Version v1.0.1 of this package has been released (thanks to @giordano for solving the mess I made trying to register v1.0.0). This version uses Turing for sampling, and the two examples, for a mixture of normals model and a stochastic volatility model, show how approximate Bayesian computing (ABC) / method of simulated moments (MSM) may be done using Turing.
The sampling is done using AdvancedMH. People sometimes ask why not NUTS. The reason is that the likelihood is computed by simulation, and it is not continuous or differentiable. However, the neural net fit gives a very good starting value for the chain, and a very good covariance estimator for a random walk proposal, so sampling is effective.