Machine LearningBagging for Other Models

Bagging for Other Models

LevelIntermediate

Duration90 mins

TopicBagging for Other Models

5 / 5

When Bagging Helps

Recognizing Bagging Opportunities

Bagging is not a universal solution—it excels in specific circumstances and provides minimal benefit in others. The ability to recognize when bagging will help (and when it won't) is a mark of ensemble expertise.

This page synthesizes everything we've learned into practical decision frameworks. You'll learn to diagnose whether your problem will benefit from bagging, anticipate the magnitude of improvement, and understand the conditions that make bagging most effective.

What You Will Learn

By the end of this page, you will understand: (1) The key conditions that make bagging effective, (2) How to diagnose high-variance situations, (3) When bagging provides minimal benefit, (4) Practical decision frameworks, and (5) Real-world case studies demonstrating bagging's impact.

Conditions for Bagging Success

Bagging works by averaging diverse predictions to reduce variance. For this to be effective, three conditions must be met:

Condition 1: High Base Model Variance

The base model must exhibit significant variance—its predictions should change noticeably with different training samples. Models with high variance include:

Decision trees (especially deep, unpruned)
Neural networks (with random initialization)
K-Nearest Neighbors with small K
Polynomial regression with high degree
Models with many features relative to samples

Models with low variance (linear regression, Naive Bayes, heavily regularized models) show little benefit from bagging—they produce similar predictions regardless of which training samples are used.

Condition 2: Low Correlation Between Models

Bagging's variance reduction is bounded by the correlation between model predictions:

$$\text{Var}_{ensemble} = \rho\sigma^2 + \frac{1-\rho}{B}\sigma^2$$

When $\rho$ is high (models agree), bagging helps little. The bootstrap sampling must create sufficient diversity.

Factors that increase model correlation (bad for bagging):

Dominant features that always drive predictions
Very large datasets (bootstrap samples overlap heavily)
Highly regularized models
Simple decision boundaries

Condition 3: Low Bias (or Acceptable Bias)

Bagging does not reduce bias. If your base model systematically underfits, averaging more underfitting models won't help. You need:

Either: A base model with low bias (complex enough to fit the pattern)
Or: Acceptable bias for your application

The Three Conditions for Bagging Success
Condition	Diagnostic	If Not Met
High variance	Predictions change with training sample	Use simpler aggregation or boost instead
Low correlation	Models disagree on substantial fraction of samples	Add feature randomization (→ Random Forest)
Low bias	Individual models fit training data well	Use more complex base model or switch to boosting

Quick Diagnostic

Train 5-10 models with different random seeds on bootstrapped samples. If their predictions largely agree (>90% agreement), bagging won't help much. If they disagree significantly (30-50% on some examples), bagging has high potential.

Diagnosing High-Variance Situations

High variance is the primary driver of bagging's benefit. Here's how to identify and quantify it:

Method 1: Repeated Cross-Validation Variance

Train your model with different random seeds or data splits. High variance in test performance indicates bagging opportunity.

Run 1: 85.2% accuracy
Run 2: 82.1% accuracy  
Run 3: 87.4% accuracy
Run 4: 83.0% accuracy
Run 5: 86.8% accuracy

Standard deviation: 2.2% → High variance, bagging likely helps

Compare to a low-variance model:

Run 1: 78.5% accuracy
Run 2: 78.7% accuracy
Run 3: 78.4% accuracy
Run 4: 78.6% accuracy
Run 5: 78.5% accuracy

Standard deviation: 0.1% → Low variance, bagging won't help

Method 2: Prediction Stability Analysis

For the same input, measure how predictions change across different model instances:

$$\text{Prediction Stability} = \frac{1}{N}\sum_{i=1}^{N} \frac{\text{std}(\hat{y}_i^{(1)}, ..., \hat{y}_i^{(B)})}{\text{mean}(\hat{y}_i)}$$

High stability coefficient = high variance = bagging opportunity.

Method 3: Learning Curve Shape

Plot validation error vs. training set size. High-variance models show:

Large gap between training and validation error
Validation error decreasing with more data
Training error near zero (overfitting)

Low-variance models show:

Small gap between training and validation
Both errors plateau at similar levels
Training error higher than zero (underfitting)

variance_diagnostics.py
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import numpy as np
from sklearn.model_selection import cross_val_score, learning_curve
from sklearn.tree import DecisionTreeClassifier
from sklearn.linear_model import LogisticRegression
import matplotlib.pyplot as plt
 
def variance_diagnostic(model_factory, X, y, n_iterations=20, cv=5):
    """
    Diagnose model variance through repeated cross-validation.
    
