# Expected Float Entropy Minimisation

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Expected Float Entropy Minimisation (EFE) is a mathematically formulated model of consciousness that follows naturally from the following underlying postulate about the nature of consciousness, in which the word interpretation means relational model when translated to the mathematical domain.

(The fundamental postulate of EFE minimisation). If we suppose that consciousness is given by an interpretation or representation of system states then, notwithstanding the possibility that a system may need to satisfy a number of requirements to be conscious, among the infinitely many possible interpretations, consciousness is given by some form of minimum expected entropy interpretation of system states that yields an experience free of unnecessary discontinuities whilst exhibiting the intrinsic structural regularities of probable system states.

The theory is formulated with the aim of explaining (up to relationship isomorphism) how the brain defines the content of consciousness at least with respect to all relationships and associations within subjective experience and the structural content comprised of such relationships. For example, one might ask how the brain defines the perceived geometry of the field of view or the perceived relationships between different colours, or between different audible frequencies. At higher structural levels there are also perceived relationships between different objects and between objects and words for example.

In 2021 the theory was shown to easily extend to include the concept and mathematical formulation of Model Unity[1], which is closely related to the unity (and disunity) of consciousness and provides a different notion of integration to that of IIT. The intention behind Model Unity is to answer questions such as, why different individuals do not have shared perception, why individual visual perception is unified and why visual perception is phenomenally very different to auditory perception, for example.

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 information theory type approaches and EFE is a form of expected conditional entropy where the condition involves 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 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. 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. The principle article (Quasi-Conscious Multivariate Systems[2]) 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[3]) in 2019. The extension to Model Unity was introduced in the article (Model Unity and the Unity of Consciousness: Developments in Expected Float Entropy Minimisation[1]) published in 2021. EFE first appeared in a publication in 2012[4].

The nomenclature “Float Entropy” comes from the notion of floating a choice of relationship parameters over a state of a system, similar to the idiom “to float an idea”. Optimisation methods are used in order to obtain the relationship parameters that minimise Expected Float Entropy. A process that performs this minimisation is itself a type of learning method.

## Overview

Relationships are ubiquitous among mathematical structures. In particular, weighted relations (also called weighted graphs and 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 functions since functions are a rather constrained type of 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. The theory involves a hierarchy of relational models and at the lowest level the primary models involve pairs of weighted relations. Expected Float Entropy minimisation is very general in scope. For example, the theory has been successfully applied in the context to image processing[2] but also applies to waveform recovery from audio data[4].

## Definitions and connections with some areas of mathematics

### Definitions

For a nonempty set ${\displaystyle S}$, a weighted relation on ${\displaystyle S}$ is a function of the form

${\displaystyle R:S^{2}\to [0,1]}$.

Such a weighted relation is called reflexive if ${\displaystyle R(a,a)=1}$ for all ${\displaystyle a\in S}$, and symmetric if ${\displaystyle R(a,b)=R(b,a)}$ for all ${\displaystyle a,b\in S}$. The set of all reflexive, symmetric weighted relations on ${\displaystyle S}$ is denoted ${\displaystyle \Psi _{S}}$.

If ${\displaystyle S}$ is the set of nodes of a system, such as a neural network, then a state of the system ${\displaystyle S_{i}}$ is given by the aggregate of the states of the nodes over some range ${\displaystyle V:=\{v_{1},v_{2},\ldots ,v_{m}\}}$ of node states. Therefore each state of the system ${\displaystyle S_{i}}$ is determined by a corresponding function ${\displaystyle f_{i}:S\to V}$. The set of all possible states of the system is denoted ${\displaystyle \Omega _{S,V}}$.

Given an element ${\displaystyle S_{i}\in \Omega _{S,V}}$, the above definitions give rise to a canonical map from ${\displaystyle \Psi _{V}}$ to ${\displaystyle \Psi _{S}}$. That is, for ${\displaystyle U\in \Psi _{V}}$, the function ${\displaystyle R\{U,S_{i}\}}$ defined by

${\displaystyle R\{U,S_{i}\}(a,b):=U(f_{i}(a),f_{i}(b))}$, for all ${\displaystyle a,b\in S}$,

is an element of ${\displaystyle \Psi _{S}}$.

