Entropy, Uncertainty, and Why Models Hallucinate

When training AI models, understanding entropy and uncertainty helps you build more reliable systems. These concepts explain why models sometimes confidently state complete nonsense.

For a deeper technical dive into this topic, read our research post: Hallucinations in LLMs: The Entropy Problem and Current Solutions

What is Entropy?

Entropy measures disorder or unpredictability in a system. In AI models, it tells us how “spread out” the probability distribution is over possible next words or tokens.

Simple Example

Imagine the model is completing: “The capital of France is…” Low Entropy (Certain):

Paris: 95% probability
Lyon: 2% probability
Marseille: 1% probability
Other: 2% probability

High Entropy (Uncertain):

Paris: 25% probability
London: 20% probability
Berlin: 20% probability
Rome: 20% probability
Other: 15% probability

Low entropy means the model is confident. High entropy means it’s considering many options as equally likely.

Entropy vs Uncertainty

These terms are related but not identical:

Concept	What It Measures	Example
Entropy	Mathematical disorder in probability distribution	How spread out token probabilities are
Uncertainty	Any lack of knowledge or predictability	Model doesn’t know it doesn’t know
Epistemic Uncertainty	What the model doesn’t know	Facts outside training data
Aleatoric Uncertainty	Inherent randomness	Multiple valid answers exist

The Entropy Paradox

You might think high entropy equals more hallucinations. Not always true.

Counter-intuitive FindingModels often hallucinate with extremely LOW entropy - they’re confidently wrong.

Why This Happens

High Entropy, Correct

Scenario: Multiple valid options“The weather might be…”

sunny (30%)
cloudy (25%)
rainy (25%)
snowy (20%)

Model correctly represents uncertainty

Low Entropy, Wrong

Scenario: Confident hallucination“The Treaty of Versailles was signed in…”

1923 (98%)
1919 (1%)
Other (1%)

Model is wrong but very confident

Types of Uncertainty in Models

1. Token-Level Uncertainty

The model is unsure about the exact next word:

# High token uncertainty
"The best programming language is..."
# Could be: Python, JavaScript, Rust, etc.

# Low token uncertainty
"Water freezes at..."
# Almost certainly: 0, zero, 32°F

2. Semantic Uncertainty

The model is unsure about meaning, not just words:

# Same meaning, different tokens
"The capital of France" = "Paris" = "City of Light"
# Traditional entropy sees 3 options
# Semantic entropy sees 1 meaning

3. Factual Uncertainty

The model doesn’t know if information is true:

# Model might generate:
"The population of Tokyo is 37.4 million"
# But has no way to verify this number
# Could be confidently wrong

Measuring Uncertainty

Traditional Entropy

Shannon entropy formula:

H(X) = -Σ p(x) log p(x)

Where p(x) is the probability of each token. Problems:

Treats all tokens as independent
Can’t distinguish between word choice and meaning
Doesn’t capture epistemic uncertainty

Semantic Entropy

A better approach that measures uncertainty over meanings:

Generate multiple outputs
Group by semantic equivalence
Calculate entropy over meaning clusters

# Traditional entropy sees 4 different outputs:
outputs = [
    "It's raining",
    "It is raining",
    "Rain is falling",
    "It's pouring"
]
# High entropy

# Semantic entropy groups them:
meanings = {
    "precipitation": ["It's raining", "It is raining", "Rain is falling", "It's pouring"]
}
# Low entropy - one meaning

Confidence Calibration

Measuring if confidence matches correctness:

Model Says	Confidence	Actually Correct	Calibration
”Paris is capital of France”	99%	Yes	Good
”Treaty signed in 1923”	98%	No	Bad
”Might be Python or Java”	50%	Uncertain	Good
”Definitely JavaScript”	95%	No	Bad

Entropy in Training

During Training

Entropy changes as models learn:

Early Training

High entropy everywhere

Model knows nothing
Random predictions
Uniform distributions

Mid Training

Mixed entropy

Learning patterns
Some confident predictions
Some uncertainty remains

Late Training

Risk of overconfidence

Low entropy on training data
May hallucinate confidently
Needs regularization

Loss Functions and Entropy

Cross-entropy loss directly relates to entropy:

# Cross-entropy loss
loss = -Σ y_true * log(y_pred)

# Encourages:
# - Low entropy when answer is clear
# - High entropy when unsure

But this doesn’t prevent confident hallucinations!

Controlling Entropy

Temperature Scaling

Temperature controls randomness in generation:

# Low temperature (0.1): More deterministic
logits = logits / 0.1
# Sharp distribution, low entropy

# High temperature (2.0): More random
logits = logits / 2.0
# Flat distribution, high entropy

Effects:

T < 0.5: Conservative, repetitive
T = 1.0: Default balance
T > 1.5: Creative, risky

Top-k and Top-p Sampling

Limit which tokens to consider:

# Top-k: Only consider k most likely tokens
top_k = 50  # Reduces effective entropy

# Top-p: Consider tokens until cumulative prob = p
top_p = 0.9  # Dynamic entropy reduction

Entropy Regularization

Add entropy constraints during training:

# Penalize too-confident predictions
loss = cross_entropy + lambda * entropy_penalty

# Where entropy_penalty encourages uncertainty
# when appropriate

Advanced Techniques

Entropy-Aware Training

Our approach at Monostate modifies attention mechanisms:

Attention_controlled = softmax(QK^T/√dk - λH(p))V

Where λ controls entropy suppression strength.

