Deep Inference Interleave

Deep Inference Interleave

Encoding of Jules Hedges' Feb 2024 Cybercat post "Passive Inference is Compositional, Active Inference is Emergent" into the ASI chromatic walk. Sits at trit=0 (coordinator/mediator) because it translates a categorical observation into a concrete triad of executable artifacts.

1. Bayesian Chain Rule

Given Markov kernels φ : X -> Y and ψ : Y -> Z and a prior π on X, the Bayesian inverse (dagger) of the composite is the reversed composite of inverses:

        (φ ; ψ)†_π  =  ψ†_{π;φ}  ;  φ†_π

LaTeX:

(\varphi \mathbin{;} \psi)^{\dagger}_{\pi}
  = \psi^{\dagger}_{\pi ; \varphi} \mathbin{;} \varphi^{\dagger}_{\pi}

This is the functoriality of inversion on the state-indexed category Kl(P)_\bullet. The prior propagates forward along φ to condition ψ†, exactly the pattern a continuation handler needs.

2. Passive (Compositional) vs Active (Emergent)

Introduce a semi-reliability drift parameter p \in [0,1]:

• p = 1 — perfectly reliable channel; Bayesian inverse is exact; chain rule holds on the nose.
• p < 1 — lossy/drifting channel; errors accumulate.

Passive inference (perception only): composition of dagger kernels. Because each stage uses the pushed-forward prior, errors are absorbed into the state index and the aggregate converges as the chain extends. Compositional.

Active inference (perception + action closing a loop): the agent's action modifies the very generative model used to invert. The fixed point p* is no longer guaranteed; small drift in p can diverge. Emergent — must be simulated, not composed.

Slogan (Hedges): "Passive inference is a functor. Active inference is a dynamical system on the space of functors."

Empirical refinement (babashka prototypes at ~/i/passive-inference-proto/, 2026-04-12):

• chain_rule.clj — passive chain-rule residual ≈ 1.8×10⁻¹⁶ on composed 1-D Gaussians. Green.
• active_conjecture.clj — the failure mode is non-identifiability, not divergence: individual factors (p,q) are unrecoverable from pushforward samples, only the composite is; drift ≈1.5–1.8 in per-factor slopes while composite stays on truth.
• sync_vs_async.clj — async gradient descent beats sync on composite error (≈0.35–0.56 vs ≈0.69–0.72 across 3 seeds). Reads as Strang-splitting-vs-Euler-step on the nonlinear coupling ψ'_{π;φ(p)}(q).
• langevin_split.clj — inverts the last result. Add Langevin noise √(2ηT)dW and joint wins (0.57–0.72 vs 0.70–0.74). Noise and splitting are substitutes. Refined conjecture: async ≥ sync at the same effective noise level; the T=0 async advantage was a finite-step artifact, not a fixed-point property.

3. Continuation = Propagator Backward Flow

Hedges writes the dagger in continuation-passing style:

k(σ ; ψ) ; ψ′_σ(q)

where σ is the current belief state, ψ′_σ is the local Bayesian inverse at σ, and k is the continuation receiving the updated belief.

This is exactly the backward arrow in a Radul–Sussman propagator network: forward cells publish evidence, backward continuations re-derive upstream cells by composing local inverses. See ~/i/zig-syrup/src/propagator.zig (neurofeedback gate variant) and ~/i/zig-syrup/src/continuation.zig (AGM belief revision as dagger accumulator).

Identification:

continuation k       <->   backward propagator fiber
local inverse ψ′_σ   <->   cell update rule at σ
prior push π;φ       <->   forward fiber (standard propagate)

4. Cortex-as-Anthill — Siegel Stack L0..L5 (Inverted)

Hedges argues the cortex is an anthill: no central controller, hierarchical layers each running local passive inference, with active inference emerging only at the aggregate. Map to the Siegel hardware/software stack inverted — sensorimotor closest to silicon, abstraction at the top-of-stack social layer:

Inversion matters: the "deepest" part of cognition is physically shallowest; the ledgered social-self is the emergent top. Active inference lives at L4–L5; passive inference dominates L0–L3.

