An approach to adaptive Bayesian computation where the acceptance rate is a Bernoulli process

Question

I am considering an approach to adaptive approximate Bayesian computation technique (ABC). The acceptance rejection algorithm is used wherein proposals from the prior are accepted if the simulated data $X_\text{sim}$ matches the observed data $X_\text{data}$:

\begin{equation} \rho(X_\text{sim},X_\text{data}) < \epsilon \end{equation}

Where $\rho(.)$ is some distance function and $\epsilon$ is the tolerance factor. The aim is to find the value of the tolerance factor $\epsilon$ so that the rate of acceptance $\alpha$ matches some target acceptance rate $\alpha_0$.

Essentially the problem is to find $\epsilon = \epsilon_0$ such that $\mathbb{E}[\alpha] = \alpha_0$.

My approach is to model the acceptances of the algorithm as a Bernoulli process where $\alpha$ is the probability of acceptance.

1. Choose the shape parameters of the beta prior distribution of the acceptance rate $\alpha$.
2. For each run of the acceptance-rejection algorithm update the posterior distribution of $\alpha$. Then draw a random sample from the posterior distribution of $\alpha$. If $\alpha$ is lower than the target $\alpha_0$ then $\epsilon$ should be increased. If $\alpha$ is higher than $\alpha_0$ then $\epsilon$ should be decreased.

The question is how should the changes to $\epsilon$ be proposed. My feeling is that $\epsilon$ should also have a prior and posterior distribution. Perhaps a model should be proposed for the relationship between these two parameters? Any suggestions on this problem would be appreciated. Additionally has this approach been considered before in the literature (if so please post a link)?

Answer 1

The classical answer in the ABC literature to this approach is to simulate pseudo-samples from the prior predictive and to set $\epsilon$ to the $\alpha$-quantile of the corresponding distances. I find it difficult to see how an adaptive approach could be more efficient there.