The operation of cortical circuits has traditionally been studied using either bottom-up or top-down approaches. The former identified dynamical mechanisms responsible for a wealth of empirical data, but without reference to computational function, while the laer extracted signatures of specific computations from neural activity, but remained agnostic as to the underlying mechanisms. Here we bridge these two approaches and study the dynamics and function of cortical circuits in a principled unifying framework. In contrast to recent optimisation-based approaches, which use highly simplified architectures and only address trial-average responses, here we train stochastic, recurrent neural circuits with realistic components that allow us to link response dynamics and variability more directly to computational function. We train networks to perform sampling-based probabilistic inference under a widely-used generative model of natural images, the Gaussian Scale Mixture (GSM) model. We first show that the GSM posterior mean grows with stimulus contrast z, superlinearly for small z and saturating for large z, while the posterior variance decreases with z. We then employ a novel, assumed density filtering-based approach to obtain the moments of activity in stochastic networks as smooth, dierentiable functions of network parameters, and match them to those of the GSM posterior for a set of training stimuli. We show that the network appropriately generalizes to novel stimuli, reproducing the scaling of means and variances with contrast. Furthermore, the networks thus obtained operate in the dynamical regime of stabilised supralinear networks (SSN) that has recently been proposed to underlie response normalization in V1. Thus, our results suggest a generic function for inhibition stabilised dynamics with a loose excitatory-inhibitory balance: they provide ideal substrates of recognition models for probabilistic inference. Conversely, our approach could also be used to infer the brain’s internal models based on observed dynamics.