文章目录

• 概率状态空间模型(Probabilistic State Space Models)
• • 概率状态空间模型的定义
  • 状态${\mathbf{x}}_k$的马尔可夫性质(*Markov property of states*)
  • 观测向量${\mathbf{y}}_k$的条件独立性(*Conditional independence of measurements*)

概率状态空间模型(Probabilistic State Space Models)

概率状态空间模型的定义

(Probabilistic state space model) A probabilistic state space model or non-linear filtering model consists of a sequence of conditional probability distributions:
$$\begin{array}{rl}{\mathbf{x}}_k& \sim p({\mathbf{x}}_k|{\mathbf{x}}_{k-1})\\ {\mathbf{y}}_k& \sim p({\mathbf{y}}_k|{\mathbf{x}}_k)\end{array}\text{\hspace{1em}(1)}$$ for $k=1,2,\cdots$，其中

• ${\mathbf{x}}_k\in {\mathbb{R}}^n$是系统在时刻$k$的状态；
• ${\mathbf{y}}_k\in {\mathbb{R}}^m$是时刻$k$的观测向量；
• $p({\mathbf{x}}_k|{\mathbf{x}}_{k-1})$是一个动态系统模型，描述了系统的随机动态变化(stochasticc dynamics)；
• $p({\mathbf{y}}_k|{\mathbf{x}}_k)$是测量模型，描述了给定状态条件下的测量模型。

并且认为该模型为马尔可夫模型，如下图所示。这意味着它有两个重要性质。

状态${\mathbf{x}}_k$的马尔可夫性质(Markov property of states)

The states { x k : k = 0 , 1 , 2 , ⋯   } \{ \boldsymbol x_k: k=0,1,2,\cdots \} {xk​:k=0,1,2,⋯} form a Markov sequence (or Markov chain if the state is discrete.). This Markov property means that ${\mathbf{x}}_k$ (and actually the whole future ${\mathbf{x}}_{k+1},{\mathbf{x}}_{k+2},\cdots$) given ${\mathbf{x}}_{k-1}$ is independent of anything that has happened before the time step $k-1$：
$$p({\mathbf{x}}_k|{\mathbf{x}}_{1:k-1},{\mathbf{y}}_{1:k-1})=p({\mathbf{x}}_k|{\mathbf{x}}_{k-1})\text{\hspace{1em}(2)}$$

这里我们需要注意性质里边没有提到${\mathbf{x}}_k$之后的状态(future state)，由上图可知，我们知道${\mathbf{x}}_k$与之后的状态/观测向量是有联系的，比如说：$\exists T>k$
$$p({\mathbf{x}}_k|{\mathbf{x}}_{1:k-1},{\mathbf{y}}_{1:T})=p({\mathbf{x}}_k|{\mathbf{x}}_{k-1},{\mathbf{y}}_{k:T})$$

Also, the past is independent of the future given the present:
$$p({\mathbf{x}}_{k-1}|{\mathbf{x}}_{k:T},{\mathbf{y}}_{k:T})=p({\mathbf{x}}_{k-1}|{\mathbf{x}}_k)\text{\hspace{1em}(3)}$$

类似地，我们也需要注意，${\mathbf{x}}_k$与之前的状态/观测向量是有联系的，比如说：
$$p({\mathbf{x}}_{k-1}|{\mathbf{x}}_{k:T},{\mathbf{y}}_{1:T})=p({\mathbf{x}}_{k-1}|{\mathbf{x}}_k,{\mathbf{y}}_{1:k-1})$$

观测向量${\mathbf{y}}_k$的条件独立性(Conditional independence of measurements)

The current measurement ${\mathbf{y}}_k$ given the current state ${\mathbf{x}}_k$ is conditionally independent of the measurement and state histories:
$$p({\mathbf{y}}_k|{\mathbf{x}}_{1:k},{\mathbf{y}}_{1:k-1})=p({\mathbf{y}}_k|{\mathbf{x}}_k)$$