# Actual Causation¶

This section demonstrates how to use PyPhi to evaluate actual causation as described in

```
>>> import pyphi
>>> from pyphi import actual, config, Direction
```

## Configuration¶

Before we begin we need to set some configuration values. The correct way of
partitioning for actual causation is using the `'ALL'`

partitions setting;
`'TRI'`

-partitions are a reasonable approximation. In case of ties the
smaller purview should be chosen. IIT 3.0 style bipartitions will give
incorrect results.

```
>>> config.PARTITION_TYPE = 'TRI'
>>> config.PICK_SMALLEST_PURVIEW = True
```

When calculating a causal account of the transition between a set of elements \(X\) at time \(t-1\) and a set of elements \(Y\) at time \(t\), with \(X\) and \(Y\) being subsets of the same system, the transition should be valid according to the system’s TPM. However, the state of \(X\) at \(t-1\) does not necessarily need to have a valid previous state so we can disable state validation:

```
>>> config.VALIDATE_SUBSYSTEM_STATES = False
```

## Computation¶

We will look at how to perform computations over the basic OR-AND network introduced in Figure 1 of the paper.

```
>>> network = pyphi.examples.actual_causation()
```

This is a standard PyPhi `Network`

so we can look at its TPM:

```
>>> pyphi.convert.state_by_node2state_by_state(network.tpm)
array([[1., 0., 0., 0.],
[0., 1., 0., 0.],
[0., 1., 0., 0.],
[0., 0., 0., 1.]])
```

The `OR`

gate is element `0`

, and the `AND`

gate is element `1`

in the
network.

```
>>> OR = 0
>>> AND = 1
```

We want to observe both elements at \(t-1\) and \(t\), with `OR`

ON and `AND`

OFF in both observations:

```
>>> X = Y = (OR, AND)
>>> X_state = Y_state = (1, 0)
```

The `Transition`

object is the core of all actual causation calculations. To
instantiate a `Transition`

, we pass it a `Network`

, the state of the network at
\(t-1\) and \(t\), and elements of interest at \(t-1\) and \(t\). Note that PyPhi
requires the state to be the state of the entire network, not just the state of
the nodes in the transition.

```
>>> transition = actual.Transition(network, X_state, Y_state, X, Y)
```

Cause and effect repertoires can be obtained for the transition. For example, as shown on the right side of Figure 2B, we can compute the effect repertoire to see how \(X_{t-1} = \{OR = 1\}\) constrains the probability distribution of the purview \(Y_t = \{OR, AND\}\):

```
>>> transition.effect_repertoire((OR,), (OR, AND))
array([[0. , 0. ],
[0.5, 0.5]])
```

Similarly, as in Figure 2C, we can compute the cause repertoire of \(Y_t = \{OR, AND = 10\}\) to see how it constrains the purview \(X_{t-1} = \{OR\}\):

```
>>> transition.cause_repertoire((OR, AND), (OR,))
array([[0.5],
[0.5]])
```

Note

In all `Transition`

methods the constraining occurence is passed as
the `mechanism`

argument and the constrained occurence is the `purview`

argument. This mirrors the terminology introduced in the IIT code.

`Transition`

also provides methods for computing cause and effect
ratios. For example, the effect ratio of \(X_{t-1} = \{OR = 1\}\) constraining
\(Y_t = \{OR\}\) (as shown in Figure 3A) is computed as follows:

```
>>> transition.effect_ratio((OR,), (OR,))
0.415037
```

The effect ratio of \(X_{t-1} = \{OR = 1\}\) constraining \(Y_t = \{AND\}\) is negative:

```
>>> transition.effect_ratio((OR,), (AND,))
-0.584963
```

And the cause ratio of \(Y_t = \{OR = 1\}\) constraining \(X_{t-1} = \{OR, AND\}\) (Figure 3B) is:

```
>>> transition.cause_ratio((OR,), (OR, AND))
0.415037
```

We can evaluate \(\alpha\) for a particular pair of occurences, as in Figure 3C.
For example, to find the irreducible effect ratio of \(\{OR, AND\} \rightarrow \{OR, AND\}\),
we use the `find_mip`

method:

```
>>> link = transition.find_mip(Direction.EFFECT, (OR, AND), (OR, AND))
```

