Intuitively, if I want to update two parameters in one step, I have to come up with a proposal that are good for both parameters. Assuming that the parameters are independent, is it correct to demonstrate this intuition by saying the following?
if the probability of accepting the proposal for one parameter is X, then the probability of accepting the proposal for two parameters is lower at $X^2$
Based on my experience running MCMC I have a hunch that this statement is not mathematically correct (i.e. the probability of accepting for two parameters can't be as low as $X^2$), but I cannot figure out why mathematically.