Layer-wise Entropy Budgets

Different layers get different entropy allowances:

Layer Type	Entropy Budget	Purpose
Early	High	Explore possibilities
Middle	Medium	Reason about options
Final	Low	Commit to answer

Semantic Entropy Probes

Extract semantic uncertainty from internal states:

5-10x faster than generating multiple outputs
Can detect hallucinations before they happen
Works with any transformer model

Practical Implications

For Training

DO

Monitor entropy during training
Use temperature scaling
Implement entropy regularization
Track confidence calibration

DON'T

Assume low entropy = correct
Ignore semantic uncertainty
Over-regularize entropy
Trust confidence scores blindly

For Inference

High-Stakes Applications:

Use lower temperature (0.3-0.7)
Implement semantic entropy checks
Require multiple consistent outputs
Add confidence thresholds

Creative Applications:

Use higher temperature (1.0-1.5)
Allow more entropy
Embrace variation
Filter bad outputs later

Connection to Hallucinations

Entropy relates to hallucinations in complex ways:

Not Just High Entropy: Hallucinations occur at all entropy levels
Confidence ≠ Correctness: Low entropy can mean confident nonsense
Semantic vs Syntactic: Word uncertainty doesn’t equal meaning uncertainty
Knowledge Gaps: No entropy signal for “I don’t know this”

Current Research

Semantic Entropy (Oxford/MIT)

Measures uncertainty over meanings rather than tokens. Shows 2-3x better hallucination detection than traditional methods.

Conformal Prediction

Provides statistical guarantees:

“With 95% confidence, the answer is in this set”
If set is too large or empty, admits uncertainty
Mathematically rigorous approach

Multi-Agent Consensus

Our fuzzy logic approach:

Multiple agents evaluate options
Consensus through fuzzy scoring
Disagreement triggers re-evaluation
Natural uncertainty handling

Best Practices

During Training

Track Multiple Metrics:
- Token entropy
- Semantic entropy
- Confidence calibration
- Factual accuracy
Use Appropriate Regularization:
- Dropout for uncertainty
- Label smoothing
- Entropy penalties
- Early stopping
Validate Uncertainty:
- Test on out-of-distribution data
- Check confidence on known errors
- Measure calibration curves

During Inference

Set Appropriate Temperature:
- Match to task requirements
- Lower for facts
- Higher for creativity
Implement Safeguards:
- Semantic entropy checks
- Multiple generation consensus
- Confidence thresholds
- Human-in-the-loop for critical decisions
Design for Uncertainty:
- Allow “I don’t know” responses
- Show confidence levels
- Provide alternatives
- Flag potential hallucinations

The Bottom Line

Entropy and uncertainty are fundamental to how models work, but they’re not simple predictors of reliability. Understanding these concepts helps you:

Build more reliable models
Detect potential hallucinations
Set appropriate generation parameters
Design better training procedures

Remember: A confident model isn’t necessarily a correct model. The key is building systems that know what they don’t know.

Watch: The RL Overfitting Problem

Models trained with reinforcement learning can become overconfident and lose the ability to say “I don’t know.” This video explains why RL-trained models sometimes become worse at handling uncertainty.

Hallucinations & Limitations

Practical guide to model limitations

Our Research

Deep dive into entropy and hallucinations

Getting Started

AI Training Fundamentals

Core Concepts

Interface Selection

​Entropy, Uncertainty, and Why Models Hallucinate

​What is Entropy?

​Simple Example

​Entropy vs Uncertainty

​The Entropy Paradox

​Why This Happens

High Entropy, Correct

Low Entropy, Wrong

​Types of Uncertainty in Models

​1. Token-Level Uncertainty

​2. Semantic Uncertainty

​3. Factual Uncertainty

​Measuring Uncertainty

​Traditional Entropy

​Semantic Entropy

​Confidence Calibration

​Entropy in Training

​During Training

​Loss Functions and Entropy

​Controlling Entropy

​Temperature Scaling

​Top-k and Top-p Sampling

​Entropy Regularization

​Advanced Techniques

​Entropy-Aware Training

​Layer-wise Entropy Budgets

​Semantic Entropy Probes

​Practical Implications

​For Training

DO

DON'T

​For Inference

​Connection to Hallucinations

​Current Research

​Semantic Entropy (Oxford/MIT)

​Conformal Prediction

​Multi-Agent Consensus

​Best Practices

​During Training

​During Inference

​The Bottom Line

​Watch: The RL Overfitting Problem

​Further Reading

Hallucinations & Limitations

Our Research

Entropy, Uncertainty, and Why Models Hallucinate

What is Entropy?

Simple Example

Entropy vs Uncertainty

The Entropy Paradox

Why This Happens

Types of Uncertainty in Models

1. Token-Level Uncertainty

2. Semantic Uncertainty

3. Factual Uncertainty

Measuring Uncertainty

Traditional Entropy

Semantic Entropy

Confidence Calibration

Entropy in Training

During Training

Loss Functions and Entropy

Controlling Entropy

Temperature Scaling

Top-k and Top-p Sampling

Entropy Regularization

Advanced Techniques

Entropy-Aware Training

Layer-wise Entropy Budgets

Semantic Entropy Probes

Practical Implications

For Training

For Inference

Connection to Hallucinations

Current Research

Semantic Entropy (Oxford/MIT)

Conformal Prediction

Multi-Agent Consensus

Best Practices

During Training

During Inference

The Bottom Line

Watch: The RL Overfitting Problem

Further Reading