This is going to be a stupid question, but I know I am missing something, so here it goes:
My textbook (Probabilistic Robotics, p.17) trivially states that Bayes' rule gives:
$ p(x |y,z) = \frac{p(y |x, z)p(x|z)}{p(y|z)} $
Are there some conditionig rule in probability that easily lets you arrive there from the simple Bayes' rule?:
$ p(x |y) = \frac{p(y |x)p(x)}{p(y)} $
B/c for me to even get close to something similar I have to do:
$$ p(x | z,y) = \frac{ p(y,z|x) p(x)} {p(y,z)} $$ $$ = \frac{ \frac{p(y|x)p(z|x)p(x)}{p(z)} } { \frac{p(y,z)}{p(z)} }$$ $$ = \frac{p(x|z)p(y|x)}{p(y|z)} $$ Where I had to use that y and z are conditionally independent...
So what am I missing?
Thank you