For ${\displaystyle U\in \Psi _{V}}$ and ${\displaystyle R\in \Psi _{S}}$, the Float Entropy of a state of the system ${\displaystyle S_{i}\in \Omega _{S,V}}$, relative to ${\displaystyle U}$ and ${\displaystyle R}$, is defined as

${\displaystyle fe(R,U,S_{i}):=\log _{2}(\#\{S_{j}\in \Omega _{S,V}\colon d(R,R\{U,S_{j}\})\leq d(R,R\{U,S_{i}\})\})}$,

where ${\displaystyle d}$ is a metric given by a matrix norm on the elements of ${\displaystyle \Psi _{S}}$ in matrix form. In the article Quasi-Conscious Multivariate Systems[2] the ${\displaystyle L_{1}}$ 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.

The Expected Float Entropy (EFE) of a system, relative to ${\displaystyle U\in \Psi _{V}}$ and ${\displaystyle R\in \Psi _{S}}$, is defined as

${\displaystyle efe(R,U,P):=\sum _{S_{i}\in \Omega _{S,V}}P(S_{i})fe(R,U,S_{i})}$,

where ${\displaystyle P}$ is the probability distribution ${\displaystyle P:\Omega _{S,V}\to [0,1]}$ 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.

According to the theory, a system (such as a subregion of the brain) defines a particular choice of ${\displaystyle U}$ and ${\displaystyle R}$ (up to a certain resolution) under the requirement that the EFE is minimized. Therefore, for a given system (i.e., for a fixed ${\displaystyle P}$), solutions in ${\displaystyle U}$ and ${\displaystyle R}$ to the equation

${\displaystyle efe(R,U,P)=\min _{R'\in \Psi _{S},\,U'\in \Psi _{V}}efe(R',U',P)}$

are the weighted relations of interest. For example, when the theory is applied to digital photographs, U gives the relationships between colours and R gives the relationships that determine the geometry of the field of view.

### Model Unity and the unity of consciousness

The theory of EFE minimisation was extended in 2021 to include the definition of Model Unity[1], which is closely related to the unity (and disunity) of consciousness and provides a different notion of integration to that of IIT.

For a given system, let ${\displaystyle {\widehat {S}}}$ denote the set of all possible ways to view the system as a collection of subsystems. Let ${\displaystyle {\mathcal {X}}\in {\widehat {S}}}$, and define

${\displaystyle \mu ({\mathcal {X}},P):=\left(\sum _{X\in {\mathcal {X}}}efe({\mathfrak {R}}_{X},{\mathfrak {U}}_{X},P_{X})\right)-efe({\mathfrak {R}},{\mathfrak {U}},P),}$

where ${\displaystyle P_{X}}$ is the marginal probability distribution for the subsystem ${\displaystyle X}$, each term is individually minimized with respect to the choice of primary models used and the last term is the minimum EFE for the whole system. Furthermore, define

${\displaystyle M(P):=\min _{{\mathcal {X}}\in {\widehat {S}}}\mu ({\mathcal {X}},P).}$

The definition of Model Unity is then as follows.

A system, with probability distribution ${\displaystyle P:\Omega _{S,V}\mapsto [0,1]}$ giving the probability of finding the system in any given state, has Model Unity if and only if ${\displaystyle M(P)\geq 0}$.

The intention behind Model Unity is to answer questions such as, why different individuals do not have shared perception, why individual visual perception is unified and why visual perception is phenomenally very different to auditory perception, for example.

### Connection with ideas in topology

In its simplest form involving only “primary relationships” (i.e. just ${\displaystyle R}$ and ${\displaystyle U}$ as shown above) EFE minimisation can also be considered as a generalisation of the 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

There are some similarities between the minimisation of Expected Float Entropy and the minimisation of surprise in Karl J. Friston’s Free energy principle. The theory is also somewhat complementary to Giulio Tononi’s Integrated information theory (IIT) which was initially developed to quantify consciousness but gave little priority to how systems may define relationships.