[Probabilitic Robotics] #02. BAYES FILTER

Befor we start, I mention that i wrote these contents refered in "Probabilitic Robotics (Sebastian Thrun)".
and i would not mention about deep theory of it. so if you are curious about deep knowledge of the contents. have a look at the book.

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In this chapter, we would check how can Bayes Filter is derived.

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Bayes rule of x, y

\[p(x|y)=\frac{p(y|x)\cdot p(x)}{p(y)}\]

p(x) : prior probability distribution
p(x|y) : posterior probability distribution
p(y) :  nomalizer
p(y|x) : inverse conditional probability

Bayes role of x, y, z

\[p(x|y,z)=\frac{p(y|x,z)\cdot p(x|z)}{p(y|z)}\]
p(x|z) : prior probability distribution
p(x|y,z) : posterior probability distribution
p(y|z) :  nomalizer
p(y|x,z) : inverse conditional probability

Bayes Filter

1. defalut form of equation

\[p(x_{t}|z_{1:t},u_{1:t})=\frac{p(z_{t}|x_{t},z_{1:t-1},u_{1:t})\cdot p(x_{t}|z_{1:t-1},u_{1:t})}{p(z_{t}|z_{1:t-1},u_{1:t})}\\=\eta\cdot p(z_{t}|x_{t},z_{1:t-1},u_{1:t})\cdot p(x_{t}|z_{1:t-1},u_{1:t})\]

x : state
z : sensor measurement
u : control input
η : nomalizer

2. if we know xt and were interested in predicting the measurement zt, no past measurement or control would provide us additional information