    Returns:
    --------
    dict with variance metrics and recommendation
    """
    scores = []
    
    for i in range(n_iterations):
        model = model_factory()
        cv_scores = cross_val_score(model, X, y, cv=cv)
        scores.extend(cv_scores)
    
    mean_score = np.mean(scores)
    std_score = np.std(scores)
    cv_coefficient = std_score / mean_score  # Coefficient of variation
    
    # Interpretation
    if std_score > 0.05:
        variance_level = "HIGH"
        recommendation = "Bagging will likely provide significant benefit"
    elif std_score > 0.02:
        variance_level = "MODERATE"
        recommendation = "Bagging may help, worth testing"
    else:
        variance_level = "LOW"
        recommendation = "Bagging unlikely to help much, consider boosting"
    
    return {
        'mean_score': mean_score,
        'std_score': std_score,
        'cv_coefficient': cv_coefficient,
        'variance_level': variance_level,
        'recommendation': recommendation,
        'all_scores': scores,
    }
 
 
def prediction_stability_analysis(models, X):
    """
    Analyze how predictions vary across model instances.
    
    Parameters:
    -----------
    models : list of fitted models
    X : test data
    
    Returns:
    --------
    stability metrics per sample
    """
    # Collect predictions
    if hasattr(models[0], 'predict_proba'):
        all_probs = np.array([m.predict_proba(X)[:, 1] for m in models])
        predictions = all_probs
    else:
        predictions = np.array([m.predict(X) for m in models])
    
    # Per-sample statistics
    sample_means = predictions.mean(axis=0)
    sample_stds = predictions.std(axis=0)
    
    # Identify high-variance samples
    high_var_mask = sample_stds > np.median(sample_stds) * 2
    
    return {
        'mean_predictions': sample_means,
        'std_predictions': sample_stds,
        'mean_std': sample_stds.mean(),
        'max_std': sample_stds.max(),
        'high_variance_samples': np.where(high_var_mask)[0],
        'high_variance_fraction': high_var_mask.mean(),
    }
 
 
def bagging_benefit_estimator(base_model_factory, X_train, y_train, 
                                X_test, y_test, max_estimators=50):
    """
    Estimate potential bagging benefit before full training.
    
    Trains progressively larger ensembles and measures improvement.
    """
    from sklearn.ensemble import BaggingClassifier
    
    estimator_counts = [1, 3, 5, 10, 20, 30, 50]
    estimator_counts = [e for e in estimator_counts if e <= max_estimators]
    
    scores = []
    
    for n_est in estimator_counts:
        if n_est == 1:
            # Single model baseline
            model = base_model_factory()
            model.fit(X_train, y_train)
            score = model.score(X_test, y_test)
        else:
            ensemble = BaggingClassifier(
                estimator=base_model_factory(),
                n_estimators=n_est,
                random_state=42,
                n_jobs=-1
            )
            ensemble.fit(X_train, y_train)
            score = ensemble.score(X_test, y_test)
        
        scores.append(score)
        print(f"n_estimators={n_est}: {score:.4f}")
    
    # Compute improvement
    single_score = scores[0]
    max_score = max(scores)
    improvement = max_score - single_score
    
    # Estimate saturation point
    improvements = [scores[i+1] - scores[i] for i in range(len(scores)-1)]
    if improvements and max(improvements) > 0:
        marginal_utility = improvements[-1] / improvements[0] if improvements[0] > 0 else 0
    else:
        marginal_utility = 0
    
    return {
        'estimator_counts': estimator_counts,
        'scores': scores,
        'single_model_score': single_score,
        'best_ensemble_score': max_score,
        'improvement': improvement,
        'marginal_utility': marginal_utility,
        'recommendation': (
            "Strong bagging benefit" if improvement > 0.03 else
            "Moderate bagging benefit" if improvement > 0.01 else
            "Limited bagging benefit"
        )
    }
 
