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Expected Float Entropy Minimisation (view source)
Revision as of 13:13, 4 May 2020
, 13:13, 4 May 2020no edit summary
Expected Float Entropy minimisation is very general in scope. For example, the theory has been successfully applied in the context to image processing<ref name="Mason2016" /> but also applies to waveform recovery from audio data<ref name="Mason2012" />.
==Definitions and connection connections with Shannon entropysome areas of mathematics==
=== Definitions===
For a nonempty set <math>S</math>, a weighted relation on <math>S</math> is a function of the form
The expression on the left is similar in form to the definition of Shannon entropy. The middle expression reveals the value to be similar to that of <math>efe(R,U,P)</math> when the probabilities in the argument of the logarithm are comparable. Indeed, <math>efe(R,U,P)</math> is an approximation of (1). The expression on the right of (1) shows the mathematical connection to Shannon entropy; the first term is the Shannon entropy <math>H</math> of the system and, with consideration of the log function, the second term has a negative value between <math>-H</math> and 0.
===Connection with ideas in topology===
In its simplest form involving only “primary relationships” (i.e. just <math>R</math> and <math>U</math> as shown above) EFE minimisation can also be considered as a generalisation of the [https://en.wikipedia.org/wiki/Initial_topology/ initial topology] (i.e. weak topology). To see this, the family of functions involved are the typical (probable) system states, the common domain of these functions is the set of system nodes (e.g. neurons, tuples of neurons or larger structures) and the common codomain is the set of node states. In the case of the initial topology a topology is already assumed on the common codomain and the initial topology is then the coarsest topology on the common domain for which the functions are continuous. In the case of EFE minimisation no structure is assumed on either the domain or codomain. Instead EFE minimisation simultaneously finds structures (for us weighted graphs, but topologies could in principle be used) on both the domain and codomain such that the functions are close (in some suitable sense) to being continuous whilst avoiding trivial solutions (such as the two element trivial topology) for which arbitrary improbable functions (system states) would also be continuous. Thus we find the primary relational structures that the system itself defines. In this context objects (visual and auditory) are present and EFE then extends to secondary relationships between such objects by involving correlation for example.
==Connection to other mathematical theories of consciousness==
