复习笔记Day121：卡尔曼滤波的推导

2023-07-28 01:46 作者:间宫_卓司 0人读过 | 我要投稿

这篇专栏只涉及到卡尔曼滤波的一个比较原始的推导，比较现代的推导见

https://zhuanlan.zhihu.com/p/166342719

首先来介绍一下卡尔曼滤波要解决的问题：对于给定的系统

$%5Cbegin%7Bcases%7D%0A%09x%5Cleft(%20k%2B1%20%5Cright)%20%3D%5CvarPhi%20x%5Cleft(%20k%20%5Cright)%5C%5C%0A%09y%5Cleft(%20k%20%5Cright)%20%3D%5CvarTheta%20x%5Cleft(%20k%20%5Cright)%5C%5C%0A%5Cend%7Bcases%7D$

其中 $x(k)$ 为实际值， $y(k)$ 为观测值，而现实中测量以及系统本身都是存在误差的，也就是说，在这个系统中，初值不再是一个数，而是服从某个分布的随机变量；观测和迭代也不可能是绝对准确的，要加上一个噪声。在这样的假设下，模型就变成了

$%5Cbegin%7Bcases%7D%0A%09x%5Cleft(%20k%2B1%20%5Cright)%20%3D%5CvarPhi%20x%5Cleft(%20k%20%5Cright)%20%2B%5Cxi%20%5Cleft(%20k%20%5Cright)%5C%5C%0A%09y%5Cleft(%20k%20%5Cright)%20%3D%5CvarTheta%20x%5Cleft(%20k%20%5Cright)%20%2B%5Ceta%20%5Cleft(%20k%20%5Cright)%5C%5C%0A%5Cend%7Bcases%7D$

其中假设了初值以及噪声都是服从正态分布的。

因为加入了噪声，所以要绝对精确的得知 $x(k)$ 的值是不可能的，所以只能去求在已知信息下，关于 $x(k)$ 的最优估计 $%5Chat%7Bx%7D%5Cleft(%20k%7Ck%20%5Cright)%20$ ，其中后一个k代表前k步的信息作为已知信息

首先需要明确的是何为最优，在我参考的教材中，对最优的定义见复习笔记Day119，同样在那篇文章里说明了，此时有 $x_0$ ，也就是说：在已知观测值 $y%5Cleft(%201%20%5Cright)%20%2Cy%5Cleft(%202%20%5Cright)%20%2C%5Ccdots%20%2Cy%5Cleft(%20n%20%5Cright)%20$ 的情况下，对 $x(k)$ 的最优估计正是限制在这些观测值下迭代条件期望。下面分几步来给出计算这个值的递推公式

注意这里的迭代关系都是线性的，故 $x(k)%2Cy(k)$ 都是服从某个多维正态分布的（注意它们是向量而不是一个数）

首先先来研究一下如果随机变量 $X$ 的概率密度函数为 $f(x)$ ，随机变量 $Y$ 的概率密度函数为 $f(y)$ ，它们的联合分布的概率密度函数为 $f(x%2Cy)$ 。那么 $X$ 在 $Y$ 的限制下的条件概率密度函数如何？，也就是去计算 $f%5Cleft(%20x%7Cy%20%5Cright)%20%3D%5Cfrac%7Bf%5Cleft(%20x%2Cy%20%5Cright)%7D%7Bf%5Cleft(%20y%20%5Cright)%7D$

下面来做一些假设，设 $X%5Csim%20N%5Cleft(%20m_x%2CR_x%20%5Cright)%2CY%5Csim%20N%5Cleft(%20m_y%2CR_y%20%5Cright)%20$ , $X%2CY$ 之间的协方差矩阵为 $R_%7Bxy%7D$ ，那么随机变量 $C%3D(X%2CY)$ 的协方差矩阵就是 $R%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_x%26%09%09R_%7Bxy%7D%5C%5C%0A%09R_%7Byx%7D%26%09%09R_y%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20$ ，于是就有下面的计算过程，计算过程其实不是很复杂，不过写的比较详细，所以看起来很长。

记 $T%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09I%26%09%09-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7D%5C%5C%0A%090%26%09%09I%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20$ ，则 $R_1%3DTR%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%26%09%090%5C%5C%0A%09R_%7Byx%7D%26%09%09R_y%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20$ ，进而

$R_%7B1%7D%5E%7B-1%7D%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09%5Cleft(%20R_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%20%5Cright)%20%5E%7B-1%7D%26%09%090%5C%5C%0A%09-R_%7By%7D%5E%7B-1%7DR_%7Byx%7D%5Cleft(%20R_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%20%5Cright)%20%5E%7B-1%7D%26%09%09R_%7By%7D%5E%7B-1%7D%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20$ ，从后面的结果来看，这样做是为了把 $y$ 单独分离出来。

