  Bayesian modelling enables us to accommodate complex forms of data and make a
comprehensive inference, but the effect of partial misspecification of the
model is a concern. One approach in this setting is to modularize the model,
and prevent feedback from suspect modules, using a cut model. After observing
data, this leads to the cut distribution which normally does not have a
closed-form. Previous studies have proposed algorithms to sample from this
distribution, but these algorithms have unclear theoretical convergence
properties. To address this, we propose a new algorithm called the Stochastic
Approximation Cut algorithm (SACut) as an alternative. The algorithm is divided
into two parallel chains. The main chain targets an approximation to the cut
distribution; the auxiliary chain is used to form an adaptive proposal
distribution for the main chain. We prove convergence of the samples drawn by
the proposed algorithm and present the exact limit. Although SACut is biased,
since the main chain does not target the exact cut distribution, we prove this
bias can be reduced geometrically by increasing a user-chosen tuning parameter.
In addition, parallel computing can be easily adopted for SACut, which greatly
reduces computation time.
