Intro and Bayes Filters

• $z_t$: measurement data
• $u_t$: control data
• $x_t$: state

Probabilistic Generative Laws (how we simplify the things) §

State transition probability §

$$p(x_t|x_{0:t-1},z_{1:t-1},u_{1:t})=p(x_t|x_{t-1},u_t)$$

This means only $u_t$ matters if we know $x_t-1$.

Measurement probability §

$$p(z_t|x_{0:t},z_{1:t-1},u_{1:t})=p(z_t|x_t)$$

This means $x_t$ is sufficient to predict measurement $z_t$.

More notations: $bel(x_t)=p(x_t|z_{1:t},u_{1:t})$ If we try to predict $x_t$ before we got measurement $z_t$, we got the posterior

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

From $\overline{bel}(x_t)$ to $bel(x_t)$ is called correction or measurement update.

Bayes Filter Algorithm §

So unlike our previous definition, we now compute $bel(x_t)$ in a recursive fashion. In line 3, we do predict: given control, sum over all the probability assuming different $x_{t-1}$ In line 3, we do measurement update, take in $z_t$ and get the conditional probability on that. Obviously some Bayes rule kicks in.