  Sensory processing is often characterized as implementing probabilistic
inference: networks of neurons compute posterior beliefs over unobserved causes
given the sensory inputs. How these beliefs are computed and represented by
neural responses is much-debated (Fiser et al. 2010, Pouget et al. 2013). A
central debate concerns the question of whether neural responses represent
samples of latent variables (Hoyer & Hyvarinnen 2003) or parameters of their
distributions (Ma et al. 2006) with efforts being made to distinguish between
them (Grabska-Barwinska et al. 2013). A separate debate addresses the question
of whether neural responses are proportionally related to the encoded
probabilities (Barlow 1969), or proportional to the logarithm of those
probabilities (Jazayeri & Movshon 2006, Ma et al. 2006, Beck et al. 2012).
Here, we show that these alternatives - contrary to common assumptions - are
not mutually exclusive and that the very same system can be compatible with all
of them. As a central analytical result, we show that modeling neural responses
in area V1 as samples from a posterior distribution over latents in a linear
Gaussian model of the image implies that those neural responses form a linear
Probabilistic Population Code (PPC, Ma et al. 2006). In particular, the
posterior distribution over some experimenter-defined variable like
"orientation" is part of the exponential family with sufficient statistics that
are linear in the neural sampling-based firing rates.
