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.
- Choose the shape parameters of the beta prior distribution of the acceptance rate $\alpha$.
- 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)?