Navigating Complexity: How Resource-Limited Agents Derive Probability and Generate Emergence

Probability, precision, and emergence aren't fundamental axioms — they're what you get when a compression-seeking agent runs into the hard limits of computation and noisy data.

The core idea is disarmingly simple: an agent trying to survive must build a compact model of the world. But Kolmogorov complexity — the length of the shortest program that reproduces a dataset — is provably uncomputable. No agent can find the optimal model. So what does a rational, resource-limited agent do instead? It maintains a portfolio of models, weighted by how well each one compresses past data and how simple each one is. That weighted portfolio is Bayesian inference, and the Solomonoff prior (shorter programs get exponentially higher weight) is Occam's razor — both derived, not assumed. The library-browsing toy example makes this concrete: an agent reading pages from the Library of Congress can't build one universal compressor on the fly, so it maintains specialized sub-models and bets on whichever fits the incoming page best.

That handles which model to trust. But there's a second, orthogonal question: when your model's prediction disagrees with what you observe, how hard should you update? This is where precision enters — defined here as inverse variance, i.e., confidence. The paper formalizes a gain term (essentially the Kalman gain) that interpolates between "trust the data" and "trust the model" based on their relative precisions. Crucially, this same gain operates on the model itself over slower timescales: persistent, structured errors reshape the model; structureless noise gets left as residual. Learning saturates naturally once the model has absorbed all the compressible regularity. The paper then grounds this in neural hardware — cross-frequency coupling in a laminar neural mass model — where slow oscillation envelopes gate the gain on fast prediction-error channels. Attention and gating fall out as the extreme cases of precision weighting (precision → 0 suppresses a channel; precision → ∞ amplifies it).

Friston's Free Energy Principle and Active Inference fit neatly inside this picture as a special case: when the agent approximates a true posterior with a tractable variational distribution, minimizing variational free energy is exactly what the bounded-agent story prescribes. The paper is careful to flag two genuine differences rather than collapsing the frameworks: KT keeps the objective function separate from the generative model (goals aren't smuggled in as prior preferences), and KT's simplicity drive is algorithmic ($2^{-K(M)}$) rather than prior-relative.

Finally, the paper gives a formal, agent-centric definition of emergence: a system is emergent for an agent when coarse-graining its microscopic description dramatically reduces apparent Kolmogorov complexity while preserving high entropy and mutual algorithmic information with the original — and when operating on that coarser model actually improves the agent's survival utility. This ties cellular automata (Rule 110, Israeli & Goldenfeld coarse-graining results) and the renormalization group into the same framework: both are instances of finding a macroscopic description that is far simpler than the microscopic one, without throwing away what matters. The upshot is that "complexity" is not a property of a system alone — it's relational, defined with respect to an observing agent and the coarse-graining that agent can apply.