从而

$%5Cbegin%7Baligned%7D%0A%09%26f(x%2Cy)%5C%5C%0A%09%26%3D(2%5Cpi%20)%5E%7B-%5Cfrac%7Bn_x%2Bn_y%7D%7B2%7D%7D(%5Cdet%20R)%5E%7B-%5Cfrac%7Bn_x%2Bn_y%7D%7B2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bl%7D%0A%09x-m_x%5C%5C%0A%09y-m_y%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%20%5E%7B%5Cmathrm%7BT%7D%7DR%5E%7B-1%7D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bl%7D%0A%09x-m_x%5C%5C%0A%09y-m_y%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%20%5Cright%5C%7D%5C%5C%0A%09%26%3D(2%5Cpi%20)%5E%7B-%5Cfrac%7Bn_x%2Bn_y%7D%7B2%7D%7D(%5Cdet%20R)%5E%7B-%5Cfrac%7Bn_x%2Bn_y%7D%7B2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bl%7D%0A%09x-m_x%5C%5C%0A%09y-m_y%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%20%5E%7B%5Cmathrm%7BT%7D%7D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_x%26%09%09R_%7Bxy%7D%5C%5C%0A%09R_%7Byx%7D%26%09%09R_y%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5E%7B-1%7D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bl%7D%0A%09x-m_x%5C%5C%0A%09y-m_y%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%20%5Cright%5C%7D%5C%5C%0A%09%26%3D(2%5Cpi%20)%5E%7B-%5Cfrac%7Bn_x%2Bn_y%7D%7B2%7D%7D(%5Cdet%20R)%5E%7B-%5Cfrac%7Bn_x%2Bn_y%7D%7B2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bl%7D%0A%09x-m_x%5C%5C%0A%09y-m_y%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%20%5E%7B%5Cmathrm%7BT%7D%7D%5Cleft(%20T%5E%7B-1%7DR_1%20%5Cright)%20%5E%7B-1%7D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bl%7D%0A%09x-m_x%5C%5C%0A%09y-m_y%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%20%5Cright%5C%7D%5C%5C%0A%09%26%3D(2%5Cpi%20)%5E%7B-%5Cfrac%7Bn_x%2Bn_y%7D%7B2%7D%7D(%5Cdet%20R)%5E%7B-%5Cfrac%7Bn_x%2Bn_y%7D%7B2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bl%7D%0A%09x-m_x%5C%5C%0A%09y-m_y%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%20%5E%7B%5Cmathrm%7BT%7D%7DR_%7B1%7D%5E%7B-1%7DT%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bl%7D%0A%09x-m_x%5C%5C%0A%09y-m_y%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%20%5Cright%5C%7D%5C%5C%0A%09%26%3D(2%5Cpi%20)%5E%7B-%5Cfrac%7Bn_x%2Bn_y%7D%7B2%7D%7D(%5Cdet%20R)%5E%7B-%5Cfrac%7Bn_x%2Bn_y%7D%7B2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bl%7D%0A%09x-m_x%5C%5C%0A%09y-m_y%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%20%5E%7B%5Cmathrm%7BT%7D%7D%20%5Cright.%5C%5C%0A%09%26%5Cleft.%20%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09%5Cleft(%20R_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%20%5Cright)%20%5E%7B-1%7D%26%09%090%5C%5C%0A%09-R_%7By%7D%5E%7B-1%7DR_%7Byx%7D%5Cleft(%20R_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%20%5Cright)%20%5E%7B-1%7D%26%09%09R_%7By%7D%5E%7B-1%7D%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09I%26%09%09-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7D%5C%5C%0A%090%26%09%09I%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bl%7D%0A%09x-m_x%5C%5C%0A%09y-m_y%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%20%5Cright%5C%7D%5C%5C%0A%09%26%3D(2%5Cpi%20)%5E%7B-%5Cfrac%7Bn_x%2Bn_y%7D%7B2%7D%7D(%5Cdet%20R)%5E%7B-%5Cfrac%7Bn_x%2Bn_y%7D%7B2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%20%5Cleft(%20x-m_x%20%5Cright)%20%5ET%5Cleft(%20R_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%20%5Cright)%20%5E%7B-1%7D%20%5Cright.%20%5Cright.%5C%5C%0A%09%26%5Cleft.%20-%5Cleft(%20y-m_y%20%5Cright)%20%5ETR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%5Cleft(%20R_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%20%5Cright)%20%5E%7B-1%7D%2C%5Cleft(%20y-m_y%20%5Cright)%20%5ETR_%7By%7D%5E%7B-1%7D%20%5Cright%5D%5C%5C%0A%09%26%5Cleft.%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09%5Cleft(%20x-m_x%20%5Cright)%20-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7D%5Cleft(%20y-m_y%20%5Cright)%5C%5C%0A%09y-m_y%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%20%5Cright%5C%7D%5C%5C%0A%09%26%5Ctriangleq%20(2%5Cpi%20)%5E%7B-%5Cfrac%7Bn_x%7D%7B2%7D%7D(%5Cdet%20R)%5E%7B-%5Cfrac%7Bn_x%7D%7B2%7D%7D%5Cexp%20%5Cleft.%20%5Cleft%5C%7B%20-%5Cfrac%7B1%7D%7B2%7D%5Cleft(%20x-m_%7Bx%7Cy%7D%20%5Cright)%20%5ETR_%7Bx%7Cy%7D%5E%7B-1%7D%5Cleft(%20x-m_%7Bx%7Cy%7D%20%5Cright)%20%5Cright.%20%5Cright%5C%7D%20f%5Cleft(%20y%20%5Cright)%5C%5C%0A%5Cend%7Baligned%7D$

