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There has been much recent interest in modifying Bayesian inference formisspecified models so that it is useful for specific purposes. One popularmodified Bayesian inference method is "cutting feedback" which can be used whenthe model consists of a number of coupled modules, with only some of themodules being misspecified. Cutting feedback methods represent the fullposterior distribution in terms of conditional and sequential components, andthen modify some terms in such a representation based on the modular structurefor specification or computation of a modified posterior distribution. The maingoal of this is to avoid contamination of inferences for parameters of interestby misspecified modules. Computation for cut posterior distributions ischallenging, and here we consider cutting feedback for likelihood-freeinference based on Gaussian mixture approximations to the joint distribution ofparameters and data summary statistics. We exploit the fact that marginal andconditional distributions of a Gaussian mixture are Gaussian mixtures to giveexplicit approximations to marginal or conditional posterior distributions sothat we can easily approximate cut posterior analyses. The mixture approachallows repeated approximation of posterior distributions for different databased on a single mixture fit, which is important for model checks which aid inthe decision of whether to "cut". A semi-modular approach to likelihood-freeinference where feedback is partially cut is also developed. The benefits ofthe method are illustrated in two challenging examples, a collective cellspreading model and a continuous time model for asset returns with jumps.

Modularized Bayesian analyses and cutting feedback in likelihood-free inference