“Low-dimensional regression-based models are constructed for each of these conditional distributions using synthetic (simulated) parameter value and summary statistic pairs, which then permit approximate Gibbs update steps (…) synthetic datasets are not generated during each sampler iteration, thereby providing efficiencies for expensive simulator models, and only require sufficient synthetic datasets to adequately construct the full conditional models (…) Construction of the approximate conditional distributions can exploit known structures of the high-dimensional posterior, where available, to considerably reduce computational overheads”

**G**uilherme Souza Rodrigues, David Nott, and Scott Sisson have just arXived a paper on approximate Gibbs sampling. Since this comes a few days after we posted our own version, here are some of the differences I could spot in the paper:

- Further references to earlier occurrences of Gibbs versions of ABC, esp. in cases when the likelihood function factorises into components and allows for summaries with lower dimensions. And even to ESP.
- More an ABC version of Gibbs sampling that a Gibbs version of ABC in that approximations to the conditionals are first constructed and then used with no further corrections.
- Inherently related to regression post-processing à la Beaumont et al. (2002) in that the regression model is the start to designing an approximate full conditional, conditional on the “other” parameters and on the overall summary statistic. The construction of the approximation is far from automated. And may involve neural networks or other machine learning estimates.
- As a consequence of the above, a preliminary ABC step to design the collection of approximate full conditionals using a single and all-purpose multidimensional summary statistic.
- Once the approximations constructed, no further pseudo-data is generated.
- Drawing from the approximate full conditionals is done exactly, possibly via a bootstrapped version.
- Handling a highly complex g-and-k dynamic model with 13,140 unknown parameters, requiring a ten days simulation.

“In certain circumstances it can be seen that the likelihood-free approximate Gibbs sampler will exactly target the true partial posterior (…) In this case, then Algorithms 2 and 3 will be exact.”

Convergence and coherence are handled in the paper by setting the algorithm(s) as noisy Monte Carlo versions, à la Alquier et al., although the issue of incompatibility between the full conditionals is acknowledged, with the main reference being the finite state space analysis of Chen and Ip (2015). It thus remains unclear whether or not the Gibbs samplers that are implemented there do converge and if they do what is the significance of the stationary distribution.