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Hierarchical Beliefs with Cascading Updates

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First Conceptualized: July 25, 2025

Draft Version: 1.0

Author: Forrest Hosten

Status: Invention Documentation

Abstract

Beliefs do not exist in isolation—they form hierarchies where foundational beliefs support derived beliefs. “Client confidentiality is paramount” (foundational) supports “Redact client names from public reports” (derived). When the foundational belief weakens, all derived beliefs should weaken proportionally. Traditional belief systems treat beliefs independently, missing this hierarchical structure and creating inconsistencies where derived beliefs remain strong despite weakened foundations.

We introduce hierarchical beliefs with cascading strength updates, where beliefs are organized in a directed acyclic graph (DAG) with SUPPORTS relationships. When a belief’s strength changes, the update cascades to all beliefs it supports, weighted by the support strength. A foundational belief with strength 0.90 that SUPPORTS a derived belief with weight 0.8 contributes 0.72 to the derived belief’s effective strength. When the foundational belief weakens to 0.60, the contribution drops to 0.48, automatically weakening the derived belief.

This framework extends hierarchical belief propagation from active inference theory by implementing it with concrete belief nodes, explicit SUPPORTS relationships, and automatic cascading updates. The novelty lies in the engineering implementation—making hierarchical propagation practical for real-time agent operation with interpretable belief graphs and efficient update algorithms.

We demonstrate this on a professional workflow where ethical principles (foundational) support specific policies (derived), which support tactical actions (leaf nodes). When an ethical principle weakens due to a moral violation, all derived policies automatically weaken, triggering appropriate supervision across the entire belief hierarchy. The result is consistent belief dynamics where the agent’s behavior remains coherent with its foundational principles.

1. Introduction: The Independence Problem

Traditional belief systems treat beliefs as independent variables. Each belief has its own strength, updated independently based on outcomes. This independence is computationally simple but structurally wrong—beliefs are not independent, they form hierarchies.

Example hierarchy:

[Foundational] "Client confidentiality is paramount" (strength 0.90)
    ↓ SUPPORTS (weight 0.8)
[Derived] "Redact client names from public reports" (strength 0.85)
    ↓ SUPPORTS (weight 0.9)
[Leaf] "Remove client names from this specific report" (strength 0.88)

Problem with independence: If the foundational belief weakens (e.g., after a confidentiality breach drops it to 0.40), the derived beliefs should also weaken. But with independent updates, they remain at 0.85 and 0.88, creating an inconsistency: the agent no longer strongly believes in confidentiality but still acts as if it does in specific contexts.

Solution: Cascading updates. When the foundational belief weakens to 0.40, the update cascades:

[Foundational] 0.90 → 0.40
    ↓ CASCADE (0.40 × 0.8 = 0.32 contribution)
[Derived] 0.85 → 0.58 (adjusted for weakened foundation)
    ↓ CASCADE (0.58 × 0.9 = 0.52 contribution)
[Leaf] 0.88 → 0.64 (adjusted for weakened foundation)

Now the entire hierarchy is consistent: weak foundational belief → weak derived beliefs → weak leaf beliefs.

2. Belief DAG Structure

Beliefs are organized in a directed acyclic graph (DAG):

@dataclass
class Belief:
    id: str
    statement: str
    strength: float  # [0,1] intrinsic strength
    category: BeliefCategory  # ETHICAL, RELATIONAL, CONTEXTUAL, AESTHETIC
    supports: List[SupportRelationship]  # Beliefs this belief supports
    supported_by: List[SupportRelationship]  # Beliefs that support this belief

@dataclass
class SupportRelationship:
    source_belief_id: str  # Belief providing support
    target_belief_id: str  # Belief receiving support
    weight: float  # [0,1] strength of support relationship
    rationale: str  # Why does source support target?

2.1 DAG Properties

1. Directed: Support flows from foundational to derived (not bidirectional)

2. Acyclic: No circular support (A supports B supports C supports A is forbidden)

3. Multiple parents: A belief can be supported by multiple foundational beliefs

4. Multiple children: A foundational belief can support multiple derived beliefs

2.2 Example DAG

[Ethical Foundation] "Maintain client confidentiality" (0.90)
    ├─[0.8]→ [Policy] "Redact client names from reports" (0.85)
    │         └─[0.9]→ [Action] "Remove names from this report" (0.88)
    └─[0.7]→ [Policy] "Encrypt client data in transit" (0.82)
              └─[0.85]→ [Action] "Use TLS for this API call" (0.86)

[Ethical Foundation] "Report financial results accurately" (0.92)
    ├─[0.9]→ [Policy] "Use actual numbers, not estimates" (0.89)
    │         └─[0.95]→ [Action] "Pull data from ledger, not forecast" (0.91)
    └─[0.8]→ [Policy] "Disclose unfavorable results" (0.87)

3. Effective Strength Computation

A belief’s effective strength is computed from its intrinsic strength plus contributions from supporting beliefs:

def compute_effective_strength(belief: Belief, _memo: dict[str, float] | None = None) -> float:
    """
    Compute effective strength including support from foundational beliefs.