 
def compare_variance_across_models(X, y):
    """
    Compare variance characteristics of different model types.
    """
    models = [
        ("Deep Tree (max_depth=None)", 
         lambda: DecisionTreeClassifier(max_depth=None)),
        ("Shallow Tree (max_depth=3)", 
         lambda: DecisionTreeClassifier(max_depth=3)),
        ("Logistic Regression", 
         lambda: LogisticRegression(max_iter=1000)),
    ]
    
    results = {}
    for name, factory in models:
        result = variance_diagnostic(factory, X, y, n_iterations=20)
        results[name] = result
        print(f"{name}:")
        print(f"  Mean: {result['mean_score']:.4f}")
        print(f"  Std:  {result['std_score']:.4f}")
        print(f"  Level: {result['variance_level']}")
        print(f"  {result['recommendation']}")
        print()
    
    return results

Rule of Thumb

If repeated cross-validation standard deviation exceeds 2-3% of the mean score, bagging is likely to help. If ensemble improvement plateaus before 20-30 trees, you've captured most of the benefit. If improvement continues growing with more trees, your base model has high variance and bagging is working.

When Bagging Doesn't Help

Recognizing when bagging won't help is as important as knowing when it will. Here are the key scenarios where bagging provides minimal benefit:

Bagging Provides Minimal Benefit When:

•Base model has low variance — Linear regression, Naive Bayes, heavily regularized models. Averaging similar predictions doesn't improve them.
•Base model has high bias — Bagging cannot fix underfitting. Shallow trees need boosting, not bagging.
•Dataset is very large — Bootstrap samples overlap almost entirely, producing nearly identical models. Diversity vanishes.
•Single feature dominates — All trees split on the same feature first, making them correlated. Consider feature randomization.
•Problem is inherently noisy — If irreducible noise dominates, variance reduction helps little. Focus on noise reduction.
•Already at optimal complexity — If your model is well-calibrated (good bias-variance balance), ensemble methods have limited upside.

Quantitative Thresholds:

Diagnostic	Bagging Unlikely to Help If
CV standard deviation	< 1% of mean
Improvement from 1→10 trees	< 0.5%
Model correlation $\rho$	> 0.8
Training-test gap	< 2% (underfit)
Feature importance concentration	Top feature > 70%

Case Study: Why Bagging a Linear Model Fails

Consider bagging logistic regression for a linearly separable problem:

Each bootstrap sample produces nearly identical coefficients
Decision boundaries differ by tiny amounts
Averaged predictions essentially equal any individual prediction
Computational cost multiplied by $B$ with no benefit

In this case, invest computation in better features or more data, not bagging.

The Diminishing Returns Trap

Watch for early saturation. If performance plateaus at 10-20 trees, adding more is wasted computation. But if you're still seeing improvement at 100 trees, your base model has very high variance—consider simplifying it slightly for efficiency.

Bagging Benefit by Problem Type

Different problem types present different opportunities for bagging. Here's what to expect based on your domain:

Expected Bagging Benefit by Problem Domain
Problem Type	Typical Benefit	Key Considerations
Tabular classification (small)	High (3-8%)	Trees have high variance; bagging works well
Tabular classification (large)	Moderate (1-3%)	Less variance due to data quantity; still helps
Tabular regression	High (5-15%)	Regression targets amplify variance; bagging very effective
Text classification	Moderate (2-5%)	Feature sparsity increases diversity
Image classification (CNN)	Moderate (1-3%)	Initialization variance enables ensemble benefit
Time series forecasting	Moderate (3-8%)	Care needed: temporal structure in bootstrap
Ranking problems	Moderate (2-4%)	Ensemble rankings more stable than individual
Anomaly detection	High (5-10%)	Boundary estimation highly variable

Domain-Specific Considerations:

Medical Diagnosis:

Bagging improves both accuracy AND calibration
Multiple models provide uncertainty estimates critical for clinical decisions
Regulatory requirements often favor interpretable ensembles
Strong benefit: 3-8% improvement typical

Financial Prediction:

High noise levels can limit bagging's benefit
But uncertainty quantification is valuable for risk management
Non-stationarity means different trees capture different regimes
Moderate benefit: 2-5% improvement typical

Computer Vision:

