Bayes' rule appears to bevery simple at first sight, but when studied deeply I find it is difficult and confusing, especially in MCMC applications when multiple parameters need to be estimated.
For example, assuming $x,y,z,t$ are four parameters, which of the following three expressions are true (or true under some specifications)?
- $P(x)P(z\mid x,y)=P(x,z\mid y)$
- $P(x\mid y)P(z\mid x,y)=P(x,z\mid y)$
- $P(x\mid y)P(z\mid x,t)=P(x,z\mid y,t)$
I usually see formulas similar with 3, but I wonder why it hold. Could someone explain it in detail?
If there are any excellent books that could help me, pls list them.