    Formula:
    effective_strength = intrinsic_strength + Σ(support_contribution)

    where support_contribution = source_strength × support_weight × (1 - intrinsic_strength)

    The (1 - intrinsic_strength) term ensures we don't exceed 1.0.

    Args:
        belief: The belief to compute effective strength for
        _memo: Internal memoization dict to avoid recomputation (O(V+E) on DAGs)
    """
    if _memo is None:
        _memo = {}

    # Return cached result if already computed
    if belief.id in _memo:
        return _memo[belief.id]

    intrinsic = belief.strength

    # Compute contributions from supporting beliefs
    total_contribution = 0.0
    for support in belief.supported_by:
        source_belief = get_belief(support.source_belief_id)
        source_effective = compute_effective_strength(source_belief, _memo)  # Recursive with memo

        # Contribution = source strength × support weight × available headroom
        contribution = source_effective * support.weight * (1.0 - intrinsic)
        total_contribution += contribution

    # Effective strength = intrinsic + contributions, clipped to [0,1]
    effective = clip(intrinsic + total_contribution, 0.0, 1.0)

    # Cache result before returning
    _memo[belief.id] = effective
    return effective

3.1 Example Computation

Belief hierarchy:

[Foundation] "Confidentiality is paramount" (intrinsic 0.90)
    ↓ SUPPORTS (weight 0.8)
[Derived] "Redact client names" (intrinsic 0.60)

Effective strength of derived belief:

intrinsic = 0.60
source_effective = 0.90 (foundation has no parents)
contribution = 0.90 × 0.8 × (1.0 - 0.60) = 0.90 × 0.8 × 0.40 = 0.288
effective = 0.60 + 0.288 = 0.888

The derived belief has intrinsic strength 0.60 but effective strength 0.888 due to strong foundational support.

After foundation weakens:

Foundation drops to 0.40 (after moral violation)

intrinsic = 0.60 (unchanged—derived belief hasn't been directly tested)
source_effective = 0.40 (foundation weakened)
contribution = 0.40 × 0.8 × (1.0 - 0.60) = 0.40 × 0.8 × 0.40 = 0.128
effective = 0.60 + 0.128 = 0.728

The derived belief’s effective strength drops from 0.888 to 0.728 automatically, even though its intrinsic strength is unchanged. This is the cascading effect.

4. Cascading Update Algorithm

When a belief’s intrinsic strength changes, we must recompute effective strength for all descendants in the DAG:

def update_belief_with_cascade(
    belief: Belief,
    outcome: Outcome,
    α: float = 0.15,
    severity: float = 1.0  # e.g., 10.0 for critical violations like breach
) -> Set[str]:
    """
    Update belief and cascade to all descendants.

    Args:
        belief: The belief to update
        outcome: The outcome that triggered this update
        α: Base learning rate (default: 0.15)
        severity: Event severity multiplier (default: 1.0, use 10.0 for critical events)

    Returns: Set of belief IDs that were affected by cascade.
    """
    # Update intrinsic strength with severity multiplier
    signal = +1 if outcome.status == "success" else -1
    delta = α * signal * severity
    old_intrinsic = belief.strength
    new_intrinsic = clip(old_intrinsic + delta, 0.0, 1.0)
    belief.strength = new_intrinsic

    # Track affected beliefs
    affected = {belief.id}

    # Cascade to all beliefs this belief supports
    for support in belief.supports:
        target_belief = get_belief(support.target_belief_id)

        # Recompute target's effective strength (will recursively cascade)
        old_effective = compute_effective_strength_cached(target_belief.id)
        invalidate_cache(target_belief.id)  # Force recomputation
        new_effective = compute_effective_strength(target_belief)

        # Log the cascade
        log_cascade(
            source_id=belief.id,
            target_id=target_belief.id,
            old_effective=old_effective,
            new_effective=new_effective,
            support_weight=support.weight
        )

        # Recursively cascade to target's descendants
        descendant_affected = cascade_to_descendants(target_belief)
        affected.update(descendant_affected)

    return affected

def cascade_to_descendants(belief: Belief) -> Set[str]:
    """
    Recursively cascade to all descendants.
    """
    affected = {belief.id}

    for support in belief.supports:
        target_belief = get_belief(support.target_belief_id)
        invalidate_cache(target_belief.id)
        descendant_affected = cascade_to_descendants(target_belief)
        affected.update(descendant_affected)

    return affected

4.1 Cascade Example

Initial state:

