# Probabilistic Robotics学习笔记: Bayes Filter 基本概念

## RECURSIVE STATE ESTIMATION

### Introduction

State estimation seeks to recover state variables from the data. Probabilistic state estimation algorithms compute belief distributions over possible world states.

$$!p(a|b)=\frac{p(b|a)\cdot p(a)}{p(b)}$$

$$!p(b)=p(b|a_1)p(a_1)da_1$$

### Bayes Filter

#### Belief Distributions

Bayes Filter 的核心在于对信念分布的操作，所谓信念分布，是在已知传感器数据，动作器动作的情况下对状态$$x_t$$的一个估计

$$!bel(x_t)=p(x|z_{1:t},u_{1:t})$$
$$!\overline{bel(x_t)}=p(x|z_{1:t-1},u_{1:t})$$

#### Bayes 的迭代规则

$$!\overline{bel(x_t)}=\int p(x_t|u_t,x_{t-1}) bel(x_{t-1})dx$$
$$!bel(x_t)=\frac{p(z_t|x_t)\overline{bel(x_t)}}{p(z_i)}$$

$$p(z_i)$$为归一化常数，我们在这里仅仅讨论上面二式的物理学意义

$$x_t$$,$$z_t$$之间的关系，通过乘因子$$p(z_t|x_t)$$和归一化后，我们得到了对于x估计的完整值。

#### 一个简单的栗子

$$!bel(x_0)=0.5$$
$$!bel(!x_0)=0.5$$

$$!p(x_t|x_t)=0.6$$
$$!p(!z_t|x_t)=0.4$$
$$!p(z_t|!x_t)=0.2$$
$$!p(!z_t|!x_t)=0.8$$

$$!p(x_t|!x_{t-1},U_t)=0.8$$
$$!p(!x_t|!x_{t-1},U_t)=0.2$$

$$!p(x_t|x_{t-1},U_t)=1$$
$$!p(!x_t|x_{t-1},U_t)=0$$

$$!\overline{bel(X_t)}=\int p(x_t|U_t,x_{t-1}) bel(x_{t-1})dx$$

$$!\overline{bel(X_t)}=\sum_{x_{t-1}} p(x_t|U_1,x_{t-1}) bel(x_{t-1})$$
$$!\overline{bel{X_t}}=p(X_t|U_t,!x_{t-1}) bel(!x_{t-1})+p(X_{t}|U_t,x_{t-1}) bel(x_{t-1})$$

$$!\overline{bel{x_1}}=bel(!x_1)=0.5$$

$$!bel(X_t)=\eta \cdot p(Z_t|X_t) \overline{bel(X_t)} = p(z_t|X_t)\overline{bel(x_t)}$$

$$!bel(x_1)=\eta 0.6\cdot 0.5=0.3\eta$$
$$!bel(!x_1)=\eta 0.2\cdot 0.5=0.1\eta$$

$$!bel(x_1)=0.75; bel(!x_1)=0.25$$

$$!overline(x_2)=0.751+0.250.8=0.95$$
$$!overline(!x_2)=00.75+0.250.2=0.05$$

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