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-driven agent hits resource limits.

The paper's central move is to ask: what happens when an agent tries to compress a noisy, diverse world but can't compute the theoretically optimal model? The answer, it turns out, is Bayesian inference. The authors use a vivid toy example: a robot working through a vast library, page by page, with only partial access to each page before it must pick a model. No single model handles novels, research papers, and Latin manuscripts equally well. The rational response is to maintain a weighted ensemble of models — and once you add the Solomonoff prior (shorter programs are exponentially more probable, formalizing Occam's razor), you've derived Bayesian inference from first principles rather than assumed it.

Precision is the paper's second contribution and the most operationally concrete. When a model's prediction disagrees with incoming data, the agent faces a fork: is this mismatch signal (update the model) or noise (ignore it)? Precision — defined as inverse variance, i.e., confidence — resolves this via a gain factor that interpolates between "trust the data" and "trust the model." The math is the scalar Kalman gain, which the authors connect explicitly to predictive coding. The key insight is that this same gain operates at two timescales: fast (updating the current estimate) and slow (revising the model itself). As the model accumulates structure, its precision grows, the gain shrinks, and structural learning saturates — the model has absorbed the compressible regularity and the rest is noise. Attention and gating, in this view, are just precision driven to extremes.

The paper then positions Karl Friston's Free Energy Principle (FEP) as a special case of this broader story rather than a competing framework. FEP's variational free energy — expected prediction error plus a KL-divergence complexity penalty — falls out naturally once you assume the agent approximates a true posterior with a tractable one. Two differences are flagged honestly: KT keeps the objective function separate from the generative model (goals aren't baked into priors), and KT's simplicity drive is algorithmic (Kolmogorov complexity) rather than prior-relative.

Finally, emergence gets a formal agent-centric definition: it occurs when a coarse-graining of microscopically incompressible data yields a macroscopic description with dramatically lower apparent complexity, non-trivial entropy, and — crucially — improved utility for the agent. This isn't just philosophical; it connects to cellular automata (Rule 110 looks random microscopically but has compressible macro-structure), the renormalization group in physics (integrating out microscopic degrees of freedom to find scale-invariant behavior), and Kolmogorov's structure function. The punchline is that emergence is partly in the eye of the beholder: it depends on which coarse-graining the agent applies.