Decision Confidence Estimator

2. Beta posterior concentration as confidence (Ch.4 §4.2.1, eq. 4.27–4.28): For a Beta(α, β) posterior:

• mean = α / (α + β) (eq. 4.27)
• Posterior is sharp (high confidence) when α + β is large
• Posterior is flat (low confidence) when α + β is small (few observations)

The relative standard error (Ch.14 §14.1, eq. 14.6) formalises this:

relative_SE = σ̂ / (μ̂ × √n)     (eq. 14.6)

High relative SE → low confidence; low relative SE → high confidence.

Composite confidence score:

C = 0.25 × c_sensor + 0.20 × c_regime + 0.30 × c_uncertainty + 0.25 × c_margin
C = max(0.0, min(1.0, C − penalty))

Step-by-Step Execution

Step 1: Retrieve Bayesian posterior uncertainty from knowledge base

search_knowledge_base(
    query="Bayesian posterior uncertainty sharpness concentration total count Dirichlet model confidence",
    k=3
)

What to extract from the result:

• Ch.16 §16.4, eq. 16.6 — belief over model parameters: b(θ) = ∏ Dir(θ^(s,a) | N(s,a)) Total count n = sum(alpha) controls posterior width; used to ground the principle that more evidence → narrower posterior → higher decision confidence
• Algorithm 16.8 — n = sum(model.D[s,a].alpha) → if n=0, return 0 (no confidence); as n grows the posterior concentrates: warehouse analogy — more sensor readings and historical moves → higher confidence in inventory estimate

Step 2: Retrieve Beta posterior mean and relative standard error

get_entity_by_number(number="4.27")

What to extract from the result:

• Ch.4 §4.2.1, eq. 4.27 — Beta(α,β) mean = α / (α+β)
• Posterior sharpness = α + β (total pseudocount); larger → more confident
• Increasing pseudocounts narrows the prior (reduces variance)

get_entity_by_number(number="14.6")

What to extract from the result:

• Ch.14 §14.1, eq. 14.6 — relative standard error: σ̂ / (μ̂ × √n)
• This is the operational inverse of confidence: high relative SE → act with caution; low relative SE → sufficient precision to execute
• Warehouse mapping: c_uncertainty = 1 − uncertainty_index is the complement of the relative uncertainty quantified by inventory-uncertainty-quantifier

Step 3: Get sensor reliability component from signal-reliability-estimator

Requires reliability_score per device and low_reliability_device_count from signal-reliability-estimator.

# c_sensor: mean reliability across all active devices ∈ [0,1]
c_sensor = mean(device.reliability_score for device in active_devices)

# penalty for LOW-reliability devices (each degrades overall confidence)
# Grounded in Ch.14 eq.14.6: low-count/noisy sensors inflate relative SE
low_rel_count = count(d for d in devices if d.reliability_class == LOW)
penalty_sensor = 0.15 × low_rel_count

If signal-reliability-estimator not run: c_sensor = 0.5 (neutral), penalty_sensor = 0.

Step 4: Get regime stability component from inventory-flow-regime-detector

Requires regime_label from inventory-flow-regime-detector.

# c_regime: stability of the inventory flow regime ∈ [0,1]
# Grounded in Ch.16 §16.4: model uncertainty is higher when the environment
# is in an uncharacterised state (REGIME_SHIFT = unknown transition model)
c_regime = {
    STABLE:               1.0,
    ELEVATED_ADJUSTMENTS: 0.6,
    REGIME_SHIFT:         0.2
}[regime_label]

If inventory-flow-regime-detector not run: c_regime = 0.8 (assume stable, add warning).

Step 5: Get inventory uncertainty component from inventory-uncertainty-quantifier

Requires uncertainty_index from inventory-uncertainty-quantifier.

# c_uncertainty: complement of uncertainty index ∈ [0,1]
# Grounded in eq. 14.6: 1 − relative_SE ≈ 1 − uncertainty_index
c_uncertainty = 1.0 - uncertainty_index

If inventory-uncertainty-quantifier not run: c_uncertainty = 0.5 (neutral), add warning.

Step 6: Get action margin component from expected-utility-action-ranker

Requires margin and decision_confidence from expected-utility-action-ranker.

# c_margin: normalised advantage margin ∈ [0,1]
# Grounded in Ch.7 §7.3 eq. 7.15: A(s,a) = Q(s,a) − U(s)
# Large margin → clear winner → high confidence in selection
c_margin = min(margin / 0.10, 1.0)   # saturates at margin ≥ 0.10

# If DS-01 returned TIE, override c_margin to 0
if decision_confidence_ds01 == TIE:
    c_margin = 0.0

If expected-utility-action-ranker not run: c_margin = 0.5 (neutral), add warning.

Step 7: Compute composite confidence score

Using the weighted sum grounded in Ch.16 §16.4 (all signals as posterior concentration contributors, weighted by their relative information value):

# Weighted composite — weights sum to 1.0
C_raw = (0.25 × c_sensor
       + 0.20 × c_regime
       + 0.30 × c_uncertainty
       + 0.25 × c_margin)

# Apply sensor penalty (Ch.14 eq.14.6: low-count signals inflate uncertainty)
total_penalty = penalty_sensor   # 0.15 per LOW-reliability device

C = max(0.0, min(1.0, C_raw - total_penalty))

Step 8: Classify confidence and recommend next step

if C >= 0.75:
    confidence_class      = CONFIDENT
    recommended_next_step = EXECUTE

elif C >= 0.50:
    confidence_class      = ADEQUATE
    recommended_next_step = EXECUTE_WITH_FLAG

elif C >= 0.25:
    confidence_class      = LOW_CONFIDENCE
    recommended_next_step = DEFER