其中最后一个等号的计算过程如下

$%5Cbegin%7Baligned%7D%0A%09%26%5Cleft(%20x-m_x%20%5Cright)%20%5ET%5Cleft(%20R_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%20%5Cright)%20%5E%7B-1%7D-%5Cleft(%20y-m_y%20%5Cright)%20%5ETR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%5Cleft(%20R_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%20%5Cright)%20%5E%7B-1%7D%5C%5C%0A%09%26%3D%5Cleft(%20%5Cleft(%20x-m_x%20%5Cright)%20%5ET-%5Cleft(%20y-m_y%20%5Cright)%20%5ETR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%20%5Cright)%20%5Cleft(%20R_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%20%5Cright)%20%5E%7B-1%7D%5C%5C%0A%09%26%3D%5Cleft(%20x-m_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7D%5Cleft(%20y-m_y%20%5Cright)%20%5Cright)%20%5ET%5Cleft(%20R_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%20%5Cright)%20%5E%7B-1%7D%5C%5C%0A%09%26%5Ctriangleq%20%5Cleft(%20x-m_%7Bx%7Cy%7D%20%5Cright)%20R_%7Bx%7Cy%7D%5E%7B-1%7D%5C%5C%0A%5Cend%7Baligned%7D$

也就是说

$%5Cfrac%7Bf%5Cleft(%20x%2Cy%20%5Cright)%7D%7Bf%5Cleft(%20y%20%5Cright)%7D%3D(2%5Cpi%20)%5E%7B-%5Cfrac%7Bn_x%7D%7B2%7D%7D(%5Cdet%20R)%5E%7B-%5Cfrac%7Bn_x%7D%7B2%7D%7D%5Cexp%20%5Cleft.%20%5Cleft%5C%7B%20-%5Cfrac%7B1%7D%7B2%7D%5Cleft(%20x-m_%7Bx%7Cy%7D%20%5Cright)%20%5ETR_%7Bx%7Cy%7D%5E%7B-1%7D%5Cleft(%20x-m_%7Bx%7Cy%7D%20%5Cright)%20%5Cright.%20%5Cright%5C%7D%20$

注意到这个式子中， $%5Cfrac%7Bf%5Cleft(%20x%2Cy%20%5Cright)%7D%7Bf%5Cleft(%20y%20%5Cright)%7D$ 同样可以视为某个服从正态分布的随机变量的概率密度函数

接下来来计算

记

$Z%5Csim%20N%5Cleft(%20m_%7Bx%7Cy%7D%2CR_%7Bx%7Cy%7D%5E%7B%7D%20%5Cright)%20%3DN%5Cleft(%20m_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7D%5Cleft(%20y-m_y%20%5Cright)%20%2CR_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%20%5Cright)%20$ ，注意这样假设把y视为了常量，那么

$%5Cmathbb%7BE%7D%20%5Cleft(%20X%7Cy%20%5Cright)%20%3D%5Cmathbb%7BE%7D%20%5Cleft(%20z%20%5Cright)%20%3Dm_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7D%5Cleft(%20y-m_y%20%5Cright)%20$

注意这里把 $y$ 视为了一个数，而 $%5Cmathbb%7BE%7D%20%5Cleft(%20X%7Cy%20%5Cright)%20$ 是 $%5Cmathbb%7BE%7D%20%5Cleft(%20X%7CY%3Dy%20%5Cright)%20$ 的缩写，所以如果把 $y$ 视为一个随机变量，那么 $%5Cmathbb%7BE%7D%20%5Cleft(%20X%7Cy%20%5Cright)%20$ 也是一个随机变量，所以干脆记成 $%5Cmathbb%7BE%7D%20%5Cleft(%20X%7CY%20%5Cright)%20$ （这块不太能讲清楚，凑合着看吧）。总之，这样就得到了

$%5Cmathbb%7BE%7D%20%5Cleft(%20X%7CY%20%5Cright)%20%3Dm_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7D%5Cleft(%20Y-m_y%20%5Cright)%20$

接下来，为了获得递推式，来计算 $%5Cmathbb%7BE%7D%20%5Cleft(%20X%7Cy%2Cz%20%5Cright)%20$ ，其中 $X%2CY%2CZ$ 都是服从正态分布的。先考虑简单的情况，即 $Y%2CZ$ 是相互独立的情况

$%5Cbegin%7Baligned%7D%0A%09%5Cmathbb%7BE%7D%20%5Cleft(%20X%7Cy%2Cz%20%5Cright)%20%26%3Dm_x-%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_%7Bxy%7D%26%09%09R_%7Bxz%7D%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_y%26%09%09%5C%5C%0A%09%26%09%09R_z%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5E%7B-1%7D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09y-m_y%5C%5C%0A%09z-m_z%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%5C%5C%0A%09%26%3Dm_x-%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_%7Bxy%7D%26%09%09R_%7Bxz%7D%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_%7By%7D%5E%7B-1%7D%26%09%09%5C%5C%0A%09%26%09%09R_%7Bz%7D%5E%7B-1%7D%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09y-m_y%5C%5C%0A%09z-m_z%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%5C%5C%0A%09%26%3D%5Cleft(%20m_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7D%5Cleft(%20y-m_y%20%5Cright)%20%5Cright)%20%2B%5C%5C%0A%09%26%5Cleft(%20m_x-R_%7Bxy%7DR_%7Bz%7D%5E%7B-1%7D%5Cleft(%20z-m_z%20%5Cright)%20%5Cright)%20-m_x%5C%5C%0A%09%26%3D%5Cmathbb%7BE%7D%20%5Cleft(%20X%7Cy%20%5Cright)%20%2B%5Cmathbb%7BE%7D%20%5Cleft(%20X%7Cz%20%5Cright)%20-m_x%5C%5C%0A%5Cend%7Baligned%7D$

和上面一样，也有 $%5Cmathbb%7BE%7D%20%5Cleft(%20X%7CY%2CZ%20%5Cright)%20%3D%5Cmathbb%7BE%7D%20%5Cleft(%20X%7CY%20%5Cright)%20%2B%5Cmathbb%7BE%7D%20%5Cleft(%20X%7CZ%20%5Cright)%20-m_x$