CNN ensembles show consistent improvements
Test-time augmentation is a form of implicit bagging
Main benefit is robustness to adversarial examples
Moderate benefit: 1-3% improvement, but reliability boost is significant

problem_type_bagging.py
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import numpy as np
from sklearn.datasets import (
    make_classification, make_regression, 
    fetch_20newsgroups_vectorized
)
from sklearn.ensemble import RandomForestClassifier, RandomForestRegressor
from sklearn.tree import DecisionTreeClassifier, DecisionTreeRegressor
from sklearn.model_selection import cross_val_score
 
def estimate_bagging_benefit_for_problem_type():
    """
    Demonstrate bagging benefit varies by problem type.
    """
    results = {}
    
    # Binary classification - balanced
    X, y = make_classification(
        n_samples=1000, n_features=20, n_informative=10,
        random_state=42
    )
    
    single_scores = cross_val_score(
        DecisionTreeClassifier(random_state=42), X, y, cv=5
    )
    ensemble_scores = cross_val_score(
        RandomForestClassifier(n_estimators=50, random_state=42), X, y, cv=5
    )
    
    results['binary_classification'] = {
        'single': single_scores.mean(),
        'ensemble': ensemble_scores.mean(),
        'improvement': ensemble_scores.mean() - single_scores.mean()
    }
    
    # Regression
    X, y = make_regression(
        n_samples=1000, n_features=20, n_informative=10,
        noise=10, random_state=42
    )
    
    single_scores = cross_val_score(
        DecisionTreeRegressor(random_state=42), X, y, cv=5,
        scoring='neg_mean_squared_error'
    )
    ensemble_scores = cross_val_score(
        RandomForestRegressor(n_estimators=50, random_state=42), X, y, cv=5,
        scoring='neg_mean_squared_error'
    )
    
    # Convert to RMSE improvement
    single_rmse = np.sqrt(-single_scores.mean())
    ensemble_rmse = np.sqrt(-ensemble_scores.mean())
    
    results['regression'] = {
        'single_rmse': single_rmse,
        'ensemble_rmse': ensemble_rmse,
        'improvement_percent': (single_rmse - ensemble_rmse) / single_rmse * 100
    }
    
    # Imbalanced classification
    X, y = make_classification(
        n_samples=1000, n_features=20, n_informative=10,
        weights=[0.9, 0.1], random_state=42  # 90-10 imbalance
    )
    
    single_scores = cross_val_score(
        DecisionTreeClassifier(random_state=42), X, y, cv=5,
        scoring='f1'
    )
    ensemble_scores = cross_val_score(
        RandomForestClassifier(n_estimators=50, random_state=42), X, y, cv=5,
        scoring='f1'
    )
    
    results['imbalanced'] = {
        'single_f1': single_scores.mean(),
        'ensemble_f1': ensemble_scores.mean(),
        'improvement': ensemble_scores.mean() - single_scores.mean()
    }
    
    # High-dimensional sparse (text-like)
    X, y = make_classification(
        n_samples=1000, n_features=1000, n_informative=100,
        random_state=42
    )
    
    single_scores = cross_val_score(
        DecisionTreeClassifier(random_state=42), X, y, cv=5
    )
    ensemble_scores = cross_val_score(
        RandomForestClassifier(n_estimators=50, random_state=42), X, y, cv=5
    )
    
    results['high_dimensional'] = {
        'single': single_scores.mean(),
        'ensemble': ensemble_scores.mean(),
        'improvement': ensemble_scores.mean() - single_scores.mean()
    }
    
    # Print summary
    print("Bagging Benefit by Problem Type:")
    print("=" * 50)
    for problem, metrics in results.items():
        print(f"\n{problem}:")
        for k, v in metrics.items():
            if isinstance(v, float):
                print(f"  {k}: {v:.4f}")
            else:
                print(f"  {k}: {v}")
    
    return results
 
 
def time_series_bagging_considerations():
    """
    Special considerations for time series bagging.
    
    Standard bagging violates temporal structure.
    Use block bootstrap or moving block bootstrap instead.
    """
    def block_bootstrap(data, block_size, n_samples):
        """
        Block bootstrap for time series.
        
