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I have a number of queries about computing the acceptance rate of samples generated from the Metropolis (symmetric random walk) algorithm empirically, that is, in the presence of burning-in and thinning of the autocorrelated samples.

To provide some detail to anchor the question, as part of an assignment for self-study, I am interested in sampling parameter vectors $\boldsymbol{\theta}$ from a posterior $p(\boldsymbol{\theta} | \mathbf{y})$ given data $\mathbf{y}$, using a symmetric proposal distribution, that is, an isotropic Gaussian proposal on the parameters $\boldsymbol{\theta}' \sim \mathcal{N}(\boldsymbol{\theta}' | \boldsymbol{\theta}, \sigma^2 \mathbf{I})$. I am also mandated to used one fixed initialisation of $\mathbf{0}$.

Now I am mandated to explore the behaviour of the Metropolis algorithm for various values of the isotropic Gaussian covariance parameter $\sigma^2$, and "thinning" parameter $t$. The thinning parameter $t$, as I understand, is a means of dealing with the autocorrelated samples generated from the Metropolis algorithm after burning it in for an appropriate number of iterations. I am required to report the empirical acceptance rate/rejection ratio for various values of the MCMC parameters $\sigma^2$ and $t$.

Given that I want to collect 5000 samples of the parameters, $\{ \boldsymbol{\theta}^{(n)} \}^N_{n=1}$ from the posterior I am mandated to do the following for say a thinning parameter of $t = 2$:

  1. Run the Metropolis algorithm for $B$ burn-in iterations, yielding $B$ iterations of the parameter vector $\{ \boldsymbol{\theta}^{(b)} \}^B_{b=1}$. I will discard these samples.

  2. Continue running the Metropolis algorithm for $T = 5000t$ iterations, where $t$ is the thinning parameter. For a thinning parameter of $t = 2$, that means I continue running the Metropolis algorithm for $T = 10000$ iterations, resulting in 10000 values of the parameter vector $\{ \boldsymbol{\theta}^{(s)} \}^T_{s=1}$

  3. For the case where the thinning parameter is $t = 2$, I only collect samples every 2nd iteration out of a total 10,000 iterations above. This should generate $N = 5000$ samples from the posterior (hopefully, depending on a raft of other diagnostics), that is, $\{ \boldsymbol{\theta}^{(n)} \}^N_{n=1}$.

Query.

I now use the term acceptance rate to refer to the proportion of samples that are accepted by the Metropolis algorithm empiricallly, and not the quantity frequently denoted as $\alpha$ used to implement the decision rule for accepting/rejecting samples. Primarily, I am soliciting further clarity on how this metric is conventionally reported. Further, whilst I am aware that the empirical acceptance rate can be "tuned", that is currently outside the remit of my assignment.

Q1. In the case that the acceptance rate is supposed to be a single number, when I report the acceptance rate of samples from the Metropolis algorithm, do I report the acceptance rate computed using a total of $T$ autocorrelated samples, or do I compute it after I have adjusted for autocorrelation via thinning, that is, using a total of $N$ samples? Are there any conventions on this?

In the former case, that would mean using $\{ \boldsymbol{\theta}^{(s)} \}^T_{s=1}$ from step 2. to compute: $$\text{acceptance rate} = \frac{1}{T} \sum^T_{s=1} \mathbb{I}(\boldsymbol{\theta}^{(s)} \text{accepted})$$ In the latter case, that would amount to using $\{ \boldsymbol{\theta}^{(n)} \}^N_{n=1}$ from step 3. to compute: $$\text{acceptance rate} = \frac{1}{N} \sum^N_{n=1} \mathbb{I}(\boldsymbol{\theta}^{(n)} \text{accepted})$$

  1. Q2. Are there conventions whereby the acceptance rate is not reported as a single one-off number, but is rather time-varying? (I have read somewhere, I think in "Bayesian Data Analysis" or "Monte Carlo Statistical Methods" that a sliding window can be used.

Some assistance would be greatly appreciated.

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The better formula for estimating acceptance rate is

$$ \text{acceptance rate} = \frac{1}{T} \sum^T_{s=1} \mathbb{I}(\boldsymbol{\theta}^{(s)} \text{accepted}) $$

because it includes more samples. Given you have 5000 samples after thinning, it should not really matter which formula you use, though.

It can be a good idea to compute the acceptance rate using a sliding window approach, as that may help you determine whether the MCMC has converged. If it has already been established elsewhere that the MCMC has converged, there is not any value in a sliding window approach, and then I would just report the acceptance rate as a single number.

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    $\begingroup$ Thank you for providing some insight on the matter. You state that computing the acceptance rate using a sliding window can serve as one (of many) indicators of MCMC convergence. Am I correct in guessing that in this case, the acceptance rate would stabilise as a function of iterations? Is there a concise explanation as to why that might be the case intuitively? Or a reference where I could explore further? $\endgroup$
    – microhaus
    Feb 26, 2021 at 19:27

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