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What is the standard way of drawing multiple parameters for a MCMC run? Say I have 9 parameters, what is the most efficient method to get all parameters to explore their distributions properly?

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First you should explore their [joint] distribution [no s] rather than [marginal] distributions. Second, it all depends on the structure of the target, i.e., whether or not, the parameters come as groups that can be simulated at once as a single vector. In case one is unsure, one parameter at a time Gibbs sampling is the default method I would recommend.

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