez (2006) Quantum quandaries: a category-theoretic perspective. In: The Structural Foundations of Quantum Gravity, D. Rickles, S. French and J. T. Saatsi (Eds), pages 240–266. Oxford University Press. arXiv:quant-ph/0404040

Then

B. Coecke, D. Pavlovic, and J. Vicary (2008) A new description of orthogonal bases. Mathematical Structures in Computer Science. To appear. arXiv:0810.0812
Original
2008
Later among the works it cites.
M. Hasegawa, M. Hofmann and G. Plotkin (2008) Finite dimensional vector spaces are complete for traced symmetric monoidal categories. Lecture Notes in Computer Science
2008
Later among the works it cites.
J. Vicary (2008) Categorical Formulation of Finite-dimensional C*-algebras. Electronic Notes in Theoretical Computer Science
Original
2008
Later among the works it cites.
H. Barnum and A. Wilce (2009) Ordered linear spaces and categories as frameworks for information-processing characterizations of quantum and classical theory. arXiv:0908.2354
Original
2009
Later among the works it cites.
B. Coecke and E. O. Paquette (2009) Categories for the practising physicist. In: New Structures for Physics, B. Coecke (ed), pages 173–286. Lecture Notes in Physics, Springer-Verlag. arXiv:0905.3010

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

2006

Cited alongside, same era.

H. Barnum, J. Barrett, M. Leifer and A. Wilce (2006) Cloning and broadcasting in generic probabilistic theories. arXiv:quant-ph/0611295

2006

Cited alongside, same era.

B. Coecke and E. O. Paquette (2006) POVMs and Naimark’s theorem without sums. Electronic Notes in Theoretical Computer Science

2006

Cited alongside, same era.

A. D. Lauda (2006) Frobenius algebras and ambidextrous adjunctions. Theory and Applications of Categories

2006

Cited alongside, same era.

M. S. Leifer (2006) Quantum dynamics as an analog of conditional probability. Physical Review A

2006

Cited alongside, same era.

J. Barrett (2007) Information processing in general probabilistic theories. Physical Review A

2007

Cited alongside, same era.

B. Coecke and D. Pavlovic (2007) Quantum measurements without sums. In: Mathematics of Quantum Computing and Technology, G. Chen, L. Kauffman and S. Lamonaco (eds), pages 567–604. Taylor and Francis. arXiv:quant-ph/0608035

2007

Cited alongside, same era.

P. Selinger (2007) Dagger compact categories and completely positive maps. Electronic Notes in Theoretical Computer Science

2007

Cited alongside, same era.

Original

2009