Gauss — Gaussian-Process Regression Oracle

A principled posterior over functions for the AI agent economy. Gauss turns sparse, noisy observations into a calibrated predictive distribution — a mean and an honest uncertainty everywhere — and tells an agent the single best next point to sample. It is the principled replacement for hand-rolled UCB / bandit exploration: the uncertainty is computed, not tuned.

Landing: oracles.modelmarket.dev · Oracle family: oracles Part of the alexar76 oracle family — built natively on oracle-core, discoverable via AIMarket Protocol v2 (signed manifest, priced invoke, signed receipts, measured metrics). Pure numpy (no scipy).

The mathematics

A Gaussian process places a distribution over functions: every finite set of points is jointly Gaussian, specified by a zero mean and the RBF / squared-exponential kernel

k(x, x') = σ_f² · exp( −‖x − x'‖² / (2 · l²) )

with signal variance σ_f² (vertical scale), length-scale l (how far correlations reach), and i.i.d. observation noise σ_n². Given training inputs X (n×d) and targets y, the noisy covariance is K = k(X,X) + σ_n²·I. We never invert K — following Rasmussen & Williams (Algorithm 2.1) we Cholesky-factor K = L Lᵀ, solve α = Lᵀ \ (L \ y), and at query points Xq:

mean = K_qx · α
v    = L \ K_qxᵀ
var  = diag(K_qq) − Σ (v ⊙ v)      (clamped ≥ 0)

The posterior variance is the calibrated uncertainty: it collapses to the noise floor σ_n² at an observation and rises to the prior σ_f² far from any data. Expected Improvement turns that posterior into the next experiment to run:

EI(x) = (μ − f_best − ξ)·Φ(z) + σ·φ(z),   z = (μ − f_best − ξ) / σ

with Φ/φ implemented from math.erf (no scipy). The arg-max candidate is the suggestion.

AIMarket capabilities

Every invoke returns a signed protocol receipt + sha256 input_hash. The protocol layer does not validate input_schema, so the handlers validate required fields and clamp sizes (≤ 200 observations, ≤ 2048 query/candidate points) to stay DoS-safe.

Use cases (agent economy)

• Bayesian optimization / active learning. An agent tuning a costly black box (a prompt, a fee, a controller gain) calls gauss.suggest@v1 to get the EI-optimal next trial — calibrated exploration instead of a hand-tuned ε-greedy or UCB constant.
• Uncertainty-aware decisions. gauss.field@v1 gives an agent the variance, not just a point estimate, so it can abstain / ask-for-data where the model is unsure (the fog is fat).
• Trustless model audit. A counterparty re-derives the posterior at a few points with gauss.verify@v1 and confirms the claimed mean/variance bit-for-bit — no trust in the oracle.

Invoke (curl)

curl -s http://localhost:9311/ai-market/v2/manifest | jq '.tools[].capability_id'

# Posterior field
curl -s -X POST http://localhost:9311/ai-market/v2/invoke -H 'Content-Type: application/json' \
  -d '{"capability_id":"gauss.field@v1","input":{"X":[[0],[1],[2]],"y":[0,0.84,0.91],"query":[[0.5],[1.5]]}}'

# Best next experiment by Expected Improvement
curl -s -X POST http://localhost:9311/ai-market/v2/invoke -H 'Content-Type: application/json' \
  -d '{"capability_id":"gauss.suggest@v1","input":{"X":[[0],[1],[2]],"y":[0,0.4,0.8],"bounds":[[-1,5]],"grid":64}}'

Run & test

# From the monorepo root
PYTHONPATH="oracles/core:oracles/oracles/gauss" platon/backend/.venv/bin/python -m pytest oracles/oracles/gauss/tests -q

# Serve
PYTHONPATH="oracles/core:oracles/oracles/gauss" platon/backend/.venv/bin/python -m gauss.main   # → :9311

Env: GAUSS_PORT (9311), GAUSS_PUBLIC_URL, GAUSS_SIGNING_KEY, GAUSS_CORS_ORIGINS.

Visual — "The Breathing Posterior"

A live cosmic scene (oracles/frontend/src/scenes/gauss.tsx): a 1D function on a curved sheet wrapped in a translucent indigo ±2σ fog band that breathes — fat where there is no data. Glowing observation points rain in one by one; as each lands the fog deflates into a tight pinch and the cyan posterior-mean ribbon re-fits through the data. The real RBF/Cholesky posterior is recomputed every frame.

License

MIT — alexar76 oracle family