# Complexity as Causal Information Integration

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## Abstract

**:**

## 1. Introduction

Therefore Integrated Information can be seen as a measure of the systems complexity. In this context it belongs to the class of theories that define complexity as to what extent the whole is more than the sum of its parts.In short, integrated information captures the information generated by causal interactions in the whole, over and above the information generated by the parts.

#### Integrated Information Measures

Therefore, measures of the strength of causal cross-connections should be based on split models, that have a graphical representation.It seems that if conditional independence judgments are by-products of stored causal relationships, then tapping and representing those relationships directly would be a more natural and more reliable way of expressing what we know or believe about the world. This is indeed the philosophy behind causal Bayesian networks.

**Definition**

**1**(Complexity)

**.**

**Property**

**1.**

**Property**

**2.**

**Property**

**3.**

**Definition**

**2**(Stochastic Interaction)

**.**

**Definition**

**3**(Geometric Integrated Information)

**.**

**Definition**

**4**(Integrated Information)

**.**

## 2. Causal Information Integration

#### 2.1. Definition

**Definition**

**5**(Causal Information Integration)

**.**

**Proposition**

**1.**

- 1.
- $\tilde{P}=\underset{m\to \infty}{lim}{Q}^{m}.$
- 2.
- For every $m\in \mathbb{N}$ there exists a distribution ${\widehat{Q}}^{m}\in \mathcal{P}(\mathcal{Z}\times {\mathcal{W}}^{m})$ that has $\mathcal{Z}$ marginals equal to ${Q}^{m}$$${Q}^{m}\left(z\right)={\widehat{Q}}^{m}\left(z\right),\phantom{\rule{1.em}{0ex}}\forall z\in \mathcal{Z}.$$Additionally ${\widehat{Q}}^{m}$ factors according to the graph corresponding to the split system$${\widehat{Q}}^{m}(z,w)=\widehat{Q}{\left(x\right)}^{m}\prod _{i=1}^{n}{\widehat{Q}}^{m}\left({y}_{i}\right|{x}_{i},w){\widehat{Q}}^{m}\left(w\right),\phantom{\rule{1.em}{0ex}}\forall (z,w)\in \mathcal{Z}\times {\mathcal{W}}^{m}.$$

**Theorem**

**1**

**.**Let q be a prime power. The smallest m for which any probability distribution on $\{1,\dots ,q\}$ can be approximated arbitrarily well as mixture of m product distributions is ${q}^{n-1}$.

#### 2.1.1. Ground Truth

**Proposition**

**2.**

**Proposition**

**3.**

#### 2.1.2. Relationships between the Different Measures

**Conjecture**

**1.**

**Example**

**1.**

**Theorem**

**2.**

#### 2.1.3. em-Algorithm

**Theorem**

**3**

**.**The minimum divergence between ${\mathcal{M}}_{W|Z}$ and ${\mathcal{E}}^{m}$ is equal to the minimum divergence between $\tilde{P}$ and ${\mathcal{M}}_{CII}^{m}$ in the visible manifold

**Proof of Theorem**

**3.**

**Proposition**

**4**

**.**The monotonic relations

**Proof of Proposition**

**4.**

**Proposition**

**5.**

#### 2.2. Comparison

#### 2.2.1. Ising Model

#### 2.2.2. Results

## 3. Discussion

## 4. Materials and Methods

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Graphical Models

**Definition**

**A1.**

**Definition**

**A2.**

**Lemma**

**A1.**

**Proof of Lemma**

**A1.**

**Definition**

**A3.**

- 1.
- Generate an ij edge as in Table A1, steps 8 and 9, between i and j on a collider trislide with an endpoint j and an endpoint in M if the edge of the same type does not already exist.
- 2.
- Generate an appropriate edge as in Table A1, steps 1 to 7, between the endpoints of every tripath with inner node in M if the edge of the same type does not already exist. Apply this step until no other edge can be generated.
- 3.
- Remove all nodes in M.

**Table A1.**Types of edge induced by tripaths with inner node m ∈ M and trislides with endpoint m ∈ M.

1 | i ← m ← j | generates | i ← j |

2 | i ← m – j | generates | i ← j |

3 | i ↔ m —j | generates | i ↔ j |

4 | i ← m → j | generates | i ↔ j |

5 | i ← m ↔ j | generates | i ↔ j |

6 | i – m ← j | generates | i ← j |

7 | i – m – j | generates | i–j |

8 | m → i – ⋯ – $\circ \leftarrow $j | generates | i ← j |

9 | m$\to i--\cdots --\circ \leftrightarrow $j | generates | i ↔ j |

**Definition**

**A4**(c-separation)

**.**

## Appendix B. Proofs

**Proof of the Relationship**

**(4).**

**Proof of Proposition**

**1.**

**Proof of Proposition**

**2.**

**Proof of Proposition**

**3.**

**Proof of Proposition**

**5.**

**Proof of Theorem**

**2.**

- they form an undirected path between A and B,
- they can form a directed path from A to B,
- they can form a directed path form B to A,
- there exists a collider,
- A and B have a common exterior influence.

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**Figure 2.**Interior and exterior influences on Y in the full and the split system corresponding to ${\mathsf{\Phi}}_{I}$.

**Figure 10.**Sketch of the relationships among $MP\left(\mathcal{Z}\right),{\mathcal{M}}_{CIS}$ and ${\mathcal{N}}_{CIS}.$

**Figure 12.**Sketch of the relationship between the manifolds corresponding to the different measures.

**Figure 15.**Ising model with 2 nodes and the differences between ${\mathsf{\Phi}}_{CIS}$ and ${\mathsf{\Phi}}_{CII}$.

**Figure 19.**Curve of one run of the em-algorithm for each $\beta $ coloured according to the distribution of W.

$\left|\mathcal{W}\right|$ | Minimum | Maximum | Arithmetic Mean |
---|---|---|---|

2 | 0.011969035529826939 | 0.5028091152589176 | 0.15263592877594967 |

3 | 0.021348311360946 | 0.5499395859771526 | 0.1538653506807848 |

4 | 0.014762084688030863 | 0.3984635189946462 | 0.15139198568055212 |

8 | 0.017334311629729246 | 0.4383731978333986 | 0.15481967618112732 |

16 | 0.024306996171092318 | 0.4238222051787452 | 0.1490336847067273 |

300 | 0.016524177216064712 | 0.47733473380366764 | 0.15493896625208842 |

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Langer, C.; Ay, N.
Complexity as Causal Information Integration. *Entropy* **2020**, *22*, 1107.
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**AMA Style**

Langer C, Ay N.
Complexity as Causal Information Integration. *Entropy*. 2020; 22(10):1107.
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**Chicago/Turabian Style**

Langer, Carlotta, and Nihat Ay.
2020. "Complexity as Causal Information Integration" *Entropy* 22, no. 10: 1107.
https://doi.org/10.3390/e22101107