Rao-Blackwell theorem
In statistics, the Rao–Blackwell theorem, sometimes referred to as the Rao-Blackwell-Kolmogorov theorem, is a result which characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar criteria. 在统计学领域, Rao–Blackwell 理论是一种结果：以将一个粗略的估计器转化为一个经过均方差或其它类似的准则优化的估计器为特征。 The Rao-Blackwell theorem states that if g(X) is any kind of estimator of a parameter θ, then the conditional expectation of g(X) given T(X), where T is a sufficient statistic, is typically a better estimator of θ, and is never worse. Sometimes one can very easily construct a very crude estimator g(X), and then evaluate that conditional expected value to get an estimator that is in various senses optimal. Rao-Blackwell 理论认为对于参数θ的任何一种估计器g(X)，在给定T(X)的条件下g(X)的条件期望是参数θ一个更好估计，这里T是一个充分统计量。有时候很容易构造g(X)的一个粗略估计器，然后计算它的条件期望值以得出一个在某种（各种）意义下是最优的估计器。 The theorem is named after Calyampudi Radhakrishna Rao and David Blackwell. The process of transforming an estimator using the Rao-Blackwell theorem is sometimes called Rao-Blackwellization. The transformed estimator is called the Rao-Blackwell estimator. 这个理论是以Calyampudi Radhakrishna Rao和David Blackwell两人的名字命名的。应用 Rao-Blackwell theorem 寻找新的更优估计的过程通常被称为Rao-Blackwellization，寻找到的新统计量被称为Rao-Blackwell 统计量。 A sufficient statistic T(X) is a statistic calculated from data X to estimate some parameter θ for which it is true that no other statistic which can be calculated from data X provides any additional information about θ. It is defined as an observable random variable such that the conditional probability distribution of all observable data X given T(X) does not depend on the unobservable parameter θ, such as the mean or standard deviation of the whole population from which the data X was taken. In the most frequently cited examples, the "unobservable" quantities are parameters that parametrize a known family of probability distributions according to which the data are distributed. 一个充分统计量T(X)是一个从数据X中去估计某些参数θ的统计量，但是没有可以从数据X中计算出来的其他统计量可以提供关于θ的额外信息。在给予T(X)的情况下所有可观测数据X的条件概率分布不依赖于不可观测的参数θ，例如样本X的总体的均值和标准差。在大多数可以引证的例子中，不可观测量是那些参数化数据X服从的已知类型概率分布的参数。 A Rao–Blackwell estimator δ1(X) of an unobservable quantity θ is the conditional expected value E(δ(X) | T(X)) of some estimator δ(X) given a sufficient statistic T(X). Call δ(X) the "original estimator" and δ1(X) the "improved estimator". It is important that the improved estimator be observable, i.e. that it not depend on θ. Generally, the conditional expected value of one function of these data given another function of these data does depend on θ, but the very definition of sufficiency given above entails that this one does not. 在给定充分统计量T(X)的条件下，估计器δ(X) 的条件期望值E(δ(X) | T(X))是不可观测量θ的一个Rao–Blackwell估计器δ1(X)，δ(X)叫做初始估计，δ1(X)叫改进的估计。重要的是改进的估计器δ1(X)是可观测的，即它不依赖参数θ。一般来说，这些数据的一个函数的条件期望值，在给于另一个这些函数的条件下，确实要依赖θ。但是“充分性定义”使得这个δ1(X)不依赖。 怎样理解充分统计量（sufficient statistic）？ 统计量是数据的函数。函数就是一种“浓缩”讯息的动作。因此, 统计量中所包含的讯息, 通常比整个样本数据所包含的来得少。例如样本的顺序统计量只包含了有哪些值出现,而不同值出现的顺序这样的讯息不见了。但统计量比样本原数据少掉的讯息可能是无关紧要的，和我们要了解的群体特性不相干, 如上述数据出现顺序在很多时候和我们关心的群体特性无关。统计量所包含关于群体特性的讯息不比原样本数据少, 就是充分统计量。 --------------------------------------------------------- 如果翻译的不正确，欢迎指出。
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If you want to estimate a state vector that has components nonlinearly related to your measurement model or your process model, you may use some numerical estimators such as point mass filter or particle filter (Monte Carlo). These estimators are non optimal because you solve the integration in the equation of the mathematical expectation in a numeric way not in analytical way.
Rao-Blackwelliz theorem says that if we have some linear components inside the state vector (state vector=nonlinear components+linear components), we can marginalize these linear components and track (estimate) them analytically (using Kalman filter) and the nonlinear components only will be estimated in a numeric way. Using such a separation will increase the accuracy of your estimator as the variance of ur estimator using Rao-Blacwellization is less than the variance of your estimator without using it. Also, Rao-blackwellization is used to overcome the problem of particle depletion when you want to estimate a constant. As Justin said you can check Rao-Blacwellization with particle filter for a start.
