THEORY-424: Mutual-information measures of individuality that read a system's next state off its present one measure association, not propagation: I(S_{t+1}; S_t) can be positive when the environment alone drives the system, and conditioning on the environment does not recover the interventional quantity, which can be zero where the conditional measure is one bit

Source

Bourrat (2024), LIT-005, European Journal for Philosophy of Science 14:9, read in full in NOTE-027. The paper argues against Krakauer et al. (2020), which the record holds as LIT-1137, unread. The counterexample to the paper’s equivalence is NOTE-027’s own.

What was actually shown

The measures. Krakauer et al. coarse-grain a Markov process into a system S and an environment E and decompose I(S_t, E_t; S_{t+1}) two ways (Eq. 5): as A* + nC, with A* = I(S_{t+1}; S_t), and as Ei + A, with A = I(S_{t+1}; S_t | E_t). High A* they call organismal individuality, high A colonial individuality, high nC environmentally determined individuality.

The confound (§4, Fig. 2). If E fully determines S at every step, S_t and S_{t+1} are correlated through E and A* can be positive with no causal link from S_t to S_{t+1}. Bourrat’s replacement, Â = I(S_{t+1}; Ŝ_t), the mutual information under an intervention on S_t (Eq. 6), is zero in that world. NOTE-027 rates this argument correct.

The equivalence that fails (§5, p. 9). The paper says that adjusting for E_t in Krakauer’s model gives  ≡ I(S_{t+1}; S_t | E_t) = A. NOTE-027’s counterexample: S_{t+1} = S_t ⊕ E_t, with S_t and E_t independent and uniform. Then A = 1 bit, but p(s_{t+1} | do(s_t)) = Σ_e p(s_{t+1} | s_t, e) p(e) = ½ for every s_t, so  = 0 under any intervention distribution. Adjustment licenses the averaged channel Σ_z p(s′ | s, z) p(z); its mutual information is a different quantity from the conditional mutual information, which averages within-stratum dependence.

What could have come out otherwise. Conditioning on E could have been enough to remove the confound and give the causal quantity. In the XOR system it gives the most dependence the system has, where intervention finds none.

What this does not say

• That  is the right measure of individuality. It is a definition. Its value depends on the distribution chosen for Ŝ_t (fn. 4: uniform, population, or a “normal” range), which the paper leaves open, so comparisons across coarse-grainings are not yet defined.
• That Bourrat’s ETI criteria work. C1 (normalised  across coarse-grainings) and C2 (low variance of  under interventions on E) are never computed on any model, and “low” has no threshold.
• That LIT-005 shows how to estimate causal individuality from observational data. Its one implementation-level step is the false equivalence, and its Fig. 3 adjustment set leaves a backdoor through E_b,T−1 open (NOTE-027, C6).
• That  separates an individual from a part of one. The paper says it does not (§7), and offers “the coarsest grain at which  is maximal” as future work.
• That Krakauer’s measures are useless. They measure predictive retention. The point is that retention and causal propagation are different things, and a high score on the first is not evidence of the second.

Connections

• QUESTION-115 asks which of persistence, integration and causal level each information criterion tracks. This constrains the answer for temporal retention: A* and A track predictive dependence, which the environment can supply, and neither is the interventional quantity. It does not say which criterion finds a collective’s boundary.
• The ΦID review (LIT-025; THEORY-497) makes the same point for its own measures: they are interventional only if p(X′ | X) is a do()-distribution, and predictive otherwise. Neither paper cites the other.
• Effective information (THEORY-222) is mutual information under a maximum-entropy intervention. Â with a uniform intervention distribution has the same structure (NOTE-027’s observation), which is one way to fill the choice fn. 4 leaves open.
• Bourrat’s earlier account (LIT-049; THEORY-607) first stratifies on the environment, as δC|E, to exclude ecologically driven coherence. This is where that stratification and its gap show up formally.

Note of 2026-10-21: the XOR system again, and what  depends on

A10 of the exchange of 2026-10-20 took this entry’s XOR system (S_{t+1} = S_t ⊕ E_t) as its main formal lesson: “Causal individuation requires a specified family of interventions and background conditions.” Checked: A = 1 bit and  = 0. A* is also 0, so the observational and interventional measures agree here, and only the conditional one differs.  is 0 under every intervention distribution on S_t while E_t is fair. With P(E = 1) = 0.2 and uniform interventions it is 1 − H(0.2) ≈ 0.278 bits. In this system, then, what  depends on is the distribution of the background variable, not of the intervention, which adds to the summary’s “ itself depends on an intervention distribution the paper leaves open”.

A13 and A14 proposed a TERM, “intervention regime”, for what is manipulated, what is held as background and how interventions are weighted. It was not filed: its content is this entry’s limit, with QUESTION-204 and QUESTION-115’s “intervention distribution”. The intervention family is one component of TERM-065.