        Rather than sampling individual points,
        sample contiguous blocks to preserve temporal structure.
        """
        n = len(data)
        n_blocks = n_samples // block_size
        
        bootstrapped = []
        for _ in range(n_blocks):
            # Random start position for block
            start = np.random.randint(0, n - block_size + 1)
            block = data[start:start + block_size]
            bootstrapped.extend(block)
        
        return np.array(bootstrapped[:n_samples])
    
    # Example usage
    np.random.seed(42)
    time_series = np.sin(np.linspace(0, 4*np.pi, 100)) + np.random.randn(100) * 0.1
    
    # Standard bootstrap (breaks temporal structure)
    standard_boot = np.random.choice(time_series, size=100, replace=True)
    
    # Block bootstrap (preserves local structure)
    block_boot = block_bootstrap(time_series, block_size=10, n_samples=100)
    
    print("Standard bootstrap loses temporal structure")
    print("Block bootstrap maintains local patterns")
    
    return {
        'original_autocorr': np.corrcoef(time_series[:-1], time_series[1:])[0,1],
        'block_boot_autocorr': np.corrcoef(block_boot[:-1], block_boot[1:])[0,1],
    }

Practical Decision Framework

Use this decision framework to determine if bagging is right for your problem:

Step 1: Characterize Your Base Model

Ask: Does my base model have high variance?

Train 5 models with different seeds
Compute prediction agreement (% samples with same prediction)
If agreement < 85%: HIGH variance → Bagging opportunity
If agreement > 95%: LOW variance → Bagging unlikely to help

Step 2: Check for Bias Issues

Ask: Is my base model underfitting?

Compute training accuracy
If training accuracy < 95% (for classification) or R² < 0.9 (regression): Likely underfitting
If underfitting: Fix with more complex model or boosting instead

Step 3: Estimate Diversity Potential

Ask: Will bootstrap samples produce diverse models?

Check feature importance concentration
If top 3 features > 80% importance: LOW diversity → Consider feature randomization
If importance spread across 10+ features: HIGH diversity → Standard bagging sufficient

Step 4: Run Quick Experiment

# 5-minute bagging test
from sklearn.ensemble import BaggingClassifier

baseline = base_model.fit(X_train, y_train).score(X_test, y_test)

bagged = BaggingClassifier(base_model, n_estimators=20).fit(X_train, y_train)
bagged_score = bagged.score(X_test, y_test)

improvement = bagged_score - baseline
print(f"Improvement: {improvement:.3f}")
# > 0.02: Significant benefit, proceed with full bagging
# 0.01-0.02: Moderate benefit, consider if compute allows
# < 0.01: Limited benefit, likely not worth it

The 5-Minute Test

Before committing to full ensemble training, run a quick test with 20 trees. If you see > 2% improvement, proceed. If not, investigate why (low variance? high bias? correlated models?) before investing more compute.

Decision Tree Summary:

Start
│
├── Is training accuracy high (>95%)?
│   ├── No → Model underfits. Use boosting or more complex model.
│   └── Yes → Continue
│
├── Is there a large train-test gap (>5%)?
│   ├── Yes → High variance! Bagging will likely help.
│   └── No → May be optimal. Continue.
│
├── 5 random seeds → prediction agreement?
│   ├── < 85% → High variance. Bagging will help!
│   └── > 95% → Low variance. Bagging minimal benefit.
│
├── Top-3 feature importance > 80%?
│   ├── Yes → Models will be correlated. Use Random Forest (feature sampling)
│   └── No → Standard bagging sufficient.
│
└── Quick 20-tree experiment → improvement?
    ├── > 2% → Proceed with full bagging
    ├── 0.5-2% → Marginal benefit. Consider if compute allows.
    └── < 0.5% → Not worth it. Try other approaches.

Real-World Case Studies

Let's examine real scenarios where bagging's impact was measured:

Case Study 1: Medical Image Classification (Skin Cancer Detection)

Setup: 25,000 dermoscopy images, 7-class classification, ResNet-50 base model

Results:

Single model: 87.2% accuracy
5-model ensemble: 89.8% accuracy (+2.6%)
10-model ensemble: 90.4% accuracy (+3.2%)

Analysis:

High improvement due to initialization variance in CNNs
Critical for clinical deployment: reduced false negative rate by 15%
Uncertainty estimates from ensemble enabled "refer to specialist" workflow

Lesson: In high-stakes applications, even 2-3% improvement matters, and uncertainty quantification is invaluable.

Case Study 2: Credit Scoring (Financial Services)

Setup: 100,000 loan applications, 50 features, binary default prediction

Results:

Logistic Regression: 78.4% AUC (no bagging benefit)
Single Decision Tree: 72.1% AUC
Random Forest (100 trees): 81.2% AUC
XGBoost: 82.5% AUC

Analysis:

Bagging logistic regression provided < 0.1% improvement
Random Forest massively improved over single tree
But boosting ultimately outperformed bagging

Lesson: Bagging helps high-variance models; low-variance models need different approaches.

