A non-commutative Bayes' theorem
- URL: http://arxiv.org/abs/2005.03886v1
- Date: Fri, 8 May 2020 07:51:01 GMT
- Title: A non-commutative Bayes' theorem
- Authors: Arthur J. Parzygnat, Benjamin P. Russo
- Abstract summary: We prove an analogue of Bayes' theorem in the joint classical and quantum context.
We further develop non-commutative almost everywhere equivalence.
We illustrate how the procedure works for several examples relevant to quantum information theory.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Using a diagrammatic reformulation of Bayes' theorem, we provide a necessary
and sufficient condition for the existence of Bayesian inference in the setting
of finite-dimensional $C^*$-algebras. In other words, we prove an analogue of
Bayes' theorem in the joint classical and quantum context. Our analogue is
justified by recent advances in categorical probability theory, which have
provided an abstract formulation of the classical Bayes' theorem. In the
process, we further develop non-commutative almost everywhere equivalence and
illustrate its important role in non-commutative Bayesian inversion. The
construction of such Bayesian inverses, when they exist, involves solving a
positive semidefinite matrix completion problem for the Choi matrix. This gives
a solution to the open problem of constructing Bayesian inversion for
completely positive unital maps acting on density matrices that do not have
full support. We illustrate how the procedure works for several examples
relevant to quantum information theory.
Related papers
- Conditional Optimal Transport on Function Spaces [53.9025059364831]
We develop a theory of constrained optimal transport problems that describe block-triangular Monge maps.
This generalizes the theory of optimal triangular transport to separable infinite-dimensional function spaces with general cost functions.
We present numerical experiments that demonstrate the computational applicability of our theoretical results for amortized and likelihood-free inference of functional parameters.
arXiv Detail & Related papers (2023-11-09T18:44:42Z) - Connecting classical finite exchangeability to quantum theory [69.62715388742298]
Exchangeability is a fundamental concept in probability theory and statistics.
We show how a de Finetti-like representation theorem for finitely exchangeable sequences requires a mathematical representation which is formally equivalent to quantum theory.
arXiv Detail & Related papers (2023-06-06T17:15:19Z) - Quantum Bayesian Inference in Quasiprobability Representations [0.0]
Bayes' rule plays a crucial piece of logical inference in information and physical sciences alike.
quantum versions of Bayes' rule have been expressed in the language of Hilbert spaces.
arXiv Detail & Related papers (2023-01-05T08:16:50Z) - Efficient simulation of Gottesman-Kitaev-Preskill states with Gaussian
circuits [68.8204255655161]
We study the classical simulatability of Gottesman-Kitaev-Preskill (GKP) states in combination with arbitrary displacements, a large set of symplectic operations and homodyne measurements.
For these types of circuits, neither continuous-variable theorems based on the non-negativity of quasi-probability distributions nor discrete-variable theorems can be employed to assess the simulatability.
arXiv Detail & Related papers (2022-03-21T17:57:02Z) - Bayesian inversion and the Tomita-Takesaki modular group [0.0]
We show that conditional expectations, optimal hypotheses, disintegrations, and adjoints of unital completely positive maps, are all instances of Bayesian inverses.
arXiv Detail & Related papers (2021-12-06T15:57:33Z) - Bayesian Bellman Operators [55.959376449737405]
We introduce a novel perspective on Bayesian reinforcement learning (RL)
Our framework is motivated by the insight that when bootstrapping is introduced, model-free approaches actually infer a posterior over Bellman operators, not value functions.
arXiv Detail & Related papers (2021-06-09T12:20:46Z) - The semiring of dichotomies and asymptotic relative submajorization [0.0]
We study quantum dichotomies and the resource theory of asymmetric distinguishability using a generalization of Strassen's theorem on preordered semirings.
We find that an variant of relative submajorization, defined on unnormalized dichotomies, is characterized by real-valued monotones that are multiplicative under the tensor product and additive under the direct sum.
arXiv Detail & Related papers (2020-04-22T14:13:26Z) - Inverses, disintegrations, and Bayesian inversion in quantum Markov
categories [0.0]
We introduce quantum Markov categories as a structure that refines and extends a synthetic approach to probability theory and information theory.
We analyze three successively more general notions of reversibility and statistical inference.
arXiv Detail & Related papers (2020-01-21T11:19:04Z) - Distributionally Robust Bayesian Quadrature Optimization [60.383252534861136]
We study BQO under distributional uncertainty in which the underlying probability distribution is unknown except for a limited set of its i.i.d. samples.
A standard BQO approach maximizes the Monte Carlo estimate of the true expected objective given the fixed sample set.
We propose a novel posterior sampling based algorithm, namely distributionally robust BQO (DRBQO) for this purpose.
arXiv Detail & Related papers (2020-01-19T12:00:33Z) - Joint measurability meets Birkhoff-von Neumann's theorem [77.34726150561087]
We prove that joint measurability arises as a mathematical feature of DNTs in this context, needed to establish a characterisation similar to Birkhoff-von Neumann's.
We also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory.
arXiv Detail & Related papers (2018-09-19T18:57:45Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.