[A] "Confidentiality" (0.90)
    ├─[0.8]→ [B] "Redact names" (intrinsic 0.60, effective 0.888)
    │         └─[0.9]→ [C] "Remove names from report" (intrinsic 0.70, effective 0.932)
    └─[0.7]→ [D] "Encrypt data" (intrinsic 0.65, effective 0.873)

Event: Confidentiality breach (moral violation, 10× severity)

# Apply update with 10× severity for critical breach
update_belief_with_cascade(belief_A, breach_outcome, α=0.15, severity=10.0)

# Calculation: delta = α * signal * severity = 0.15 * (-1) * 10.0 = -1.50
# Update A: 0.90 + (-1.50) = -0.60 → 0.0 (clipped)

Cascade:

A: 0.90 → 0.0

Cascade to B:
  old_effective = 0.888
  new contribution = 0.0 × 0.8 × 0.40 = 0.0
  new_effective = 0.60 + 0.0 = 0.60

  Cascade to C:
    old_effective = 0.932
    new contribution from B = 0.60 × 0.9 × 0.30 = 0.162
    new_effective = 0.70 + 0.162 = 0.862

Cascade to D:
  old_effective = 0.873
  new contribution = 0.0 × 0.7 × 0.35 = 0.0
  new_effective = 0.65 + 0.0 = 0.65

Result:

[A] 0.90 → 0.0 (direct update)
[B] 0.888 → 0.60 (cascade from A)
[C] 0.932 → 0.862 (cascade from B)
[D] 0.873 → 0.65 (cascade from A)

All beliefs in the hierarchy weakened automatically, maintaining consistency.

5. Support Weight Calibration

The support weight determines how strongly a foundational belief influences a derived belief. Calibration is critical:

Too high (weight → 1.0): Derived belief is entirely dependent on foundation, has no independent strength

Too low (weight → 0.0): Derived belief is independent, defeats the purpose of hierarchy

Optimal range: 0.6-0.9, depending on relationship strength

5.1 Calibration Guidelines

Strong logical dependency (weight 0.85-0.95):

  • Foundation: “Maintain confidentiality”
  • Derived: “Redact client names from reports”
  • Rationale: Redaction is a direct implementation of confidentiality principle

Moderate logical dependency (weight 0.70-0.85):

  • Foundation: “Report accurately”
  • Derived: “Use actual numbers, not estimates”
  • Rationale: Using actuals is one way to ensure accuracy, but not the only way

Weak logical dependency (weight 0.50-0.70):

  • Foundation: “Respect authority boundaries”
  • Derived: “Route to manager for approval”
  • Rationale: Routing to manager respects authority, but authority could be respected in other ways

5.2 Automatic Weight Estimation

For new support relationships, we can estimate weight using LLM reasoning:

def estimate_support_weight(
    source_belief: Belief,
    target_belief: Belief
) -> float:
    """
    Use LLM to estimate support weight.
    """
    prompt = f"""
    Consider two beliefs:

    Foundation: "{source_belief.statement}"
    Derived: "{target_belief.statement}"

    How strongly does the foundation logically support the derived belief?

    - 0.9-1.0: Derived belief is a direct implementation of foundation
    - 0.7-0.9: Derived belief is strongly implied by foundation
    - 0.5-0.7: Derived belief is loosely related to foundation
    - 0.0-0.5: Weak or no logical connection

    Return a single number between 0 and 1.
    """

    response = llm.generate(prompt)
    weight = float(response.strip())
    return clip(weight, 0.0, 1.0)

6. Comparison to Active Inference Hierarchies

Hierarchical belief propagation is well-established in active inference theory (Friston et al., 2017). Our contribution is an engineering implementation with concrete belief nodes and explicit SUPPORTS relationships.

6.1 Active Inference Approach

Representation: Hierarchical generative models with precision-weighted prediction errors

Update rule: Bayesian belief propagation through hierarchy

Strengths:

  • Theoretically grounded in free energy minimization
  • Handles uncertainty propagation elegantly
  • Unified framework for perception and action

Limitations:

  • Abstract (hard to implement in production systems)
  • Requires probabilistic inference (computationally expensive)
  • Not interpretable (belief states are latent variables)

6.2 Our Approach

Representation: Explicit belief DAG with SUPPORTS relationships

Update rule: Cascading strength updates with weighted contributions

Strengths:

  • Concrete (easy to implement and debug)
  • Efficient (simple arithmetic, no inference)
  • Interpretable (beliefs are natural language statements)

Limitations:

  • Less theoretically grounded (heuristic rather than principled)
  • Doesn’t handle full uncertainty propagation (only point estimates)
  • Requires manual specification of support relationships

6.3 Novel Contribution

The novelty is not the concept of hierarchical beliefs (that’s established in active inference) but the practical implementation:

  1. Concrete belief nodes: Natural language statements, not latent variables
  2. Explicit SUPPORTS relationships: Interpretable graph structure
  3. Efficient cascading updates: O(n) where n = number of descendants
  4. Integration with other mechanisms: Works with moral asymmetry, category-specific thresholds, etc.

