  There has been much recent interest in modifying Bayesian inference for
misspecified models so that it is useful for specific purposes. One popular
modified Bayesian inference method is "cutting feedback" which can be used when
the model consists of a number of coupled modules, with only some of the
modules being misspecified. Cutting feedback methods represent the full
posterior distribution in terms of conditional and sequential components, and
then modify some terms in such a representation based on the modular structure
for specification or computation of a modified posterior distribution. The main
goal of this is to avoid contamination of inferences for parameters of interest
by misspecified modules. Computation for cut posterior distributions is
challenging, and here we consider cutting feedback for likelihood-free
inference based on Gaussian mixture approximations to the joint distribution of
parameters and data summary statistics. We exploit the fact that marginal and
conditional distributions of a Gaussian mixture are Gaussian mixtures to give
explicit approximations to marginal or conditional posterior distributions so
that we can easily approximate cut posterior analyses. The mixture approach
allows repeated approximation of posterior distributions for different data
based on a single mixture fit, which is important for model checks which aid in
the decision of whether to "cut". A semi-modular approach to likelihood-free
inference where feedback is partially cut is also developed. The benefits of
the method are illustrated in two challenging examples, a collective cell
spreading model and a continuous time model for asset returns with jumps.
