Editing Openai/69596681-3f7c-8006-b9fd-01d9d8191763 (section)

=== Assistant: Proceeding. ===
Proceeding. Below is METHODS BLOCK 10 — BAYESIAN EVIDENCE, expanded to maximum depth using machine-learning–oriented formalism (marginal likelihood as model scoring, complexity control without training, hard thresholds, and irreversible decision logic). This block is the only place where Bayesian reasoning appears, and it is strictly evaluative.

METHODS BLOCK 10 — BAYESIAN EVIDENCE

OBJECTIVE
Compute a deterministic marginal likelihood (Bayesian evidence) for the frozen, identifiable model as an Occam-penalized score, and compare it against a fixed baseline to decide collapse. This block performs model evaluation, not inference. No posterior learning is permitted.

PRECONDITIONS
• DataValidity 𝒬(D)=VALID
• StructuralViolationFlag evaluated
• Feasibility outcome ∈ {OK,FORCED}
• Identifiable parameter vector θ_id ∈ ℝ^{k_min}
• Likelihood Kernel ℒ(r|θ_id) defined and normalized
• No collapse triggered earlier except for attribution-only execution

If any precondition fails, evidence computation is either skipped (collapse already forced) or marked INVALID.

PRIOR SPECIFICATION (FROZEN)
Define prior π(θ_id):

• Support exactly equals projected manifold M_IR,id
• π(θ_id)=0 outside support
• π normalized: ∫ π(θ_id)dθ_id = 1
• π fixed pre-run (A2)
• No data-dependent hyperparameters

Permitted forms:
• Uniform over bounded domain
• Fixed Gaussian with frozen covariance
• Any closed-form distribution declared in Block 3

Forbidden:
• Empirical Bayes
• Learned priors
• Adaptive shrinkage

ML analogy: fixed regularizer, not learned.

EVIDENCE DEFINITION
Define Bayesian evidence (marginal likelihood):

Z = ∫ ℒ(D|θ_id) π(θ_id) dθ_id

Integral taken over ℝ^{k_min} with respect to Lebesgue measure.

Properties:
• Z ∈ ℝ⁺
• Finite iff likelihood normalized and prior proper
• Invariant under reparameterization of θ_id

LOG EVIDENCE
Define log evidence:

lnZ = log(Z)

Numerical evaluation must preserve absolute normalization; dropping constants forbidden.

BASELINE EVIDENCE
Define baseline model B₀ with frozen likelihood ℒ₀ and prior π₀.
Baseline may represent:
• Null hypothesis
• Minimal reference model
• Previously certified IR fixed point

Baseline is fixed per MethodHash.

Compute lnZ₀ analogously.

EVIDENCE STATISTIC
Define evidence difference:

ΔlnZ = lnZ − lnZ₀

Interpretation:
• ΔlnZ < 0 ⇒ baseline preferred
• ΔlnZ > 0 ⇒ current model preferred

No subjective interpretation (e.g., “strong”, “decisive”) allowed.

THRESHOLD LOGIC
Define frozen collapse threshold Λ (scalar, fixed pre-run).

Decision rule:
If ΔlnZ < Λ ⇒ EvidenceFailure = 1
Else EvidenceFailure = 0

Λ may be zero or negative. No tuning allowed.

ML analogy: hard decision boundary in model-score space.

NO POSTERIOR USAGE
Explicitly forbidden:
• Posterior mean or MAP
• Credible intervals
• Uncertainty quantification
• Predictive posterior sampling

Posterior exists only implicitly inside the integral defining Z.

COMPLEXITY PENALIZATION
Occam penalty arises automatically via integration over θ_id:

• Larger parameter volume ⇒ lower Z unless supported by likelihood
• Redundant parameters already removed (Block 7)
• No explicit regularization term added

ML analogy: exact marginal likelihood replaces AIC/BIC/MDL without approximation.

NUMERICAL REQUIREMENTS
• Integration method supplied by Block 11 only
• Deterministic convergence
• Absolute error tolerance δ_Z fixed
• Relative error tolerance forbidden (can mask collapse)

If convergence fails ⇒ INVALID.

EDGE CASES
• If k_min = 0:
 Z = ℒ(D|θ_id=∅) (degenerate integral)
• If StructuralViolationFlag=1:
 Z may be computed for attribution but EvidenceFailure cannot rescue collapse
• If Feasibility outcome=FORCED:
 Integration occurs only over θ_id

FAILURE MODES
• Improper prior ⇒ INVALID
• Non-convergent integral ⇒ INVALID
• Numerical instability ⇒ INVALID
• Any attempt to condition Λ on data ⇒ INVALID

ARTIFACT EMISSION
Emit immutable artifacts:
• lnZ
• lnZ₀
• ΔlnZ
• Λ
• EvidenceFailure flag
• Integration diagnostics hash

PIPELINE ROLE
EvidenceFailure feeds directly into:
• Global Collapse (Block 12)
• Failure Attribution (Block 15)
• Certification logic (Block 16)

ML INTERPRETATION
This block implements Bayesian model selection without learning:

• Likelihood = fixed scoring rule
• Prior = fixed complexity budget
• Evidence = exact generalization score
• Threshold = hard classifier boundary

No training loop. No validation split. No hyperparameter search.

END METHODS BLOCK 10.

If you confirm, the next block will be METHODS BLOCK 11 — INTEGRATION CONTRACT (ARTIFACT #3), expanded into nested sampling mechanics, determinism constraints, convergence guarantees, and auditability—again at maximum depth.

© Robert R. Frost
2026-01-03