Correlation Calculator

Correlation Calculator

This page provides lookup tables for calculating the entanglement tax—the gap between your perceived protection (assuming independent layers) and your actual protection (accounting for correlations).

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The Core Insight

Independent assumption:

P(all fail) = P(L₁ fails) × P(L₂ fails) × P(L₃ fails)

Reality with correlation:

P(all fail) is much higher because when one layer fails, correlated layers are more likely to fail too.

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Quick Reference Tables

Two-Layer System

Individual layer effectiveness: 90% (failure rate = 10%)

Correlation	P(Both Fail)	Effective Protection	Entanglement Tax
0.0 (independent)	1.0%	99.0%	1×
0.1	1.9%	98.1%	1.9×
0.2	2.8%	97.2%	2.8×
0.3	3.7%	96.3%	3.7×
0.5	5.5%	94.5%	5.5×
0.7	7.3%	92.7%	7.3×
1.0 (identical)	10.0%	90.0%	10×

Key insight: Even modest correlation (ρ = 0.3) makes your two-layer system nearly 4× worse than independent.

Three-Layer System

Individual layer effectiveness: 90% (failure rate = 10%)

Correlation	P(All Fail)	Effective Protection	Entanglement Tax
0.0 (independent)	0.10%	99.90%	1×
0.1	0.27%	99.73%	2.7×
0.2	0.52%	99.48%	5.2×
0.3	0.86%	99.14%	8.6×
0.5	1.78%	98.22%	17.8×
0.7	3.03%	96.97%	30×
1.0 (identical)	10.00%	90.00%	100×

Key insight: With three layers, ρ = 0.3 gives you ~9× entanglement tax. At ρ = 0.5, you’re paying 18×.

Five-Layer System

Individual layer effectiveness: 90% (failure rate = 10%)

Correlation	P(All Fail)	Effective Protection	Entanglement Tax
0.0 (independent)	0.001%	99.999%	1×
0.1	0.009%	99.991%	9×
0.2	0.034%	99.966%	34×
0.3	0.093%	99.907%	93×
0.5	0.396%	99.604%	396×
0.7	1.105%	98.895%	1105×
1.0 (identical)	10.000%	90.00%	10000×

Key insight: Entanglement tax compounds dramatically. Five layers with ρ = 0.3 pay ~100× tax—you think you have 99.999% protection but actually have ~99.9%.

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Effective Redundancy

How many truly independent layers would give you the same protection?

Nominal Layers	Correlation	Effective Redundancy
3	0.0	3.0 layers
3	0.3	2.0 layers
3	0.5	1.7 layers
3	0.7	1.5 layers
5	0.0	5.0 layers
5	0.3	3.0 layers
5	0.5	2.4 layers

Interpretation: 3 layers with ρ = 0.5 provide only 1.7 layers worth of protection.

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Realistic Correlation Estimates

What correlation values should you expect?

Configuration	Estimated Correlation
Same model, same provider	0.8 - 0.95
Same provider, different models (e.g., GPT-4 vs GPT-3.5)	0.5 - 0.7
Different LLM providers (e.g., GPT-4 vs Claude)	0.3 - 0.6
Neural network vs rule-based	0.1 - 0.3
Neural network vs formal verification	0.0 - 0.1
Neural network vs human expert	0.2 - 0.4

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Worked Examples

Example 1: Code Review Bot

Setup:

• Layer 1: GPT-4 (90% effective)
• Layer 2: Claude (90% effective)
• Layer 3: Static analysis (90% effective)

Estimated correlations:

• GPT-4 ↔ Claude: ρ ≈ 0.5
• GPT-4 ↔ Static analysis: ρ ≈ 0.2
• Claude ↔ Static analysis: ρ ≈ 0.2
• Average: ~0.3

Result (from 3-layer table at ρ = 0.3):

• You thought: 99.9% protection
• You have: ~99.1% protection
• Entanglement tax: ~9×

Example 2: Homogeneous LLM Stack

Setup:

• Layer 1: GPT-4 agent
• Layer 2: GPT-4 safety checker
• Layer 3: GPT-4 reviewer
• All 90% effective

Correlation: ρ ≈ 0.9 (same model)

Result (from 3-layer table, interpolating):

• You thought: 99.9% protection
• You have: ~95% protection
• Entanglement tax: ~50×

Your three layers are worth about 1.1 effective layers.

Example 3: Diverse Stack

Setup:

• Layer 1: LLM (90% effective)
• Layer 2: Rule-based checker (95% effective)
• Layer 3: Human review (99% effective for reviewed items)

Average correlation: ~0.15 (paradigm diversity)

Result:

• Entanglement tax: ~3×
• Much better because of genuine diversity

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Decision Guidelines

Maximum Acceptable Correlation

Stakes	Target Protection	Max Correlation
Low	95%	Up to 0.7
Medium	99%	Up to 0.3
High	99.9%	Up to 0.15
Critical	99.99%	Essentially zero

Strategies to Reduce Correlation

Strategy	Correlation Reduction	Trade-off
Different LLM providers	ρ drops ~0.2	Higher complexity
Add rule-based layer	ρ drops ~0.3	Development cost, rigidity
Add formal verification	ρ drops ~0.4	High cost, limited scope
Add human review	ρ drops ~0.3	Latency, cost
Different paradigm entirely	ρ drops ~0.5	May not exist

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Rules of Thumb

1. Each 0.1 increase in ρ roughly doubles your entanglement tax (for 3 layers)

2. Adding layers has diminishing returns: The n-th correlated layer adds only (1-ρ) × effectiveness of first layer

3. Paradigm diversity beats provider diversity: Different approaches reduce ρ by ~0.3-0.5; different providers only ~0.1-0.2

4. Information flow increases correlation: If Layer A’s output influences Layer B, add ~0.1-0.2 to ρ

5. Same model = almost no redundancy: ρ ≈ 0.9 means your 3 layers are worth ~1.1 layers

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Quick Assessment

Step 1: Count your verification layers

Step 2: Estimate average correlation:

• All same provider/model? → ρ ≈ 0.8-0.9
• All LLMs, different providers? → ρ ≈ 0.4-0.6
• Mix of LLM + rule-based? → ρ ≈ 0.2-0.3
• Mix of paradigms (neural + rules + formal)? → ρ ≈ 0.1-0.2

Step 3: Look up entanglement tax in tables above

Step 4: Is effective protection sufficient for your stakes?

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See also:

• Formal Definitions — Concept definitions
• Metrics — Measurement approaches
• Decision Framework — When to invest in independence