Entropic Inference and Backward Renormalization Group priors

Maximum Entropy inference (ME) is a unique inference engine. Starting from prior distributions,
codifying previously available information, ME permits the update to new
probability distributions, incorporating information as it becomes available.
Its ever expanding
scope of applications fuels the need to improve the construction of informative priors
that lead to improved inferential results. In this paper we are interested in
general inverse problems which can benefit from a multiscale approach. We construct a
systematic method of transferring information from coarser to finer
resolutions based on Renormalization Group transformations.
It permits building informative priors in finer scales from posteriors
in coarser scales. This can be done since, under some conditions, Renormalization Group transformations
in the space of hyperparameters can be inverted. These are a class of Markov embeddings.
These priors are then updated using renormalized data into posteriors by ME.
ME updating with constraints in the form of data measurements is
equivalent to Bayes updating but permits repited use of renormalized data at different
scales.
The resulting inference method, Backward RG (BRG) priors, is tested by doing simulations of a
functional Magnetic Resonance imaging (fMRI) experiment. Its results are
compared with a Bayesian approach working in the finest
available resolution. Using BRpriors sources can be partially identified even
when signal to noise ratio levels are up to $\sim -25$dB improving
vastly on the single step Bayesian approach. For low levels of noise
the BRprior is not an improvement over the single scale Bayesian method.
Analysis of the histograms of hyperparameters can show how to
distinguish if the method is failing, due to very high levels of noise, or
whether the identification of the sources is, at least partially possible.

Acknowledgment: Presentation supported by Fapesp