接下来讨论 $Y%2CZ$ 相关的情况，实际上有结论 $%5Cmathbb%7BE%7D%20%5Cleft(%20X%7CY%2CZ%20%5Cright)%20%3D%5Cmathbb%7BE%7D%20%5Cleft(%20X%7CY%2C%5Cwidetilde%7BZ%7D%5Cleft(%20Y%20%5Cright)%20%5Cright)%20$ ，这里的 $%5Cwidetilde%7BZ%7D%5Cleft(%20Y%20%5Cright)%20$ 指的是 $Z$ 减去了Z在Y条件下的最优预测值，即 $Z-%5Cmathbb%7BE%7D%20%5Cleft(%20Z%7CY%20%5Cright)%20$ 。这个结论的证明如下

$%5Cbegin%7Baligned%7D%0A%09%26%5Cmathbb%7BE%7D%20%5Cleft(%20X%7Cy%2Cz%20%5Cright)%5C%5C%0A%09%26%3Dm_x-%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_%7Bxy%7D%26%09%09R_%7Bxz%7D%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_y%26%09%09R_%7Byz%7D%5C%5C%0A%09R_%7Bzy%7D%26%09%09R_z%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5E%7B-1%7D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09y-m_y%5C%5C%0A%09z-m_z%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%5C%5C%0A%09%26%3Dm_x-%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_%7Bxy%7D%26%09%09R_%7Bxz%7D%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_y%26%09%09R_%7Byz%7D%5C%5C%0A%09R_%7Bzy%7D%26%09%09R_z%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5E%7B-1%7D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09y-m_y%5C%5C%0A%09z-m_z%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%5C%5C%0A%09%26%3Dm_x-%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_%7Bxy%7D%26%09%09R_%7Bxz%7D%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09%5Cleft(%20R_y-R_%7Byz%7DR_%7Bz%7D%5E%7B-1%7DR_%7Bzy%7D%20%5Cright)%20%5E%7B-1%7D%26%09%090%5C%5C%0A%09-R_%7Bz%7D%5E%7B-1%7DR_%7Bzy%7D%5Cleft(%20R_y-R_%7Byz%7DR_%7Bz%7D%5E%7B-1%7DR_%7Bzy%7D%20%5Cright)%20%5E%7B-1%7D%26%09%09R_%7Bz%7D%5E%7B-1%7D%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09I%26%09%09-R_%7Byz%7DR_%7Bz%7D%5E%7B-1%7D%5C%5C%0A%090%26%09%09I%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09y-m_y%5C%5C%0A%09z-m_z%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%5C%5C%0A%09%26%3Dm_x-%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_%7Bxy%7D%26%09%09R_%7Bxz%7D%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09%5Cleft(%20R_y-R_%7Byz%7DR_%7Bz%7D%5E%7B-1%7DR_%7Bzy%7D%20%5Cright)%20%5E%7B-1%7D%26%09%090%5C%5C%0A%09-R_%7Bz%7D%5E%7B-1%7DR_%7Bzy%7D%5Cleft(%20R_y-R_%7Byz%7DR_%7Bz%7D%5E%7B-1%7DR_%7Bzy%7D%20%5Cright)%20%5E%7B-1%7D%26%09%09R_%7Bz%7D%5E%7B-1%7D%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09y-m_y-R_%7Byz%7DR_%7Bz%7D%5E%7B-1%7D%5Cleft(%20z-m_z%20%5Cright)%5C%5C%0A%09z-m_z%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%5C%5C%0A%09%26%3Dm_x-%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_%7Bxy%7D%26%09%09R_%7Bxz%7D%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_%7Bx%7Cy%7D%5E%7B-1%7D%26%09%090%5C%5C%0A%09-R_%7Bz%7D%5E%7B-1%7DR_%7Bzy%7DR_%7Bx%7Cy%7D%5E%7B-1%7D%26%09%09R_%7Bz%7D%5E%7B-1%7D%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09y-%5Cmathbb%7BE%7D%20%5Cleft(%20y%7Cz%20%5Cright)%5C%5C%0A%09z-m_z%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%5C%5C%0A%09%26%3Dm_x-%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09R_%7Bxy%7D%26%09%09R_%7Bxz%7D%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09R_%7Bx%7Cy%7D%5E%7B-1%7D%5Cleft(%20y-%5Cmathbb%7BE%7D%20%5Cleft(%20y%7Cz%20%5Cright)%20%5Cright)%5C%5C%0A%09-R_%7Bz%7D%5E%7B-1%7DR_%7Bzy%7DR_%7Bx%7Cy%7D%5E%7B-1%7D%5Cmathbb%7BE%7D%20%5Cleft(%20y%7Cz%20%5Cright)%20%2BR_%7Bz%7D%5E%7B-1%7D%5Cleft(%20z-m_z%20%5Cright)%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%5C%5C%0A%09%26%3Dm_x-R_%7Bxy%7DR_%7Bx%7Cy%7D%5E%7B-1%7D%5Cleft(%20y-%5Cmathbb%7BE%7D%20%5Cleft(%20y%7Cz%20%5Cright)%20%5Cright)%20-R_%7Bxz%7D%5Cleft(%20-R_%7Bz%7D%5E%7B-1%7DR_%7Bzy%7DR_%7Bx%7Cy%7D%5E%7B-1%7D%5Cleft(%20y-%5Cmathbb%7BE%7D%20%5Cleft(%20y%7Cz%20%5Cright)%20%5Cright)%20%2BR_%7Bz%7D%5E%7B-1%7D%5Cleft(%20z-m_z%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3D%5Cleft(%20m_x-R_%7Bxz%7DR_%7Bz%7D%5E%7B-1%7D%5Cleft(%20z-m_z%20%5Cright)%20%5Cright)%20%2B%5Cleft(%20m_x-%5Cleft(%20R_%7Bxy%7D-R_%7Bxz%7DR_%7Bz%7D%5E%7B-1%7DR_%7Bzy%7D%20%5Cright)%20R_%7Bx%7Cy%7D%5E%7B-1%7D%5Cleft(%20y-%5Cmathbb%7BE%7D%20%5Cleft(%20y%7Cz%20%5Cright)%20%5Cright)%20%5Cright)%20-m_x%5C%5C%0A%5Cend%7Baligned%7D$

