Causal emergence and effective information

Biological writing already sorts phenomena by scale. Ramstead, Badcock and Friston (2018) describe the free-energy principle as applied to the micro-scale, such as dendritic self-organisation and morphogenesis, to intermediate scales such as cultural ensembles, and to the macro-scale, such as natural selection.¹ In that account scale marks the span of phenomena a theory covers. The same year, Moore, Valentini, Walker and Levin released the Inform software library. It includes effective information among its less common measures and credits Hoel and colleagues (2013) for it.²

Hoel and Levin (2020) state the idea for networks. Replacing a subgraph with a macro-node always lowers the number of nodes. Causal emergence is the case where the grouped macro-scale has more EI than the micro-scale before grouping.³ They treat the gain as the benefit of moving up in scale. They tie it to how far the dimension reduction has minimized noise.⁴ Their method searches across possible groupings to improve EI as much as possible. Random groupings are poor candidates, so only macro-nodes that raise EI are kept.

Hoel and Levin apply the search to a gene regulatory network. The state-space falls from 32 states to 18, and EI rises to 2.9 bits. They report that over 40% of the network takes part in the macro-scale and forms macro-nodes.⁵ The network is deterministic, so they attribute the whole gain to lower degeneracy. Degeneracy falls from 2.21 bits at the micro-scale to 1.26 bits at the macro-scale.⁶ They add that initial states can then be retrodicted more easily from output or steady states.

McMillen, Walker and Levin (2022) present EI as a measure of causal interaction between an experimental manipulation and a system.⁷ To measure it, one sets the state of each variable and records the effect on future states.⁸ They compare this to a reverse genetic screen. They give the Xenopus instructor cells as an example. Depolarizing a specific subset of epidermal cells makes the whole embryo hyperpigmented.⁹ They describe this as EI between the manipulated cells and the pigmentation. Depolarizing other cells would show little mutual information with pigmentation.