Case Study 3: E-commerce Product Categorization

Setup: 2 million products, 500 categories, text + numeric features

Results:

Single gradient boosted tree: 89.3% accuracy
Bagged GBT (5 models): 90.1% accuracy (+0.8%)
Single Random Forest: 88.7% accuracy
Time: RF trained in 20 min (parallel), GBT ensemble took 3 hours

Analysis:

Bagging boosted models provided modest improvement
Training time trade-off significant at scale
For this use case, single well-tuned model was preferred

Lesson: At large scale, the compute cost of ensembles must be weighed against accuracy gains.

Case Study Summary
Domain	Base Model	Improvement from Bagging	Verdict
Medical imaging	Deep CNN	+3.2%	Essential for deployment
Credit scoring (LogReg)	Logistic Regression	<0.1%	Not worth it
Credit scoring (Trees)	Decision Tree	+9.1%	Highly effective
E-commerce (large scale)	Gradient Boosting	+0.8%	Marginal, high compute cost
Fraud detection	Random Forest	+4.2%	Effective for rare event
Time series forecasting	ARIMA ensemble	+12%	Very effective (block bootstrap)

Key Insight from Case Studies

Bagging's benefit is highly problem-dependent. The same technique can provide essential improvements in one domain and negligible gains in another. Always validate on your specific data rather than assuming universal benefit.

Summary: When Bagging Helps

Knowing when bagging will help—and when it won't—separates ensemble practitioners from ensemble experts. Let's consolidate the key insights:

Key Takeaways

•Three conditions — Bagging needs high base variance, low model correlation, and acceptable bias. Missing any one limits benefit.
•Diagnose before committing — Use repeated CV, prediction stability, and quick experiments to estimate potential benefit.
•Low variance = no benefit — Bagging linear models or heavily regularized models is typically wasted computation.
•High bias needs boosting — If the model underfits, averaging more underfitting models doesn't help.
•Problem type matters — Regression often benefits more than classification; high-dimensional often benefits more than low.
•Always validate empirically — Theoretical expectations should be confirmed with quick experiments on your specific data.

Module Complete!

With this page, you've completed Module 6: Bagging for Other Models. You've learned:

Bagged Decision Trees — The canonical bagging application
Bagged Neural Networks — Extending bagging to deep learning
Model Aggregation Strategies — How to combine predictions effectively
Bagging vs Boosting — When each paradigm excels
When Bagging Helps — Practical decision frameworks

You're now equipped to apply bagging intelligently across model families, diagnose when it will help, and implement effective ensemble strategies for real-world problems.

Module Complete

Congratulations! You've mastered bagging for diverse model types. You understand not just how to implement bagging, but when and why it works—the mark of true ensemble expertise. Next, explore boosting techniques in Chapter 16 to complete your ensemble methods toolkit.

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Machine LearningBagging for Other Models

Bagging for Other Models

LevelIntermediate

Duration90 mins

TopicBagging for Other Models

5 / 5

When Bagging Helps

Recognizing Bagging Opportunities

What You Will Learn

Conditions for Bagging Success

Bagging works by averaging diverse predictions to reduce variance. For this to be effective, three conditions must be met:

Condition 1: High Base Model Variance

The base model must exhibit significant variance—its predictions should change noticeably with different training samples. Models with high variance include:

Decision trees (especially deep, unpruned)
Neural networks (with random initialization)
K-Nearest Neighbors with small K
Polynomial regression with high degree
Models with many features relative to samples

Condition 2: Low Correlation Between Models

Bagging's variance reduction is bounded by the correlation between model predictions:

$$\text{Var}_{ensemble} = \rho\sigma^2 + \frac{1-\rho}{B}\sigma^2$$

When $\rho$ is high (models agree), bagging helps little. The bootstrap sampling must create sufficient diversity.