其中第三个等号和之前条件概率密度函数的推导思路是一样的，第五个等号开始把 $y$ 视为了随机变量，不过是随机变量还是常量其实无所谓了吧···反正这里只是一个简化计算的符号

为了得到结论，只需要注意到

$%5Cbegin%7Baligned%7D%0A%09%26%5Cmathbb%7BE%7D%20%5Cleft(%20X%5Cleft(%20Y-m_y-R_%7Byz%7DR_%7Bz%7D%5E%7B-1%7D%5Cleft(%20Z-m_z%20%5Cright)%20%5Cright)%20%5ET%20%5Cright)%5C%5C%0A%09%26%3D%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft(%20X-m_x%20%5Cright)%20%5Cleft(%20Y-m_y-R_%7Byz%7DR_%7Bz%7D%5E%7B-1%7D%5Cleft(%20Z-m_z%20%5Cright)%20%5Cright)%20%5ET%20%5Cright)%5C%5C%0A%09%26%3DR_%7Bxy%7D-R_%7Bxz%7DR_%7Bz%7D%5E%7B-1%7DR_%7Bzy%7D%5C%5C%0A%5Cend%7Baligned%7D$

所以总而言之，上面的式子确实说明了

$%5Cmathbb%7BE%7D%20%5Cleft(%20X%7CY%2CZ%20%5Cright)%20%3D%5Cmathbb%7BE%7D%20%5Cleft(%20X%7CZ%20%5Cright)%20%2B%5Cmathbb%7BE%7D%20%5Cleft(%20X%7C%5Cwidetilde%7BY%7D%5Cleft(%20Z%20%5Cright)%20%5Cright)%20-m_x$

因为同时也有 $%5Cmathbb%7BE%7D%20%5Cleft(%20X%7CZ%2C%5Cwidetilde%7BY%7D%5Cleft(%20Z%20%5Cright)%20%5Cright)%20%3D%5Cmathbb%7BE%7D%20%5Cleft(%20X%7CZ%20%5Cright)%20%2B%5Cmathbb%7BE%7D%20%5Cleft(%20X%7C%5Cwidetilde%7BY%7D%5Cleft(%20Z%20%5Cright)%20%5Cright)%20-m_x$ ，而 $Z%2C%5Cwidetilde%7BY%7D%5Cleft(%20Z%20%5Cright)%20$ 之间是相互独立的，所以上面的过程也可以看成把 $Y$ 与 $Z$ 无关的部分提取了出来再做利用先去的结论

有了这些结论，接下来就可以计算 $%5Chat%7Bx%7D%5Cleft(%20k%7Ck%20%5Cright)%20%3D%5Cmathbb%7BE%7D%20%5Cleft(%20x%5Cleft(%20k%20%5Cright)%20%7Cy%5Cleft(%201%20%5Cright)%20%2Cy%5Cleft(%202%20%5Cright)%20%2C%5Ccdots%20%2Cy%5Cleft(%20k%20%5Cright)%20%5Cright)%20$ 了

$%5Cbegin%7Baligned%7D%0A%09%26%5Cmathbb%7BE%7D%20%5Cleft(%20x%5Cleft(%20k%20%5Cright)%20%7Cy%5Cleft(%201%20%5Cright)%20%2Cy%5Cleft(%202%20%5Cright)%20%2C%5Ccdots%20%2Cy%5Cleft(%20k%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3D%5Cmathbb%7BE%7D%20%5Cleft(%20x%5Cleft(%20k%20%5Cright)%20%7Cy%5Cleft(%201%20%5Cright)%20%2Cy%5Cleft(%202%20%5Cright)%20%2C%5Ccdots%20%2Cy%5Cleft(%20k-1%20%5Cright)%20%5Cright)%20%2B%5Cmathbb%7BE%7D%20%5Cleft(%20x%5Cleft(%20k%20%5Cright)%20%7C%5Cwidetilde%7By%7D%5Cleft(%20k%7Ck%20%5Cright)%20%5Cright)%20-m_0%5C%5C%0A%09%26%3D%5Cmathbb%7BE%7D%20%5Cleft(%20%5CvarPhi%20x%5Cleft(%20k-1%20%5Cright)%20%2B%5Cxi%20%5Cleft(%20k-1%20%5Cright)%20%7Cy%5Cleft(%201%20%5Cright)%20%2C%5Ccdots%20%2Cy%5Cleft(%20k-1%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%2B%5Cmathbb%7BE%7D%20%5Cleft(%20x%5Cleft(%20k%20%5Cright)%20%7C%5Cwidetilde%7By%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%5Cright)%20-m_0%5C%5C%0A%09%26%3D%5CvarPhi%20%5Cmathbb%7BE%7D%20%5Cleft(%20x%5Cleft(%20k-1%20%5Cright)%20%7Cy%5Cleft(%201%20%5Cright)%20%2C%5Ccdots%20%2Cy%5Cleft(%20k-1%20%5Cright)%20%5Cright)%20%2B%5Cmathbb%7BE%7D%20%5Cleft(%20x%5Cleft(%20k%20%5Cright)%20%7C%5Cwidetilde%7By%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%5Cright)%20-m_0%5C%5C%0A%09%26%3D%5CvarPhi%20%5Cmathbb%7BE%7D%20%5Cleft(%20x%5Cleft(%20k-1%20%5Cright)%20%7Cy%5Cleft(%201%20%5Cright)%20%2C%5Ccdots%20%2Cy%5Cleft(%20k-1%20%5Cright)%20%5Cright)%20%2BR_%7Bx%5Cleft(%20k%20%5Cright)%20%5Cwidetilde%7By%7D%5Cleft(%20k%7Ck-1%20%5Cright)%7DR_%7B%5Cwidetilde%7By%7D%5Cleft(%20k%7Ck-1%20%5Cright)%7D%5E%7B-1%7D%5Cwidetilde%7By%7D%5Cleft(%20k%7Ck-1%20%5Cright)%5C%5C%0A%5Cend%7Baligned%7D$

