Conditional independence#

The whole framework rests on one assumption about the causal model, introduced on the substrate page: the units of a substrate are conditionally independent given the previous state. Each unit’s next state depends only on the previous state of the substrate, not on what the other units happen to do at the same step:

\[ p(\bar{u} \mid u) = \prod_{i=1}^{n} p(\bar{u}_i \mid u). \]

The assumption follows from what a substrate is taken to be. A substrate is a complete causal model: it contains every variable that influences its units, so any dependence between them can be traced to a state inside the model rather than to something left out (Albantakis et al., 2019). Time in the model is explicit and discrete, and units act on one another only from one step to the next, never within a step. Its transition probabilities are defined by intervention: each state is imposed with the do-operator and the next state observed, so the units are physical in the sense that each can be observed and manipulated on its own (Albantakis et al., 2023). Put together, once the whole previous state is fixed there is nothing left for two units’ next states to share. Any residual correlation between them would have to come from a common influence within the same step, which a complete model rules out. The joint transition therefore factors into one term per unit.

This is what lets a substrate be described by per-unit transition probabilities (a state-by-node transition probability matrix) rather than by joint transitions, and it is what allows a partition to be applied to individual connections: noising one unit’s inputs changes that unit’s factor and leaves the others as they were. The assumption is the second equation of the IIT 4.0 formalism (Albantakis et al., 2023, Eq. 2), and PyPhi requires it of every substrate.

PyPhi enforces it#

A transition probability matrix that violates conditional independence describes instantaneous causality — units influencing each other within a single step — which indicates a missing exogenous variable. PyPhi rejects such a matrix. Consider two units that stay put when they agree and flip with probability one-half when they disagree; their joint (state-by-state) transitions are not conditionally independent:

import pyphi

pyphi.config.progress_bars = False

pyphi.examples.cond_depend_tpm()

array([[1. , 0. , 0. , 0. ],
       [0. , 0.5, 0.5, 0. ],
       [0. , 0.5, 0.5, 0. ],
       [0. , 0. , 0. , 1. ]])

Building a Substrate from it raises an error rather than silently accepting an ill-defined causal model:

try:
    pyphi.Substrate(pyphi.examples.cond_depend_tpm())
except pyphi.exceptions.ConditionallyDependentError as error:
    print(error)

TPM is not conditionally independent.
See the conditional independence example in the documentation for more info.

Recovering an independent representation#

Every state-by-node matrix corresponds to a unique conditionally independent state-by-state matrix. Converting a conditionally dependent matrix to state-by-node form and back reveals the independent representation PyPhi would assume — the two units become independent, each flipping with probability one-half:

from pyphi import convert

sbn = convert.state_by_state2state_by_node(pyphi.examples.cond_depend_tpm())
convert.state_by_node2state_by_state(sbn)

array([[1.  , 0.  , 0.  , 0.  ],
       [0.25, 0.25, 0.25, 0.25],
       [0.25, 0.25, 0.25, 0.25],
       [0.  , 0.  , 0.  , 1.  ]])

The dependence in the original matrix can always be restored by adding the missing variable explicitly. cond_independ_tpm() does this for the same two units, introducing a third unit whose state decides whether they flip; the resulting three-unit substrate satisfies conditional independence and is accepted. Deterministic transitions are always conditionally independent, so deterministic substrates never run into this constraint.

References#

  • Albantakis L, Marshall W, Hoel E, Tononi G (2019). What caused what? A quantitative account of actual causation using dynamical causal networks. Entropy 21(5): 459.

  • Albantakis L, Barbosa L, Findlay G, Grasso M, et al. (2023). Integrated information theory (IIT) 4.0. PLOS Computational Biology 19(10): e1011465.