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Due to properties such as learning, the brain is very biased toward certain system states and therefore determines typical system states and, in theory, a probability distribution over the set of all system states. This opens up the possibility of applying [https://en.wikipedia.org/wiki/Information_theory/ information theory] type approaches and EFE has some similarities with conditional Shannon entropy except the condition involved is comprised of relationship parameters. EFE is a measure of the expected amount of information required to specify the state of a system (such as an artificial or [https://en.wikipedia.org/wiki/Neural_circuit/ biological neural network]) beyond what is already known about the system from the relationship parameters. For certain non-uniformly random systems, particular choices of the relationship parameters are isolated from other choices in the sense that they give much lower Expected Float Entropy values and, therefore, the system defines relationships. In the context of these relationships a brain state acquires meaning in the form of the relational content of the corresponding experience. The principle article (Quasi-Conscious Multivariate Systems<ref name=Mason2016>Mason, J. W. (2016), Quasi-conscious multivariate systems. Complexity, 21: 125-147. doi:10.1002/cplx.21720</ref>) on this mathematical theory was published in 2015 and was followed by the article (From Learning to Consciousness: An Example Using Expected Float Entropy Minimisation<ref name=Mason2019>Mason, J. W. (2019), From Learning to Consciousness: An Example Using Expected Float Entropy Minimisation. Entropy, 21, 60. doi:10.3390/e21010060</ref>) in 2019. EFE first appeared in a publication in 2012<ref name=Mason2012>Mason, J. W. (2013), Consciousness and the structuring property of typical data. Complexity, 18: 28-37. doi:10.1002/cplx.21431</ref>.

Relationships are ubiquitous among mathematical structures. In particular, weighted relations (also called weighted graphs and [https://en.wikipedia.org/wiki/Weighted_network/ weighted networks]) are very general mathematical objects and, in the finite case, are often handled as adjacency matrices. They are a generalisation of graphs and include all [https://en.wikipedia.org/wiki/Function_(mathematics)/ functions] since functions are a rather constrained type of [https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)/ graph]. It is also the case that [[consciousness]] is awash with relationships; for example, red has a stronger relationship to orange than to green, relationships between points in our field of view give rise to geometry, some smells are similar whilst others are very different, and there’s an enormity of other relationships involving many senses such as between the sound of someone’s name, their visual appearance and the timbre of their voice. Expected Float Entropy includes weighted relations as parameters and, for certain non-uniformly random systems, certain choices of weighted relations are isolated from other choices in the sense that they give much lower Expected Float Entropy values. Therefore, systems such as the brain define relationships and, according to the theory, in the context of these relationships a brain state acquires meaning in the form of the relational content of the corresponding experience.

Given an element <math>S_{i}\in\Omega_{S,V}</math>, the above definitions give rise to a [https://en.wikipedia.org/wiki/Canonical_map/ canonical map] from <math>\Psi_{V}</math> to <math>\Psi_{S}</math>. That is, for <math>U\in\Psi_{V}</math>, the function <math>R\{U,S_{i}\}</math> defined by

where <math>d</math> is a metric given by a [https://en.wikipedia.org/wiki/Matrix_norm/ matrix norm] on the elements of <math>\Psi_{S}</math> in matrix form. In the article Quasi-Conscious Multivariate Systems<ref name="Mason2016" /> the <math>L_{1}</math> norm is used. The article also includes a more general definition of Float Entropy called Multirelational Float Entropy and the nodes of the system can be larger structures than individual neurons.

where <math>P</math> is the [https://en.wikipedia.org/wiki/Probability_distribution/ probability distribution] <math>P:\Omega_{S,V}\to [0,1]</math> determined by the bias of the system due to the long term effect of the system’s inherent learning paradigms in response to external stimulus.

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.

There are some similarities between the minimisation of Expected Float Entropy and the minimisation of surprise in [https://en.wikipedia.org/wiki/Karl_J._Friston/ Karl J. Friston]’s [https://en.wikipedia.org/wiki/Free_energy_principle/ Free energy principle]. The theory is also somewhat complementary to [https://en.wikipedia.org/wiki/Giulio_Tononi/ Giulio Tononi]’s [https://en.wikipedia.org/wiki/Integrated_information_theory/ Integrated information theory] (IIT) which was initially developed to quantify consciousness but gave little priority to how systems may define relationships.

* [https://en.wikipedia.org/wiki/Information_theory/ Information theory]* [https://en.wikipedia.org/wiki/Integrated_information_theory/ Integrated information theory]* [https://en.wikipedia.org/wiki/Free_energy_principle/ Free energy principle]

* [https://en.wikipedia.org/wiki/Hard_problem_of_consciousness/ Hard problem of consciousness]* [https://en.wikipedia.org/wiki/Mind%E2%80%93body_problem/ Mind–body problem]* [https://en.wikipedia.org/wiki/Philosophy_of_mind/ Philosophy of mind]