最优贝叶斯滤波

贝叶斯推理

贝叶斯定理

$$p(\mathbf{x}|\mathbf{y} ) = \frac{p(\mathbf{x},\mathbf{y})}{p(\mathbf{y})}$$

再次利用该式可以重写为

$$p(\mathbf{x}|\mathbf{y} ) = \frac{p(\mathbf{y}|\mathbf{x})p(\mathbf{x})}{p(\mathbf{y})}$$

通常称$p(\mathbf{x})$为先验分布，$p(\mathbf{y}|\mathbf{x})$为最大似然估计分布，$p(\mathbf{x}|\mathbf{y})$为后验分布。

目标跟踪中的贝叶斯递推

$\mathbf{S}_k$为$k$时刻的广义目标状态。
联合概率密度函数$p(\mathbf{S}^k) =p(\mathbf{S}_0,\cdots, \mathbf{S}_k)$。
传感器输出$\mathbf{y}^{k}=(\mathbf{y}_{1}, \mathbf{y}_{2}, \cdots, \mathbf{y}_{k})$。

由贝叶斯定理

$$p\left(\mathbf{S}^{k} \mid \mathbf{y}^{k}\right)=\frac{p\left(\mathbf{y}^{k} \mid \mathbf{S}^{k}\right) p\left(\mathbf{S}^{k}\right)}{p\left(\mathbf{y}^{k}\right)}$$

归一化因子计算：

$$p\left(\mathbf{y}^{k}\right)=\int_{\mathbf{S}_{k}} \cdots \int_{\mathbf{S}_{0}} p\left(\mathbf{y}^{k} \mid \mathbf{S}_{k}, \cdots, \mathbf{S}_{o}\right) p\left(\mathbf{S}_{k}, \cdots, \mathbf{S}_{o}\right) \mathrm{d} \mathbf{S}_{k} \cdots \mathrm{d} \mathbf{S}_{o}$$

贝叶斯递推：

$$\begin{aligned} & p\left(\mathbf{y}^{k} \mid \mathbf{S}^{k}\right)=p\left(\mathbf{y}_{k}, \mathbf{y}^{k-1} \mid \mathbf{S}^{k}\right) \\ & =p\left(\mathbf{y}_{k} \mid \mathbf{y}^{k-1}, \mathbf{S}^{k}\right) p\left(\mathbf{y}^{k-1} \mid \mathbf{S}^{k}\right) =p\left(\mathbf{y}_{k} \mid \mathbf{y}^{k-1}, \mathbf{S}^{k}\right) p\left(\mathbf{y}^{k-1} \mid \mathbf{S}^{k-1}\right)\\ & p\left(\mathbf{S}^{k}\right) =p\left(\mathbf{S}_{k}, \mathbf{S}^{k-1}\right)=p\left(\mathbf{S}_{k} \mid \mathbf{S}^{k-1}\right) p\left(\mathbf{S}^{k-1}\right) \\ & p\left(\mathbf{y}^{k}\right) =p\left(\mathbf{y}_{k}, \mathbf{y}^{k-1}\right)=p\left(\mathbf{y}_{k} \mid \mathbf{y}^{k-1}\right) p\left(\mathbf{y}^{k-1}\right) \end{aligned}$$

带入

$$p\left(\mathbf{S}^{k} \mid \mathbf{y}^{k}\right)=\frac{p\left(\mathbf{y}_{k} \mid \mathbf{y}^{k-1}, \mathbf{S}^{k}\right) p\left(\mathbf{S}_{k} \mid \mathbf{S}^{k-1}\right)}{p\left(\mathbf{y}_{k} \mid \mathbf{y}^{k-1}\right)}\left\{\frac{p\left(\mathbf{y}^{k-1} \mid \mathbf{S}^{k-1}\right) p\left(\mathbf{S}^{k-1}\right)}{p\left(\mathbf{y}^{k-1}\right)}\right\}$$ $$p\left(\mathbf{S}^{k} \mid \mathbf{y}^{k}\right)=\frac{p\left(\mathbf{y}_{k} \mid \mathbf{y}^{k-1}, \mathbf{S}^{k}\right) p\left(\mathbf{S}_{k} \mid \mathbf{S}^{k-1}\right)}{p\left(\mathbf{y}_{k} \mathbf{y}^{k-1}\right)} p\left(\mathbf{S}^{k-1} \mid \mathbf{y}^{k-1}\right)$$ $$p\left(\mathbf{S}^{k} \mid \mathbf{y}^{k}\right)=\frac{p\left(\mathbf{y}_{k} \mid \mathbf{S}_{k}\right)}{p\left(\mathbf{y}_{k} \mid \mathbf{y}^{k-1}\right)} p\left(\mathbf{S}_{k} \mid \mathbf{S}_{k-1}\right) p\left(\mathbf{S}^{k-1} \mid \mathbf{y}^{k-1}\right)$$

通过推导可以得到

$$p\left(\mathbf{S}_{k} \mid \mathbf{y}^{k}\right)=\frac{1}{p\left(\mathbf{y}_{k} \mid \mathbf{y}^{k-1}\right)} p\left(\mathbf{y}_{k} \mid \mathbf{S}_{k}\right) p\left(\mathbf{S}_{k} \mid \mathbf{y}^{k-1}\right)$$

即就是

$$p\left(\mathbf{S}_{k} \mid \mathbf{y}^{k}\right)=\frac{1}{p\left(\mathbf{y}_{k} \mid \mathbf{y}^{k-1}\right)} p\left(\mathbf{y}_{k} \mid \mathbf{S}_{k}\right) \int_{\mathbf{s}_{k-1}} p\left(\mathbf{S}_{k} \mid \mathbf{S}_{k-1}\right) p\left(\mathbf{S}_{k-1} \mid \mathbf{y}^{k-1}\right) \mathrm{d} \mathbf{S}_{k-1}$$