其中第三个等号是因为 $%5Ceta(k-1)$ 与 $y%5Cleft(%201%20%5Cright)%20%2C%5Ccdots%20%2Cy%5Cleft(%20k-1%20%5Cright)%20$ 都是无关的，下面就来计算一下第三个等号的第二项的各个系数，首先有

$%5Cbegin%7Baligned%7D%0A%09%26%5Cwidetilde%7By%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%3Dy%5Cleft(%20k%20%5Cright)%20-%5Cmathbb%7BE%7D%20%5Cleft(%20y%5Cleft(%20k%20%5Cright)%20%7Cy%5Cleft(%201%20%5Cright)%20%2C%5Ccdots%20%2Cy%5Cleft(%20k-1%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3Dy%5Cleft(%20k%20%5Cright)%20-%5Cmathbb%7BE%7D%20%5Cleft(%20%5CvarTheta%20x%5Cleft(%20k%20%5Cright)%20%2B%5Cxi%20%5Cleft(%20k%20%5Cright)%20%7Cy%5Cleft(%201%20%5Cright)%20%2C%5Ccdots%20%2Cy%5Cleft(%20k-1%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3Dy%5Cleft(%20k%20%5Cright)%20-%5CvarTheta%20%5Cmathbb%7BE%7D%20%5Cleft(%20x%5Cleft(%20k%20%5Cright)%20%7Cy%5Cleft(%201%20%5Cright)%20%2C%5Ccdots%20%2Cy%5Cleft(%20k-1%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3D%5CvarTheta%20x%5Cleft(%20k%20%5Cright)%20%2B%5Ceta%20%5Cleft(%20k%20%5Cright)%20-%5CvarTheta%20%5Cmathbb%7BE%7D%20%5Cleft(%20x%5Cleft(%20k%20%5Cright)%20%7Cy%5Cleft(%201%20%5Cright)%20%2C%5Ccdots%20%2Cy%5Cleft(%20k-1%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3D%5CvarTheta%20%5Cwidetilde%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%2B%5Ceta%20%5Cleft(%20k%20%5Cright)%5C%5C%0A%5Cend%7Baligned%7D$

那么

$%5Cbegin%7Baligned%7D%0A%09%26R_%7Bx%5Cleft(%20k%20%5Cright)%20%5Cwidetilde%7By%7D%5Cleft(%20k%7Ck-1%20%5Cright)%7D%5C%5C%0A%09%26%3D%5Cmathbb%7BE%7D%20%5Cleft(%20x%5Cleft(%20k%20%5Cright)%20%5Cleft(%20%5Cwidetilde%7By%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%5Cright)%20%5ET%20%5Cright)%20%3D%5C%5C%0A%09%26%3D%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft(%20x%5Cleft(%20k%20%5Cright)%20-m_%7Bx%5Cleft(%20k%20%5Cright)%7D%20%5Cright)%20%5Cleft(%20%5CvarTheta%20%5Cwidetilde%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%2B%5Ceta%20%5Cleft(%20k%20%5Cright)%20%5Cright)%20%5ET%20%5Cright)%5C%5C%0A%09%26%3D%5Cwidetilde%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%5Cwidetilde%7Bx%7D%5ET%5Cleft(%20k%7Ck-1%20%5Cright)%20%5CvarTheta%20%5ET%5C%5C%0A%5Cend%7Baligned%7D$

其中第三个等号是因为

$%5Cbegin%7Baligned%7D%0A%09%26%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft(%20x-%5Cmathbb%7BE%7D%20%5Cleft(%20x%7Cy%20%5Cright)%20%5Cright)%20%5Cmathbb%7BE%7D%20_%7B%7D%5E%7BT%7D%5Cleft(%20x%7Cy%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3D%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft(%20x-m_x%2BR_%7Bxy%7DR_%7By%7D%5E%7B-1%7D%5Cleft(%20y-m_y%20%5Cright)%20%5Cright)%20%5Cleft(%20m_x-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7D%5Cleft(%20y-m_y%20%5Cright)%20%5Cright)%20%5ET%20%5Cright)%5C%5C%0A%09%26%3D%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft(%20x-m_x%20%5Cright)%20%5Cleft(%20y-m_y%20%5Cright)%20%5ETR_%7By%7D%5E%7B-1%7DR_%7Byx%7D-R_%7Bxy%7DR_%7By%7D%5E%7B-1%7D%5Cleft(%20y-m_y%20%5Cright)%20%5Cleft(%20y-m_y%20%5Cright)%20%5ETR_%7By%7D%5E%7B-1%7DR_%7Byx%7D%20%5Cright)%5C%5C%0A%09%26%3D0%5C%5C%0A%5Cend%7Baligned%7D$

