Complexity as Causal Information Integration

Abstract

Complexity measures in the context of the Integrated Information Theory of consciousness try to quantify the strength of the causal connections between different neurons. This is done by minimizing the KL-divergence between a full system and one without causal cross-connections. Various measures have been proposed and compared in this setting. We will discuss a class of information geometric measures that aim at assessing the intrinsic causal cross-influences in a system. One promising candidate of these measures, denoted by ΦCIS, is based on conditional independence statements and does satisfy all of the properties that have been postulated as desirable. Unfortunately it does not have a graphical representation, which makes it less intuitive and difficult to analyze. We propose an alternative approach using a latent variable, which models a common exterior influence. This leads to a measure ΦCII, Causal Information Integration, that satisfies all of the required conditions. Our measure can be calculated using an iterative information geometric algorithm, the em-algorithm. Therefore we are able to compare its behavior to existing integrated information measures.

How Morphological Computation Shapes Integrated Information in Embodied Agents

“…Hence T measures the connections between C i t and C j j+1 with i, j ∈ {1, 2} and i = j. A proof of the closed form solution can be found in Langer and Ay (2020). All the following measures can be proven in a similar way.…”

“…A proof of the closed form solution can be found in Langer and Ay ( 2020 ). All the following measures can be proven in a similar way.…”

“…It aims at measuring the strength of the connections among different nodes across different points in time, in other words, the connections that integrate the information. Since every influence on C t +1 is known in our setting, we are able to use the measure Φ T proposed in Langer and Ay ( 2020 ). This measure is defined in the following way and depicted as (b) in Figure 5 .…”

“…Measures for integrated information differ depending on the version of the theory they are referring to and on the framework they are defined in. We discussed a branch of these measures building on information geometry in Langer and Ay ( 2020 ). In this article we will use the measure that we propose in Langer and Ay ( 2020 ) in the case of a known environment, as defined in section 2.3.1.…”

“…We discussed a branch of these measures building on information geometry in Langer and Ay ( 2020 ). In this article we will use the measure that we propose in Langer and Ay ( 2020 ) in the case of a known environment, as defined in section 2.3.1. As advocated by the authors of Mediano et al ( 2021 ) we will treat the integrated information measure as a complexity measure and therefore as a way to quantify the relevant information flow in the controller.…”