Note also that since for this particular case the assumptions used for the derivation of (1) hold, it can be written

$$b_{i j}(1)={\frac{s_{j i}}{\sum_{k}s_{k i}}}$$

## 3.3 Effective Information Of Subsets

For the definition of integrated information it is required to define how to measure effective information for subsets of a given network X. Let A ⊆ X. When X is in state xi we denote with πA(xi) = Axithe state of A. Let At be the random variable representing state of A at instant t. We can define for A state transition matrix AS and state backward-transition matrix AB in analogy with the general case as

$$A_{S i j}\triangleq p(A_{t+1}=a_{j}\,|\,A_{t}=a_{i})$$
and
$${}^{A}b_{i j}(t)\triangleq p(A_{t-1}=a_{j}\,|\,A_{t}=a_{i})$$

Both can be obtained from S e p(·) after some long but straightforward computations. Intuitively and informally speaking, the computation is based on summing transition probabilities over all states of X
which are equivalent with respect to subset A, averaged with their state probabilities.

Now, all definitions introduced for a network X can be applied to any of its subset of nodes A by substituting in the previous formulas S, B, and X respectively with AS,
AB, and A. We then obtain

$$e i(t,A,a_{h})\triangleq D_{K L}({}^{A}B_{h}(t)||A_{t-1})$$
$$({\mathfrak{I}})$$
ABh(t)||At−1) (3)

## 4 Integrated Information

We are now ready to formally define integrated information, that is the quantity of information generated in a system transitioning from one state to the next by the causal interaction of its parts, above and beyond the quantity of information generated independently by each of its parts.

Given a system X let V ⊆ X and {Mk} a partition of V in m subsets. Let Mk(t) be the random variables describing the state of the k-th component of the partition at instant t. Let X be in state xi at instant t. Then V at the same instant is in state Vxi and the k-th component is in state Mk xi. In the following we use vh and µk as a shorthand for Vxi and Mk xi, respectively.

Partition-dependent integrated information is first defined for a subset V as a function of partition
{Mk}, time instant t, and current state vh as

$$\phi(t,V,\{M_{k}\},v_{h})\triangleq e i(t,V,v_{h})-\sum_{k=1}^{m}e i(t,M_{k},\mu_{k})$$

Value computed by this formula clearly depends on the considered partition. Tipically, an unbalanced partition produces a lower value of φ (see [1]). Hence the following normalization function is introduced

$$(4)$$

N(t, V, {Mk}, vh) ⊜ (m − **1) min**
k
{H(Mk(t))}