NOISY INFORMATION AND THE BAYESIAN BRAIN

A deterministic [see the note] environment is expected to be ruled by the cause-and-effect relationship among the events occurring upon the space-time dimensions and the laws of  classical mechanics and of mathematical analysis would be able to explain the whole state-of-the world. Absence of variability, i.e. the perfect reproducibility of the events from a given set of inputs, thus characterizes the deterministic condition. Indeed coming across invariable states not only is not interesting from an informative and statistical point of view, but it seems also non-plausible in a biological sense. In fact, the context where the living beings perform their lives is characterized by several sources of innovation that make uncertainty the dominant factor. Moreover, the perfect understanding of a system does not hold either in a closed, axiomatic system (as established by the Gödel’s theorems http://www.bbc.co.uk/programmes/b00dshx3) or in a real physical system (as proven by the Heisenberg uncertainty principle  http://www.encyclopediaofmath.org/index.php/Uncertainty_principle). Because uncertainty is considered a state of limited knowledge about the future outcome of a system, then it seems “natural” to deal with it in terms of probability. So, to measure the uncertainty we must take a set of possible states with their own probabilities of occurrence, in other terms we need to rely on probability distributions. As consequence, a growing trend in theoretical neuroscience looks at the brain as a Bayesian system. [1-5]. The function of this system would be to deduce the causes of sensory inputs in an optimal way. In order to carry out such a function, the nervous system would encode probabilistic models. These models would be updated by neural processing of sensory information using Bayesian inference [6]. The identification and weighting of uncertainties is the main content of the decision theory, that has involved since the 20^th century social, behavioral and mathematical sciences (namely, economics, psychology and statistics). From the interdisciplinary bases of the decision theory, it has arisen the neuroeconomics as a field of research of the decision-making (DM) process where neuroscience gives the neurophysiological framework to the DM problem.

Uncertainty in the DM arises from noisy inputs or the noisy perception of the inputs (the brain is inherently noisy) and from the lack of complete knowledge of the outcomes. Thus also DM under uncertainty needs probabilistic rules and it may be handled by means of the bayesian theory.

By going into some concepts of the bayesian inference will provide the key to understanding a possible efficient mechanism of information processing of the brain.

Bayesian brain is a notion that explains the brain's cognitive abilities based on statistical principles. It is, therefore, used to refer to the ability of the nervous system to operate in situations of uncertainty in a mode that is close to the optimal one prescribed by bayesian statistics. According to this conception, it is frequently assumed that the nervous system maintains internal probabilistic models that are updated by neural processing of sensory information using methods approximating those of Bayesian probability [4, 7]. There exists large consensus in the literature to link bayesian scheme to the function of the brain. For example in psychophysics, where many aspects of human perceptual or motor behavior are modeled with bayesian statistics. This approach, with its emphasis on behavioral outcomes as the ultimate expressions of neural information processing, is also known for modeling sensory and motor decisions using bayesian decision theory [8-11]. Besides, many theoretical studies ask how the nervous system could implement bayesian algorithms [12]. Statistical inference is the process of drawing conclusions about an unknown distribution from data generated by that distribution. Bayesian inference is a type of statistical inference where data (or new information) is used to update the probability that a hypothesis is true, that is, a system is said to perform Bayesian inference if it updates the probability that a hypothesis H is true given some data D by applying the Bayes’ rule: p(H|D)=p(D|H)×p(H)/p(D). In other words, this equation states that the probability of the hypothesis given the data (P(H|D)) is the probability of the data given the hypothesis (P(D|H)) times the prior probability of the hypothesis (P(H)) divided by the probability of the data (P(D)). Thus, the Bayesian inference relies on representations of the conditional probability density functions rather than on single estimates of the parameter values. This enables the Bayesian system to keep, at each stage of local computation, a representation of all the possible values of the parameters along with their related probabilities. The major implication for the theoretical neuroscience stands at the bayesian coding hypothesis for which the brain encodes information by probabilistic rules in a two stages path: firstly, it performs bayesian inference to evaluate the state-of-the-world and to drive the actions; secondly, it represents the sensory information in terms of probability density functions [4, 6]. In this way the information is efficiently processed over the time and space, by integrating different sensory cues and modalities and by propagating the information from one stage to another during the DM action.      

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