How Information Transfer works: interpretation of Information Contents in Bayes Theorem. Understanding Negative Information

In a given space of models or hypothesis the individual information content of each of them is considered as opposed to the Shannon ́s entropy that measures the average information content of the mentioned space. Single model information contents allow expressing, in the Information Theory framework, Bayes ́ Theorem and Shannon ́s Information Functions (like the Kullback-Leibler divergence or the Mutual Information), that are often expressed in the Probability framework. In particular expressing Bayes ́ Theorem in terms of the information contents associated to its probabilities allows understanding how bits of information, introduced in the system by an observation, are transferred to each of the models in the space. It is shown how, from a single observation not one, but two causal information sources are generated: the Information Content Associated to the Evidence that always introduces positive information, and the Information Content Associated to the Bayes ́ Likelihood that always introduces negative bits; therefore the evidence contributes to increase the probability of occurrence of the model and the likelihood to decrease it; depending on the net value of the difference between these two mentioned information contents, the information that arrives to a given model will be positive or negative. Thus, we propose a novel metric, given by the difference of the two mentioned information contents called transfer information content which measures the information transferred to each of the single models in the space. The resolution of the Monty Hall Problem (MHP) and some of its variants in the Information Theory framework proposed allows to confirm the validity of the formulas derived and to understand the counterintuitive and theoretically problematic concept of negative information. The implications of the concepts introduced in terms of information transfer to the emergent field of Local Information Dynamics, to Computational Neuroscience (particularly to Directed Information Theory and Neural Coding), and to Psychology, Active Learning, Decision Making or Philosophy between others, are proposed as further work.