描绘经典和量子贝叶斯推断
Picturing classical and quantum Bayesian inference
摘要：
我们介绍了一个图形框架贝叶斯推断，是足够普遍的，以适应不仅是标准的情况下，但最近的建议，量子贝叶斯推断的理论，其中一个认为密度算子，而不是概率分布的代表程度的信念。图解框架是在对称monoidal范畴的图形语言和紧凑的结构和Frobenius结构，其中贝叶斯反演归结为一个适当的紧凑结构的换位。我们表征经典的贝叶斯推断的图形属性，并表明，我们的方法消除了一些纯粹的传统元素，出现在其共同的表示，如是否程度的信念表示的概率或熵量。我们还介绍了一个量子演算，其中的Frobenius结构是非交换的，并表明它可以容纳Leifer的演算的“条件密度算子”。条件独立的概念也推广到我们的图形设置，我们做了一些初步的贝叶斯网络理论的连接。最后，我们演示了如何构建一个图形贝叶斯演算内的任何匕首紧凑的范畴。
英文摘要：
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.
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