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I am new to Metropolis sampling, here is a question that confuses me. Assume that there are two sets of variables $a$ and $b$ we want to sample. Let $X$ denote the observations and $p(X|a,b)$ denote the likelihood. $T$ is the max iterations. Which of the followings are correct?

Option 1:

for t in 1...T:
    new_a = ... //propose new variables a
    p = min(1,p(X|new_a,b)/p(X|a,b))
    thresh  = ... //a random number generated from a uniform [0,1]
    if thresh < p:
       a = new_a //accept new_a
    new_b = ... //propose new variables b
    p = min(1,p(X|a,new_b)/p(X|a,b))
    thresh  = ... //a random number generated from a uniform [0,1]
    if thresh < p:
       b = new_b //accept new_b

Option 2:

for t in 1...T:
    new_a = ... //propose new variables a
    new_b = ... //propose new variables b
    p = min(1,p(X|new_a,new_b)/p(X|a,b))
    thresh  = ... //a random number generated from a uniform [0,1]
    if thresh < p:
       a = new_a, b = new_b //accept both a and b

Essentially, option 1 samples $a$ and $b$ separately. In each iteration, it samples $a$ first, then it uses the value of $a$ (either accepted new value or the old value) to sample $b$. Option 2 samples $a$ and $b$ together, they are either both accepted or rejected.

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Simulating both components $a$ and $b$ from the prior at the same time and accepting with probability $$1 \wedge \dfrac{p(X\mid a^\text{new},b^\text{new})}{p(X\mid a,b)}$$ is a regular (and valid) format of the Metropolis-Hastings algorithm.

Simulating each component $a$ and $b$ sequentially from the conditional priors $\pi(a|b)$ and $\pi(b|a)$ and accepting with probabilities $$1 \wedge \dfrac{p(X\mid a^\text{new},b)}{p(X\mid a,b)} \quad\text{and}\quad 1 \wedge \dfrac{p(X\mid a,b^\text{new})}{p(X\mid a,b)}$$ is a (valid) form of the Metropolis-Hastings-within-Gibbs algorithm.

Note that simulating from the prior is in general a very inefficient choice (from the obvious reason that the goal is to generate from the posterior to the potential of missing regions of high likelihood values).

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