Sensory Integration and Density Estimation
The integration of partially redundant information from multiple sensors is a standard computational problem for agents interacting with the world. In man and other primates, integration has been shown psychophysically to be nearly optimal in the sense of error minimization. An influential generalization of this notion of optimality is that populations of multisensory neurons should retain all the information from their unisensory afferents about the underlying, common stimulus [1]. More recently, it was shown empirically that a neural network trained to perform latent-variable density estimation, with the activities of the unisensory neurons as observed data, satisfies the information-preservation criterion, even though the model architecture was not designed to match the true generative process for the data [2]. We prove here an analytical connection between these seemingly different tasks, density estimation and sensory integration; that the former implies the latter for the model used in [2]; but that this does not appear to be true for all models.
This work investigates a generative model with latent variables. This model is assumed to perfectly generate some observed distribution, which was originally generated from some other "true" model with latent variables. Specifically, this work finds sufficient conditions so that any reasonable inference procedure done to infer the latent variables in the generative model from the observed variables would retain the information contained in these observed variables about the "true" latent variables.Quality: The core derivations (section 3.2+appendix) appear to be correct.Clarity: The organization of this manuscript seems rather fragmented and confusing, for a purely theoretical paper. It would benefit from clearly and explicitly stating in the beginning: (1) the general probabilistic model (2) all the assumptions, and (3) the main result. Also, some parts could be significantly shortened or removed (e.g. the first few paragraphs of section 2.1).Originality: As far as I am aware, these results are novel.Significance: It is not very clear to me how strong are these results. Most importantly, The assumptions of perfect observed density reconstruction, and of an identical noise model seem rather strong and limiting.Minor comments:1)Something seems to be missing from Eq. 3 ( \in what set?).2)Line 240 EHF->EFH%%% Edited after author feedback %%%Following the author response, I now agree that the "identical noise model" assumption is not as weak as I thought. This is just equivalent to the assumption that X "the state of the world" is deterministically converted into the direct input to the sensory neurons. Though this is probably not true, I agree that this can be considered as a reasonable first approximation.The relaxation of the other assumption ("perfect observed density reconstruction") is not yet clear to me. The authors should clarify this important point in the final version of the paper.