# I am confused about Bayes' rule in MCMC

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)?

1. $P(x)P(z\mid x,y)=P(x,z\mid y)$
2. $P(x\mid y)P(z\mid x,y)=P(x,z\mid y)$
3. $P(x\mid y)P(z\mid x,t)=P(x,z\mid y,t)$

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.

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The generalization of the formula $$\Pr(A \cap B) = \Pr(A \mid B)\Pr(B)$$ for a conditional probability $\Pr(\cdot \mid C)$ is the following one: $$\Pr(A \cap B \mid C) = \Pr(A \mid B\cap C)\Pr(B \mid C).$$
Your formula number 2 is analogous to the above formula (with $P(\cdot \mid y)$ substituted for $\Pr(\cdot \mid C)$), and it is the correct one.