Picturing classical and quantum Bayesian inference
Picturing classical and quantum Bayesian inference
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DOI：
10.1007/s11229-011-9917-5
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发表时间：
2011-02
期刊：
影响因子：
1.5
通讯作者：
B. Coecke;R. Spekkens
中科院分区：
文献类型：
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作者：
B. Coecke;R. Spekkens
我们为贝叶斯推理引入了一个图形框架，该框架具有足够的普遍性，不仅适用于标准情况，而且适用于最近提出的量子贝叶斯推理理论，其中考虑密度算子而不是概率分布作为可信度的代表。图解框架是用对称一元范畴和紧致结构以及其中的Frobenius结构的图形语言表述的，其中贝叶斯反演归结为相对于适当的紧致结构的转置。我们用图形属性来描述经典贝叶斯推理，并证明我们的方法消除了出现在常见表示中的一些纯粹的传统元素，例如信念程度是否由概率或熵量表示。我们还引入了一种类量子演算，其中Frobenius结构是非交换的，并表明它可以适应Leifer的“条件密度算子”演算。条件独立的概念也被推广到我们的图形设置中，并与贝叶斯网络理论建立了一些初步的联系。最后，我们演示了如何在任意短紧范畴内构造一个图解贝叶斯演算。
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.