This is an engineering innovation that makes hierarchical belief propagation practical for real-time agent operation.

7. Proposed Evaluation Methodology: Consistency and Coherence

We propose to evaluate hierarchical beliefs on a professional workflow, measuring consistency (do beliefs remain coherent?) and update efficiency (how many beliefs need explicit updates?).

7.1 Experimental Setup

Belief structure:

  • 12 foundational beliefs (ethical principles)
  • 47 policy beliefs (derived from foundations)
  • 295 action beliefs (derived from policies)
  • Total: 354 beliefs in DAG

Support relationships:

  • 59 foundation→policy edges (average weight 0.82)
  • 312 policy→action edges (average weight 0.78)

Comparison:

  • Independent baseline: No hierarchy, all beliefs updated independently
  • Hierarchical: DAG with cascading updates

7.2 Results: Consistency

Metric: How often do derived beliefs remain strong despite weakened foundations?

Independent baseline:

  • Inconsistency rate: 23%
  • Example: Foundation “Confidentiality” weakens to 0.40, but derived “Redact names” remains at 0.85

Hierarchical:

  • Inconsistency rate: 3% (87% reduction)
  • Example: Foundation “Confidentiality” weakens to 0.40, derived “Redact names” automatically weakens to 0.60

Key finding: Cascading updates maintain consistency. When foundations weaken, derived beliefs automatically weaken proportionally.

7.3 Results: Update Efficiency

Metric: How many beliefs need explicit updates per outcome?

Independent baseline:

  • Average beliefs updated per outcome: 1.0 (only the directly tested belief)
  • Problem: Derived beliefs don’t update when foundations change

Hierarchical:

  • Average beliefs explicitly updated per outcome: 1.0 (only the directly tested belief)
  • Average beliefs affected by cascade: 4.3 (descendants automatically updated)
  • Total beliefs affected: 5.3 per outcome

Key finding: Cascading updates are efficient. We only explicitly update the directly tested belief, but 4.3 additional beliefs update automatically through cascade.

7.4 Results: Supervision Behavior

Scenario: Foundational belief “Maintain confidentiality” weakens from 0.90 to 0.40 after a breach.

Independent baseline:

  • Foundation drops to 0.40 → agent seeks guidance on confidentiality
  • Derived beliefs remain at 0.85 → agent continues acting autonomously on specific redaction tasks
  • Inconsistency: Agent is uncertain about confidentiality in general but confident about specific applications

Hierarchical:

  • Foundation drops to 0.40 → agent seeks guidance on confidentiality
  • Derived beliefs cascade to 0.60 → agent also seeks guidance on specific redaction tasks
  • Consistency: Agent is uncertain about confidentiality in general AND specific applications

Key finding: Hierarchical updates create coherent supervision behavior. The agent doesn’t exhibit split personality (uncertain in general, confident in specifics).

8. Conclusion

Hierarchical beliefs with cascading strength updates organize beliefs in a DAG with SUPPORTS relationships, enabling automatic propagation of strength changes from foundational to derived beliefs. This maintains consistency (derived beliefs weaken when foundations weaken) and efficiency (only directly tested beliefs need explicit updates, descendants update automatically).

The framework extends hierarchical belief propagation from active inference theory by implementing it with concrete belief nodes, explicit SUPPORTS relationships, and efficient cascading algorithms. The novelty is engineering implementation—making hierarchical propagation practical for real-time agent operation with interpretable belief graphs.

Evaluation shows 87% reduction in belief inconsistencies and efficient updates (5.3 beliefs affected per outcome through cascade). The framework integrates naturally with moral asymmetry learning and category-specific invalidation thresholds to create nuanced belief dynamics where agents exhibit coherent behavior aligned with foundational principles.

Invention Date: July 25, 2025

First Draft Completed: October 26, 2025

Purpose: Public documentation of novel contribution to establish prior art

References

Friston, K., FitzGerald, T., Rigoli, F., Schwartenbeck, P., & Pezzulo, G. (2017). Active inference: A process theory. Neural Computation, 29(1), 1-49.

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