# Deep Inference Interleave > Compositional passive inference vs emergent active inference (Hedges Feb 2024) — chain rule, continuations, Siegel-stack cortex mapping, GF(3) triad across monad-bayes / nashator / zig-syrup. - Skill: `plurigrid/deep-inference-interleave` (Agent Skill, multi-file: 2 files) - Install (CLI): `npx skillmds@latest add plurigrid/deep-inference-interleave` - Raw SKILL.md: https://api.skillmd.com/api/skills/plurigrid/deep-inference-interleave/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: AI & ML - Author: plurigrid (https://skillmd.com/u/plurigrid) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/plurigrid/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: | Siegel L | Cortex role | Stack analogue | |---------:|-------------------------|---------------------------------------| | L0 | sensorimotor / V1 / M1 | **fab / PCB** (photons, volts) | | L1 | early sensory binding | firmware / microcode | | L2 | modality-specific cortex| OS / drivers | | L3 | association cortex | application runtime | | L4 | prefrontal / planning | L2 rollup / sequencer | | L5 | narrative / social self | **mainnet** (consensus, ledgered self)| 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`): ```haskell 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). | Port | Path | Role | |------|------|------| | Python/NumPy analytic + MC | `~/i/deep-inference-prototype/chain_rule_verify.py` | chain rule numeric check | | Python/NumPy active demo | `~/i/deep-inference-prototype/active_vs_passive.py` | Hedges/Smithe divergence conjecture | | Haskell/monad-bayes SMC | `~/i/deep-inference-prototype/ChainRule.hs` | SMC 2048 particles, 5 z-grid | | TypeScript/vitest | `~/i/nashator/src/pushed_prior.ts` + `.test.ts` | pushed-prior cell in propagator network | | Zig/ziglang test | `~/i/zig-syrup/src/backward_fiber.zig` | `Fiber` + `ChainFiber`, matches Radul-Sussman backward flow | ## 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).