A Bayesian Characterization of Relative Entropy
A Bayesian Characterization of Relative Entropy
复制标题
相对熵的贝叶斯表征
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
T. Fritz
中科院分区：
文献类型：
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作者：
J. Baez;T. Fritz
我们给出了相对熵的一个新的刻画，也称为Kullback-Leibler散度。我们使用了一些与概率论相关的有趣类别。特别地，我们考虑范畴FinStat，其中对象是具有概率分布的有限集，而态射是保测函数$f：X o Y$连同随机右逆$S：Y o X$。函数$f$可以被认为是一个测量过程，而S提供了一个假设，即在给定测量结果的情况下被测量系统的状态。给定这些数据，我们可以通过将$Y$上的概率分布沿着$S$向前推来定义$X$上的概率分布相对于给定的“先验”的熵。如果这些分布一致，我们说$S$是“最优的”。我们证明了从FinStat到当$S$是最优时消失的加性么半群$[0，inty]$的任何凸线性下半连续函子一定是这个相对熵的标量倍数.我们的证明独立于所有早期的刻画，但灵感来自于Petz的工作。
We give a new characterization of relative entropy, also known as the Kullback-Leibler divergence. We use a number of interesting categories related to probability theory. In particular, we consider a category FinStat where an object is a finite set equipped with a probability distribution, while a morphism is a measure-preserving function $f: X o Y$ together with a stochastic right inverse $s: Y o X$. The function $f$ can be thought of as a measurement process, while s provides a hypothesis about the state of the measured system given the result of a measurement. Given this data we can define the entropy of the probability distribution on $X$ relative to the "prior" given by pushing the probability distribution on $Y$ forwards along $s$. We say that $s$ is "optimal" if these distributions agree. We show that any convex linear, lower semicontinuous functor from FinStat to the additive monoid $[0,infty]$ which vanishes when $s$ is optimal must be a scalar multiple of this relative entropy. Our proof is independent of all earlier characterizations, but inspired by the work of Petz.
影响因子：
2.7
作者：
Baez, John C.;Fritz, Tobias;Leinster, Tom
通讯作者：
Leinster, Tom