现在记 $%5Cwidetilde%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%5Cwidetilde%7Bx%7D%5ET%5Cleft(%20k%7Ck-1%20%5Cright)%20%3DP%5Cleft(%20k%7Ck-1%20%5Cright)%20$ ，则 $R_%7Bx%5Cleft(%20k%20%5Cright)%20%5Cwidetilde%7By%7D%5Cleft(%20k%7Ck-1%20%5Cright)%7D%3DP%5Cleft(%20k%7Ck-1%20%5Cright)%20%5CvarTheta%20%5ET$

同理

$%5Cbegin%7Baligned%7D%0A%09%26R_%7B%5Cwidetilde%7By%7D%5Cleft(%20k%20%5Cright)%7D%3D%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft(%20%5CvarTheta%20%5Cwidetilde%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%2B%5Ceta%20%5Cleft(%20k%20%5Cright)%20%5Cright)%20%5Cleft(%20%5CvarTheta%20%5Cwidetilde%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%2B%5Ceta%20%5Cleft(%20k%20%5Cright)%20%5Cright)%20%5ET%20%5Cright)%5C%5C%0A%09%26%3D%5CvarTheta%20P%5Cleft(%20k%7Ck-1%20%5Cright)%20%5CvarTheta%20%5ET%2BR_2%5C%5C%0A%5Cend%7Baligned%7D$

现在尚且需要得到 $P%5Cleft(%20k%7Ck-1%20%5Cright)%20$ 的递推关系式，因为

$%5Cbegin%7Baligned%7D%0A%09%26%5Cwidetilde%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%3Dx%5Cleft(%20k%20%5Cright)%20-%5Cwidehat%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%5C%5C%0A%09%26%3D%5CvarPhi%20x%5Cleft(%20k-1%20%5Cright)%20%2B%5Cxi%20%5Cleft(%20k-1%20%5Cright)%20-%5Cmathbb%7BE%7D%20%5Cleft(%20x%5Cleft(%20k%20%5Cright)%20%7Cy%5Cleft(%201%20%5Cright)%20%2C%5Ccdots%20y%5Cleft(%20n-1%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3D%5CvarPhi%20x%5Cleft(%20k-1%20%5Cright)%20%2B%5Cxi%20%5Cleft(%20k-1%20%5Cright)%20-%5Cmathbb%7BE%7D%20%5Cleft(%20x%5Cleft(%20k-1%20%5Cright)%20%2B%5Cxi%20%5Cleft(%20k-1%20%5Cright)%20%7Cy%5Cleft(%201%20%5Cright)%20%2C%5Ccdots%20y%5Cleft(%20n-1%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3D%5CvarPhi%20%5Ctilde%7Bx%7D%5Cleft(%20k-1%7Ck-1%20%5Cright)%20%2B%5Cxi%20%5Cleft(%20k-1%20%5Cright)%5C%5C%0A%5Cend%7Baligned%7D$

所以

$%5Cbegin%7Baligned%7D%0A%09%26P%5Cleft(%20k%7Ck-1%20%5Cright)%20%3D%5Cwidetilde%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%5Cwidetilde%7Bx%5ET%7D%5Cleft(%20k%7Ck-1%20%5Cright)%5C%5C%0A%09%26%3D%5Cleft(%20%5CvarPhi%20%5Ctilde%7Bx%7D%5Cleft(%20k-1%7Ck-1%20%5Cright)%20%2B%5Cxi%20%5Cleft(%20k-1%20%5Cright)%20%5Cright)%20%5Cleft(%20%5CvarPhi%20%5Ctilde%7Bx%7D%5Cleft(%20k-1%7Ck-1%20%5Cright)%20%2B%5Cxi%20%5Cleft(%20k-1%20%5Cright)%20%5Cright)%20%5ET%5C%5C%0A%09%26%3D%5CvarPhi%20%5Ctilde%7Bx%7D%5Cleft(%20k-1%7Ck-1%20%5Cright)%20%5Ctilde%7Bx%7D%5ET%5Cleft(%20k-1%7Ck-1%20%5Cright)%20%5CvarPhi%20%5ET%2BR_1%5C%5C%0A%09%26%5Ctriangleq%20%5CvarPhi%20P%5Cleft(%20k-1%7Ck-1%20%5Cright)%20%5CvarPhi%20%5ET%2BR_1%5C%5C%0A%5Cend%7Baligned%7D$

这样一来，就得到了一个递推关系：在已知 $%5Chat%7Bx%7D%5Cleft(%20k-1%7Ck-1%20%5Cright)%20$ 的情况下，可以计算出 $P%5Cleft(%20k-1%7Ck-1%20%5Cright)%20$ ，进而可以计算出 $P%5Cleft(%20k%7Ck-1%20%5Cright)%20$ ，据此又可以算出 $R_%7Bx%5Cleft(%20k%20%5Cright)%20%5Cwidetilde%7By%7D%5Cleft(%20k%20%5Cright)%7D%2CR_%7B%5Cwidetilde%7By%7D%5Cleft(%20k%20%5Cright)%7D%5E%7B-1%7D$ ，此外根据 $%5Chat%7Bx%7D%5Cleft(%20k-1%7Ck-1%20%5Cright)%20$ 还可以计算出 $%5Cwidetilde%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20$ ，进而算出 $%5Cwidetilde%7By%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20$ ，有了这些，就可以求出 $%5Chat%7Bx%7D%5Cleft(%20k%7Ck%20%5Cright)%20$ 了，这样就完成了一次迭代

