Source
- Bruineberg, Dołęga, Dewhurst & Baltieri (2020 preprint), LIT-603, read in NOTE-447: §3.2 (Pearl blankets), §4.2 (the soup re-run), §5.1 and Figure 7 (Friston blankets on an arbitrary network).
- Friston (2013), LIT-526, read in NOTE-421: §3 (how the blanket of the simulated soup was found).
The claim, derived
Take a directed acyclic graph G over variables V, and any distribution that factorises over it as ∏ p(x_v | pa(v)). For a set A ⊂ V, let
mb(A) = (pa(A) ∪ ch(A) ∪ pa(ch(A))) ∖ A,
the parents of A, its children, and its children’s other parents.
A is conditionally independent of R = V ∖ (A ∪ mb(A)) given mb(A). The factors of the joint that mention any variable in A are p(x_a | pa(a)) for a in A, and p(x_c | pa(c)) for children c of A. These factors mention only variables in A ∪ mb(A). Every other factor is constant in x_A. So p(x_A | x_mb, x_R) is proportional to a product that does not contain x_R, and the independence follows. This is Pearl’s local Markov property written for a set, and the same argument works for undirected graphs with the neighbours of A as its blanket.
Nothing in the derivation used a property of A. It holds for every subset. So every A with R non-empty yields a partition into internal (A), blanket (mb(A)) and external (R) states. Friston’s further split of the blanket into sensory states (parents of A) and active states (children of A) is likewise defined once A is given. The formalism takes the internal set as an input. Its output is a blanket. It never outputs the internal set.
What the sources show
Bruineberg et al. state it and draw it. They apply the four-way labelling to an arbitrary 18-node network. In their words, the partition “cannot simply be applied to any graphical model without first making some additional assumptions”. Labelling x10 as internal yields one blanket, and labelling x9 yields a different one (Figure 7). The Friston blanket “does not define what is inside and what is outside (or at least not without further assumptions), but can rather only be identified once we have already made this choice (by labelling one node as ‘internal’)”. Neither blanket’s sensory states coincide with the network’s observed variables. NOTE-447 rates this C4, strong for arbitrary graphs.
Friston’s own procedure shows it. In the soup of LIT-526, the internal set was not found by looking for a blanket. Spectral graph theory picked the eight most densely coupled subsystems from a time-averaged adjacency matrix, and the blanket was then traced as their parents, children and co-parents (NOTE-421, Key results). The selection did the individuating work. A clustering criterion and a value of k did it, and neither belongs to the Markov-blanket formalism. Bruineberg et al. reproduce this from Friston’s code (§4.2).
What this does not say
- It does not say that blankets are useless or that the free energy principle is false. The derivation is the reason blankets are useful in variational inference, and it leaves Lemma 2.1 of LIT-526 untouched.
- It does not cover every later definition. Recent work defines a blanket by sparse coupling in the flow, or in the Hessian of the log-density at nonequilibrium steady state. Bruineberg et al. note this and do not analyse it, and the record has read it only as Raja et al. summarise it (LIT-598, Eqs. 4–5), where the partition is again given before its conditions are checked. Whether some formulation also selects the internal set is open. The open question in NOTE-447 asks the same thing.
- It does not say that an added selection criterion must be arbitrary. A criterion such as densest coupling, or Friston’s conjectured minimum blanket entropy (C8 in NOTE-421), could be principled. The claim is only that such a criterion is added to the formalism, and that whatever it is does the individuating.
- It is about individuation, not about any of the four unities of ADR-024. “Internal states” here are a set of variables, not a self, an agent or a subject.
Connections
- THEORY-073 extends this account to its wider and Proposed conclusion: in practice the boundary comes from modelling choices, and a blanket cannot represent a boundary the system produces.
- Closure of constraints (LIT-582) takes the other route. It defines the boundary by which constraints the system produces, not by a set chosen beforehand. THEORY-072 uses it for the demarcation of organisms.
- The Yoneda lemma (THEORY-032) has the same shape in another formal setting. The relation determines its relata only among objects that are already given.