> ## Documentation Index
> Fetch the complete documentation index at: https://docs.prepst.com/llms.txt
> Use this file to discover all available pages before exploring further.

# Bayesian Knowledge Tracing

> Advanced mastery tracking using Bayesian inference

## What is Bayesian Knowledge Tracing?

Bayesian Knowledge Tracing (BKT) is a probabilistic framework that models student knowledge as a hidden state. Every answer provides evidence about the student's true understanding, allowing us to update our beliefs about their mastery in real-time.

## The Four BKT Parameters

<ParamField name="P(L0)" default="0.25" required>
  Prior knowledge probability - our initial belief about student mastery before any observations
</ParamField>

<ParamField name="P(T)" default="0.10" required>
  Learning rate (transition probability) - probability that a student learns the skill after attempting a problem
</ParamField>

<ParamField name="P(G)" default="0.25" required>
  Guess probability - probability of a correct answer without mastery
</ParamField>

<ParamField name="P(S)" default="0.10" required>
  Slip probability - probability of an incorrect answer despite having mastery
</ParamField>

## How BKT Works

### Step 1: Prior Belief

When we first encounter a skill, we initialize mastery with P(L0):

```python theme={null}
initial_mastery = 0.25  # We assume 25% chance student knows the skill
```

### Step 2: Posterior Update

After each answer, we update our belief using Bayesian inference:

```python theme={null}
def bayesian_update(prior, is_correct, p_guess, p_slip):
    if is_correct:
        # Student got it right
        likelihood = (prior * (1 - p_slip)) + ((1 - prior) * p_guess)
        posterior = (prior * (1 - p_slip)) / likelihood
    else:
        # Student got it wrong
        likelihood = (prior * p_slip) + ((1 - prior) * (1 - p_guess))
        posterior = (prior * p_slip) / likelihood
    return posterior
```

### Step 3: Learning Transition

We account for potential learning during the attempt:

```python theme={null}
def apply_learning_transition(posterior, p_learn):
    # The student might have learned after seeing the problem
    return posterior + (p_learn * (1 - posterior))
```

## Implementation Example

```python theme={null}
from cognition_engine import CognitionEngine

engine = CognitionEngine(supabase_url, supabase_key)

# Track an answer
result = await engine.track_answer(
    user_id="student_123",
    skill_id="algebra_linear",
    is_correct=True,
    time_spent_seconds=45,
    confidence_score=4
)

print(f"Mastery before: {result['mastery_before']:.3f}")
print(f"Mastery after: {result['mastery_after']:.3f}")
print(f"Velocity: {result['velocity']:.4f}")
```

## Key Features

### Adaptive Parameters

BKT parameters can be customized per skill based on domain characteristics:

```python theme={null}
# Math problems might have different parameters than reading
math_params = {
    "prior_knowledge": 0.20,      # Lower initial assumption
    "learn_rate": 0.12,           # Faster learning
    "guess_probability": 0.15,    # Lower guessing
    "slip_probability": 0.08       # Lower mistakes
}
```

### Plateau Detection

The engine automatically detects learning plateaus:

```python theme={null}
if result['plateau_detected']:
    print("Learning plateau detected - intervention recommended")
    # Trigger personalized help or change difficulty
```

### Velocity Tracking

Learning velocity measures the rate of mastery improvement:

```python theme={null}
velocity = result['velocity']
if velocity > 0.1:
    print("Accelerating - keep pushing!")
elif velocity < -0.05:
    print("Declining - consider review")
```

## Cognitive Efficiency Integration

BKT mastery probability is enhanced with cognitive efficiency metrics:

<AccordionGroup>
  <Accordion title="Time-Based Calibration">
    Students who answer quickly with high confidence should show different mastery than those who struggle through guesses.
  </Accordion>

  <Accordion title="Confidence Scores">
    Self-reported confidence (1-5) provides additional signal about true understanding beyond correctness alone.
  </Accordion>

  <Accordion title="Hesitation Patterns">
    Answer changes and time spent provide insights into cognitive effort and uncertainty.
  </Accordion>
</AccordionGroup>

## BKT vs. Traditional Tracking

<ComparisonTable>
  <ComparisonRow feature="Knowledge Model" whatOursDoes="Binary (know/doesn't know)" whatOursShouldDo="Probabilistic (belief distribution)" />

  <ComparisonRow feature="Update Frequency" whatOursDoes="After completion" whatOursShouldDo="Every single answer" />

  <ComparisonRow feature="Confidence" whatOursDoes="No confidence measure" whatOursShouldDo="Probability-based confidence" />

  <ComparisonRow feature="Learning Detection" whatOursDoes="No model of learning process" whatOursShouldDo="Explicit learning transitions" />
</ComparisonTable>

## Best Practices

<Warning>
  Avoid overfitting parameters to small datasets. Use default parameters for most skills and only customize when you have substantial data.
</Warning>

<Tip>
  Combine BKT mastery probability with velocity analysis and cognitive efficiency for the most complete picture of student understanding.
</Tip>

<Info>
  BKT works best when students attempt at least 5-10 problems per skill. Below this threshold, mastery estimates are less reliable.
</Info>

## BKT in Our Multi-Model Architecture

BKT is the foundation of our analytics engine, but we combine it with advanced models:

<Info>
  **BKT provides the "what"** - interpretable mastery tracking for dashboards and analytics
</Info>

For deeper insights, we also use:

* **SAKT (Self-Attentive KT)**: Attention weights show cognitive patterns
* **DKT-Forget**: Models the forgetting curve for spaced repetition
* **DTransformer**: Dynamic models that account for time gaps and spacing effects

<Tip>
  See our [Technical Deep Dive](/guides/deep-dive) guide for details on how we combine BKT with these advanced models.
</Tip>

## Research Background

BKT is based on research from Carnegie Mellon University's Human-Computer Interaction Institute. Our implementation extends the classic model with:

* **Cognitive efficiency integration**: Time and confidence weighting
* **Plateau detection**: Automatic intervention triggers
* **Velocity analysis**: Momentum and acceleration tracking
* **Adaptive parameters**: Skill-specific tuning

## Next Steps

<CardGroup cols={2}>
  <Card title="Learning Velocity" icon="gauge" href="/guides/velocity">
    Understand how we analyze learning momentum
  </Card>

  <Card title="Cognitive Efficiency" icon="chart-mixed" href="/guides/efficiency">
    Learn about multi-dimensional efficiency metrics
  </Card>
</CardGroup>