# Do I need to evaluate acceptance rates in Metropolis within Gibbs algorithm?

Consider the Gibbs sampler

1. Sample $$\theta' \sim p(\theta|\tau, D)$$
2. Sample $$\tau' \sim p(\tau|\theta', D)$$

where $$\theta,\tau$$ parameters of the data $$D$$. Now assume that we can only sample from $$p(\tau|\theta', D)$$ using a Metropolis step, for example because its normalization constant is unknown but we can evaluate the unnormalized density. In the Metropolis algorithm a proposal sample is drawn from a proposal distribution, such as the normal distribution. The step size is in this case determined by the variance of the normal, which is a tuning parameter for the acceptance rate of the Metropolis sampler.

At a given step in the Gibbs sampler, we could sample a large number of times from $$p(\tau|\theta', D)$$ i.e. holding $$\theta'$$ constant across repeated draws. Then we could estimate the acceptance rate by checking which proportion of proposals was accepted. However in the Metropolis-within-Gibbs sampler suggested above it is not common to draw a large number of samples from $$p(\tau|\theta', D)$$ at a given step; instead one sample is drawn by Metropolis and then the Gibbs sampler continues at the next iteration. Suppose that this next iteration yields $$\theta''$$ (two dashes), then the next Metropolis step in the second iteration will sample from $$p(\tau|\theta'', D)$$.

Now my specific question is: the optimal tuning parameters for the proposal distributions used for sampling from $$p(\tau|\theta', D)$$ at the first and $$p(\tau|\theta'', D)$$ at the second iteration may indeed be different. Assuming I can only set the tuning parameter once, how should I monitor acceptance rates in this Metropolis-Gibbs sampler? Is it enough to do this once across draws of $$\tau$$?