We propose a new way of defining entropy of a system, which gives a general form which may be nonextensive as Tsallis entropy, but is linearly dependent on component entropies, like Renyi entropy, which is extensive. This entropy has a conceptually novel but simple origin and is mathematically easy to define by a very simple expression, though the probability distribution resulting from optimizing it gives rather complex, which is compared numerically with the other entropies. It may, therefore, appear as the right candidate in a physical situation where the probability distribution does not suit any of the previously defined forms.PACS numbers: 02.50 Ey, 05.90 +m, 89.90 +n
We study properties of popular, near-uniform, priors for learning undersampled probability distributions on discrete nonmetric spaces and show that they lead to disastrous results. However, an Occam-style phase space argument allows us to salvage the priors and turn the problems into a surprisingly good estimator of entropies of discrete distributions.
We investigate the possible origin of hierarchical structures in complex systems describable in terms of a finite and small number of parameters which control the behavioral pattern at each level of organization. We argue that the limitation on the number of important parameters at each stage is a reflection of the fact that Thom's classifications of catastrophes, i.e., qualitative changes, involve only a few parameters. In addition, we also point out that even in systems with a large number of components, only a few may be of statistically great significance, just as in Zipf's law the quantitative measure of the important collections is inversely proportional to the rank. We then consider the concept of relative degeneracies coming from change of resolving power, at various scales, which too would vindicate the procedure of coarse-graining in building up hierarchical organizations. We suggest that, similar to the group-theoretical annihilation of dangling tensor indices due to symmetry to minimize energy, even in more inexact contexts such as in biology and the social sciences, similar attempts by the system to reduce frustration may lead to cluster formation, which are semi-closed, and let leakage interactions come into play at larger scales.
The role of perception in conscious behavior and decision-making is examined. The effect of spatial and temporal stochasticity in the acquisition of beliefs is discussed. The idea of an agent as a locally strongly coupled group of states leads to the creation of energy minima in an interaction potential landscape. The interaction of such agent states and environment states acting at different levels of complexity and scale, subject to stochastically expressed interaction interfaces, may lead to asymmetry in perceptions. Agents possessing different perception related beliefs are then connected in a social network.
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