📄 卡尔曼滤波-来自wiki（2）_mopish ranger.htm
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<DIV class=tit>卡尔曼滤波-来自Wiki（2）</DIV>
<DIV class=date>2008-08-06 23:01</DIV>
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<P><A
name=.E4.B8.8E.E9.80.92.E5.BD.92Bayesian.E4.BC.B0.E8.AE.A1.E4.B9.8B.E9.97.B4.E7.9A.84.E5.85.B3.E7.B3.BB></A></P>
<H2><SPAN class=editsection>[<A title=编辑本节
href="http://zh.wikipedia.org/w/index.php?title=%E5%8D%A1%E5%B0%94%E6%9B%BC%E6%BB%A4%E6%B3%A2&action=edit&section=13">编辑</A>]</SPAN>
<SPAN class=mw-headline>与递归Bayesian估计之间的关系</SPAN></H2>
<P>假设真正的状态是无法观察的<A class=new title=马尔科夫过程
href="http://zh.wikipedia.org/w/index.php?title=%E9%A9%AC%E5%B0%94%E7%A7%91%E5%A4%AB%E8%BF%87%E7%A8%8B&action=edit">马尔科夫过程</A>，测量结果是从隐性马尔科夫模型观察到的状态。</P>
<DL>
<DD><A class=new title=Image:HMMKalmanFilterDerivation.png
href="http://zh.wikipedia.org/w/index.php?title=Special:Upload&wpDestFile=HMMKalmanFilterDerivation.png">Image:HMMKalmanFilterDerivation.png</A></DD></DL>
<P>根据马尔科夫假设，真正的状态仅受最近一个状态影响而与其它以前状态无关。</P>
<DL>
<DD><IMG class=tex
alt="p(\textbf{x}_k|\textbf{x}_0,\dots,\textbf{x}_{k-1}) = p(\textbf{x}_k|\textbf{x}_{k-1})"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/04b9679161c7616c59d7eae3dd49e495.png"></DD></DL>
<P>与此类似，在时刻 <EM>k</EM> 测量只与当前状态有关而与其它状态无关。</P>
<DL>
<DD><IMG class=tex
alt="p(\textbf{z}_k|\textbf{x}_0,\dots,\textbf{x}_{k}) = p(\textbf{z}_k|\textbf{x}_{k} )"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/626ee96fdc4b8befd2259d2f76dd864e.png"></DD></DL>
<P>根据这些假设，隐性马尔科夫模型所有状态的概率分布可以简化为：</P>
<DL>
<DD><IMG class=tex
alt="p(\textbf{x}_0,\dots,\textbf{x}_k,\textbf{z}_1,\dots,\textbf{z}_k) = p(\textbf{x}_0)\prod_{i=1}^k p(\textbf{z}_i|\textbf{x}_i)p(\textbf{x}_i|\textbf{x}_{i-1})"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/0511e8acb9a1d4dc598ae1819138fa37.png"></DD></DL>
<P>However, when the Kalman filter is used to estimate the state
<STRONG>x</STRONG>, the probability distribution of interest is that
associated with the current states conditioned on the measurements up to
the current timestep. (This is achieved by marginalizing out the previous
states and dividing by the probability of the measurement set.)
但是，当卡尔曼滤波器用来估计状态<STRONG>x</STRONG>的时候，</P>
<P>This leads to the <EM>predict</EM> and <EM>update</EM> steps of the
Kalman filter written probabilistically. The probability distribution
associated with the predicted state is product of the probability
distribution associated with the transition from the (<EM>k</EM> − 1)th
timestep to the <EM>k</EM>th and the probability distribution associated
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