Asymptotic scaling properties of the posterior mean and variance in the Gaussian scale mixture model

The Gaussian scale mixture model (GSM) is a simple yet powerful probabilistic generative model of natural image patches. In line with the well-established idea that sensory processing is adapted to the statistics of the natural environment, the GSM has also been considered a model of the early visual system, as a reasonable"first-order"approximation of the internal model that the primary visual cortex (V1) implements. According to this view, neural activities in V1 represent the posterior distribution under the GSM given a particular visual stimulus. Indeed, (approximate) inference under the GSM has successfully accounted for various nonlinearities in the mean (trial-average) responses of V1 neurons, as well as the dependence of (across-trial) response variability with stimulus contrast found in V1 recordings. However, previous work almost exclusively relied on numerical simulations to obtain these results. Thus, for a deeper insight into the realm of possible behaviours the GSM can (and cannot) exhibit and predict, here we present analytical derivations for the limiting behaviour of the mean and (co)variance of the GSM posterior at very low and very high contrast levels. These results should guide future work exploring neural circuit dynamics appropriate for implementing inference under the GSM.