This returns a `AcRepertoireIrreducibilityAnalysis`

object, with a number of
useful properties. This particular MIP is reducible, as we can see by checking
the value of \(\alpha\):

```
>>> link.alpha
0.0
```

The `partition`

property shows the minimum information partition that
reduces the occurence and candidate effect:

```
>>> link.partition
∅ OR AND
─── ✕ ─── ✕ ───
∅ OR AND
```

Let’s look at the MIP for the irreducible occurence \(Y_t = \{OR, AND\}\) constraining \(X_{t-1} = \{OR, AND\}\) (Figure 3D). This candidate causal link has positive \(\alpha\):

```
>>> link = transition.find_mip(Direction.CAUSE, (OR, AND), (OR, AND))
>>> link.alpha
0.169925
```

To find the actual cause or actual effect of a particular occurence, use the
`find_actual_cause`

or `find_actual_effect`

methods:

```
>>> transition.find_actual_cause((OR, AND))
CausalLink
α = 0.1699 [OR, AND] ◀━━ [OR, AND]
```

## Accounts¶

The complete causal account of our transition can be computed with the
`account`

function:

```
>>> account = actual.account(transition)
>>> print(account)
Account (5 causal links)
***********************************
Irreducible effects
α = 0.415 [OR] ━━▶ [OR]
α = 0.415 [AND] ━━▶ [AND]
Irreducible causes
α = 0.415 [OR] ◀━━ [OR]
α = 0.415 [AND] ◀━━ [AND]
α = 0.1699 [OR, AND] ◀━━ [OR, AND]
```

We see that this function produces the causal links shown in Figure 4. The
`Account`

object is a subclass of `tuple`

, and can manipulated the same:

```
>>> len(account)
5
```

## Irreducible Accounts¶

The irreducibility of the causal account of our transition of interest can be evaluated using the following function:

```
>>> sia = actual.sia(transition)
>>> sia.alpha
0.169925
```

As shown in Figure 4, the second order occurence \(Y_t = \{OR, AND = 10\}\) is destroyed by the MIP:

```
>>> sia.partitioned_account
Account (4 causal links)
**************************
Irreducible effects
α = 0.415 [OR] ━━▶ [OR]
α = 0.415 [AND] ━━▶ [AND]
Irreducible causes
α = 0.415 [OR] ◀━━ [OR]
α = 0.415 [AND] ◀━━ [AND]
```

The partition of the MIP is available in the `cut`

property:

```
>>> sia.cut
KCut CAUSE
∅ OR AND
─── ✕ ─── ✕ ───
∅ OR AND
```

To find all irreducible accounts within the transition of interest, use
`nexus`

:

```
>>> all_accounts = actual.nexus(network, X_state, Y_state)
```

This computes \(\mathcal{A}\) for all permutations of of elements in \(X_{t-1}\) and
\(Y_t\) and returns a `tuple`

of all `AcSystemIrreducibilityAnalysis`

objects
with \(\mathcal{A} > 0\):

```
>>> for n in all_accounts:
... print(n.transition, n.alpha)
Transition([OR] ━━▶ [OR]) 2.0
Transition([AND] ━━▶ [AND]) 2.0
Transition([OR, AND] ━━▶ [OR, AND]) 0.169925
```

The `causal_nexus`

function computes the maximally irreducible account for
the transition of interest:

```
>>> cn = actual.causal_nexus(network, X_state, Y_state)
>>> cn.alpha
2.0
>>> cn.transition
Transition([OR] ━━▶ [OR])
```

## Disjunction of conjunctions¶

If you are interested in exploring further, the disjunction of conjunctions network from Figure 7 is provided as well:

```
>>> network = pyphi.examples.disjunction_conjunction_network()
>>> cn = actual.causal_nexus(network, (1, 0, 1, 0), (0, 0, 0, 1))
```

The only irreducible transition is from \(X_{t-1} = C\) to \(Y_t = D\), with \(\mathcal{A}\) of 2.0:

```
>>> cn.transition
Transition([C] ━━▶ [D])
>>> cn.alpha
2.0
```