5. GF(3) Triad — Σ = 0

The three agents in this chromatic walk realise the chain rule concretely:

trit  agent         artifact                         role in (φ;ψ)†
----  ------------  -------------------------------  ---------------------------
 -1   validator     monad-bayes RMSMC                computes ψ†_{π;φ} numerically
  0   coordinator   nashator coplay                  mediates prior push π ; φ
 +1   generator     zig-syrup propagator             writes φ†_π continuation

Sum: (-1) + 0 + (+1) = 0 in GF(3). The triad is conservative: the composed dagger is reconstructed distributively; no single agent holds the full inverse. This mirrors the "anthill" thesis one level up — the inference apparatus is itself decentralised.

6. References

• Toby St Clere Smithe. Mathematical Foundations for a Compositional Account of the Bayesian Brain. PhD thesis, Oxford, 2023.
• Dylan Braithwaite, Toby St Clere Smithe, Jules Hedges. The Compositional Structure of Bayesian Inference. MFPS / arXiv:2305.06112.
• Jules Hedges. Passive Inference is Compositional, Active Inference is Emergent. Cybercat Institute blog, Feb 2024.
• Karl Friston et al. Active Inference: The Free Energy Principle in Mind, Brain, and Behavior. MIT Press, 2022.
• Radul & Sussman. The Art of the Propagator. MIT CSAIL TR, 2009.

7. Baby Prototype — Numeric Chain-Rule Verification in monad-bayes

Goal: verify (φ;ψ)†_π = ψ†_{π;φ} ; φ†_π numerically for two 1-D Gaussian kernels.

Setup.

• Prior π = N(0, 1) on X = R.
• φ : X -> Y, y | x ~ N(a*x + b, σ_φ²) with a=1.0, b=0.0, σ_φ=0.5.
• ψ : Y -> Z, z | y ~ N(c*y + d, σ_ψ²) with c=1.0, d=0.0, σ_ψ=0.7.
• Composite χ = φ;ψ : X -> Z.

Two paths, must agree.

1. Direct inverse: sample x0 ~ π, push through χ to get z, condition, draw x | z via analytic Gaussian posterior χ†_π(z).
2. Chain inverse: draw x | z by first sampling y | z via ψ†_{π;φ}(z) (posterior under pushed prior N(b, a² + σ_φ²)), then x | y via φ†_π(y).

monad-bayes sketch (tweag/monad-bayes, Control.Monad.Bayes.Sampler):

import Control.Monad.Bayes.Class
import Control.Monad.Bayes.Sampler.Strict
import Control.Monad.Bayes.Weighted

phi, psi :: MonadDistribution m => Double -> m Double
phi x = normal x 0.5
psi y = normal y 0.7

-- Path 1: direct posterior of X | Z=z_obs under chi = phi;psi
direct z = do
  x <- normal 0 1
  y <- phi x
  score (normalPdf y 0.7 z)
  pure x

-- Path 2: chained posterior, Y sampled from pushed prior then inverted
chained z = do
  y <- normal 0 (sqrt (1 + 0.25))  -- pushed prior N(0, a^2 + sigma_phi^2)
  score (normalPdf y 0.7 z)
  x <- normal 0 1
  score (normalPdf x 0.5 y)
  pure x

Run both via SMC or importance sampling, compare posterior means and variances for several z_obs. Chain rule ⇒ distributions must match within Monte Carlo error.

Extension hook. Replace ψ with an action-conditioned kernel ψ_a where a = policy(σ) depends on the posterior state. Chain rule breaks — you have entered active inference. This is the cleanest baby demo of the passive/active phase boundary.

8. Prototype Quartet (all TODOs unlocked 2026-04-12)

One invariant (φ;ψ)†_π = ψ†_{π;φ} ; φ†_π implemented across four languages, matching parameters (a=1.3, b=0.7, σ_φ=0.5, σ_ψ=0.4, π=N(0,1)), GF(3) trits φ(−1) ⊗ ψ(+1) → composite(0).

9. Legacy TODO (archived)

• TODO: implement the monad-bayes prototype above under ~/i/monad-bayes-asi-interleave/prototypes/chain-rule/.
• TODO: expose backward_fiber in zig-syrup/src/propagator.zig as a first-class continuation matching section 3's type.
• TODO: wire nashator coplay to emit the pushed prior π;φ as a shared cell both validator and generator can read.
• TODO: write the active-inference divergence demo (drift p below threshold, watch fixed-point bifurcate).