写到这里其实已经推导完了，不过和课本上的格式不太一样，为了规范还是按课本上的来吧

记 $K%5Cleft(%20k%7Ck%20%5Cright)%20%3DR_%7Bx%5Cleft(%20k%20%5Cright)%20%5Cwidetilde%7By%7D%5Cleft(%20k%20%5Cright)%7DR_%7B%5Cwidetilde%7By%7D%5Cleft(%20k%20%5Cright)%7D%5E%7B-1%7D$ ，那么之前的递推式就可以写成

$%5Cbegin%7Baligned%7D%0A%09%26%5Cwidehat%7Bx%7D%5Cleft(%20k%7Ck%20%5Cright)%5C%5C%0A%09%26%3D%5CvarPhi%20%5Cwidehat%7Bx%7D%5Cleft(%20k-1%7Ck-1%20%5Cright)%20%2BK%5Cleft(%20k%7Ck%20%5Cright)%20%5Cleft(%20%5Cwidetilde%7By%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3D%5CvarPhi%20%5Cwidehat%7Bx%7D%5Cleft(%20k-1%7Ck-1%20%5Cright)%20%2BK%5Cleft(%20k%7Ck%20%5Cright)%20%5Cleft(%20y%5Cleft(%20k%20%5Cright)%20-%5CvarTheta%20%5CvarPhi%20%5Cwidehat%7Bx%7D%5Cleft(%20k-1%7Ck-1%20%5Cright)%20%5Cright)%5C%5C%0A%5Cend%7Baligned%7D$

其中第二个等号由之前 $%5Cwidetilde%7By%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20$ 的推导的第三个等号可以得到

而

$K%5Cleft(%20k%7Ck%20%5Cright)%20%3DR_%7Bx%5Cleft(%20k%20%5Cright)%20%5Cwidetilde%7By%7D%5Cleft(%20k%20%5Cright)%7DR_%7B%5Cwidetilde%7By%7D%5Cleft(%20k%20%5Cright)%7D%5E%7B-1%7D%3DP%5Cleft(%20k%7Ck-1%20%5Cright)%20%5CvarTheta%20%5ET%5Cleft(%20%5CvarTheta%20P%5Cleft(%20k%7Ck-1%20%5Cright)%20%5CvarTheta%20%5ET%2BR_2%20%5Cright)%20%5E%7B-1%7D$

因为

$%5Cbegin%7Baligned%7D%0A%09%5Cwidehat%7Bx%7D%5Cleft(%20k%7Ck%20%5Cright)%20%26%3D%5Cwidehat%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%2BK%5Cleft(%20k%7Ck%20%5Cright)%20%5Cleft(%20%5Ctilde%7By%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3D%5Cwidehat%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%2BK%5Cleft(%20k%7Ck%20%5Cright)%20%5Cleft(%20%5CvarTheta%20%5Ctilde%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%2B%5Ceta%20%5Cleft(%20k%20%5Cright)%20%5Cright)%5C%5C%0A%5Cend%7Baligned%7D$

故

$%5Cbegin%7Baligned%7D%0A%09%26%5Cwidetilde%7Bx%7D%5Cleft(%20k%7Ck%20%5Cright)%20%3Dx%5Cleft(%20k%20%5Cright)%20-%5Cwidehat%7Bx%7D%5Cleft(%20k%7Ck%20%5Cright)%5C%5C%0A%09%26%3Dx%5Cleft(%20k%20%5Cright)%20-%5Cleft(%20%5Cwidehat%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%2BK%5Cleft(%20k%7Ck%20%5Cright)%20%5Cleft(%20%5CvarTheta%20%5Ctilde%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20%2B%5Ceta%20%5Cleft(%20k%20%5Cright)%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3D%5Cleft(%20I-K%5Cleft(%20k%7Ck%20%5Cright)%20%5CvarTheta%20%5Cright)%20%5Ctilde%7Bx%7D%5Cleft(%20k%7Ck-1%20%5Cright)%20-K%5Cleft(%20k%7Ck%20%5Cright)%20%5Ceta%20%5Cleft(%20k%20%5Cright)%5C%5C%0A%5Cend%7Baligned%7D$

所以

$%5Cbegin%7Baligned%7D%0A%09%26P(k%5Cmid%20k)%3D%5Cmathbb%7BE%7D%20%5Cleft(%20%5Ctilde%7Bx%7D(k%5Cmid%20k)%5Ctilde%7Bx%7D%5E%7B%5Cmathrm%7BT%7D%7D(k%5Cmid%20k)%20%5Cright)%5C%5C%0A%09%3D%26%5BI-K(k%5Cmid%20k)%5CTheta%20%5DP(k%5Cmid%20k-1)%5BI-K(k%5Cmid%20k)%5CTheta%20%5D%5E%7B%5Cmathrm%7BT%7D%7D%5C%5C%0A%09%26%2BK(k%5Cmid%20k)R_2K(k%5Cmid%20k)%5C%5C%0A%5Cend%7Baligned%7D$