Fisher Information学习笔记

如何理解Fisher Information? 定义: 表示空间上的分布的参数族,每一个代表了一种分布。设PθP_\thetaPθ​的密度为pθp_\thetapθ​,那么Fisher informantion可以表示成这样的矩阵: 这里,score函数是θ\thetaθ处的对数似然的梯度(隐式取决于X)。Eθ表示对Pθ的期望。直观地说,Fisher信息捕获了梯度的variability。一个family分布的score函数有很高的varia... 阅读详情

Fisher Infomation的意义

Fisher Information 顾名思义,就是用来衡量样本数据的信息量的,通常我们有一组样本,我们在机器学习中需要估计出样本的分布,我们是利用样本所具有的信息量来估计参数的,样本中具有的信息量越多,估计的参数越准,样本的分布估计的就越接近真实分布,这里的信息量就是用Fisher Information来表示的。

什么样本信息量比较大?

我们用样本发生的概率来衡量样本本身所携带的信息量,如果样本发生的概率比较大,那么说明我们在这个样本上可以学习到的东西不多,例如机器学习中,样本一上来概率就都是1,那么参数w就基本学习不出了,相反,如果样本发生的概率比较小,就可以认为该样本携带的信息很多。

Fisher Information目标

对于最大似然估计(Maximum Likelihood Estimation)的基本思想。对于随机变量X~f(x|w),直觉上,当参数w取到的值接近准确值时,似然函数的值应该很大,所以当参数w取到准确值时,似然函数的值应该取到最大值,或者(对数)似然函数的一阶导数为0。
我们定义对数似然函数log(x|w) = log(f(x|w)),令它的导数为0,这有公式log′(x|w)=∂logf(x|w)∂w=f(x|w)′f(x|w)
根据上文,如果log′(x|w)非常接近于0,那么我们基本学习不到太多跟参数w有关的知识,换句话说模型基本不会更新了;相反,如果|log′(x|w)|很大,或者说|log′(x|w)2|很大,那么样本就提供了比较多的关于参数w的信息。因此,我们可以用|log′(x|w)2|来衡量提供的信息(information)。但是X是随机变量,所以我们就考虑|log′(x|w)2|

Fisher Information(费雪信息)详解 Fisher Information 是一种衡量“随机观测样本携带的未知参数θ\thetaθ的信息量”的方法,其中θ\thetaθ为待估计的参数。假定观测随机变量序列为X1X2X3X1​X2​...X3​,且都服从概率分布fX;θf(X;\theta)fX;θLX;θ∏i1nfXi;θLX;θi1∏n​fXi​;θ对数似然函数对θ\thetaθ求导并令一阶导数为0,则可以得到θ。 阅读详情

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