Factors that increase model correlation (bad for bagging):

Dominant features that always drive predictions
Very large datasets (bootstrap samples overlap heavily)
Highly regularized models
Simple decision boundaries

Condition 3: Low Bias (or Acceptable Bias)

Bagging does not reduce bias. If your base model systematically underfits, averaging more underfitting models won't help. You need:

Either: A base model with low bias (complex enough to fit the pattern)
Or: Acceptable bias for your application

The Three Conditions for Bagging Success
Condition	Diagnostic	If Not Met
High variance	Predictions change with training sample	Use simpler aggregation or boost instead
Low correlation	Models disagree on substantial fraction of samples	Add feature randomization (→ Random Forest)
Low bias	Individual models fit training data well	Use more complex base model or switch to boosting

Quick Diagnostic

Diagnosing High-Variance Situations

High variance is the primary driver of bagging's benefit. Here's how to identify and quantify it:

Method 1: Repeated Cross-Validation Variance

Train your model with different random seeds or data splits. High variance in test performance indicates bagging opportunity.

Run 1: 85.2% accuracy
Run 2: 82.1% accuracy  
Run 3: 87.4% accuracy
Run 4: 83.0% accuracy
Run 5: 86.8% accuracy

Standard deviation: 2.2% → High variance, bagging likely helps

Compare to a low-variance model:

Run 1: 78.5% accuracy
Run 2: 78.7% accuracy
Run 3: 78.4% accuracy
Run 4: 78.6% accuracy
Run 5: 78.5% accuracy

Standard deviation: 0.1% → Low variance, bagging won't help

Method 2: Prediction Stability Analysis

For the same input, measure how predictions change across different model instances:

$$\text{Prediction Stability} = \frac{1}{N}\sum_{i=1}^{N} \frac{\text{std}(\hat{y}_i^{(1)}, ..., \hat{y}_i^{(B)})}{\text{mean}(\hat{y}_i)}$$

High stability coefficient = high variance = bagging opportunity.

Method 3: Learning Curve Shape

Plot validation error vs. training set size. High-variance models show:

Large gap between training and validation error
Validation error decreasing with more data
Training error near zero (overfitting)

Low-variance models show:

Small gap between training and validation
Both errors plateau at similar levels
Training error higher than zero (underfitting)

variance_diagnostics.py
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import numpy as np
from sklearn.model_selection import cross_val_score, learning_curve
from sklearn.tree import DecisionTreeClassifier
from sklearn.linear_model import LogisticRegression
import matplotlib.pyplot as plt
 
def variance_diagnostic(model_factory, X, y, n_iterations=20, cv=5):
    """
    Diagnose model variance through repeated cross-validation.
    
    Returns:
    --------
    dict with variance metrics and recommendation
    """
    scores = []
    
    for i in range(n_iterations):
        model = model_factory()
        cv_scores = cross_val_score(model, X, y, cv=cv)
        scores.extend(cv_scores)
    
    mean_score = np.mean(scores)
    std_score = np.std(scores)
    cv_coefficient = std_score / mean_score  # Coefficient of variation
    
    # Interpretation
    if std_score > 0.05:
        variance_level = "HIGH"
        recommendation = "Bagging will likely provide significant benefit"
    elif std_score > 0.02:
        variance_level = "MODERATE"
        recommendation = "Bagging may help, worth testing"
    else:
        variance_level = "LOW"
        recommendation = "Bagging unlikely to help much, consider boosting"
    
    return {
        'mean_score': mean_score,
        'std_score': std_score,
        'cv_coefficient': cv_coefficient,
        'variance_level': variance_level,
        'recommendation': recommendation,
        'all_scores': scores,
    }
 
 
def prediction_stability_analysis(models, X):
    """
    Analyze how predictions vary across model instances.
    
    Parameters:
    -----------
    models : list of fitted models
    X : test data
    
    Returns:
    --------
    stability metrics per sample
    """
    # Collect predictions
    if hasattr(models[0], 'predict_proba'):
        all_probs = np.array([m.predict_proba(X)[:, 1] for m in models])
        predictions = all_probs
    else:
        predictions = np.array([m.predict(X) for m in models])
    
    # Per-sample statistics
    sample_means = predictions.mean(axis=0)
    sample_stds = predictions.std(axis=0)
    
    # Identify high-variance samples
    high_var_mask = sample_stds > np.median(sample_stds) * 2
    
    return {
        'mean_predictions': sample_means,
        'std_predictions': sample_stds,
        'mean_std': sample_stds.mean(),
        'max_std': sample_stds.max(),
        'high_variance_samples': np.where(high_var_mask)[0],
        'high_variance_fraction': high_var_mask.mean(),
    }
 
 
def bagging_benefit_estimator(base_model_factory, X_train, y_train, 
                                X_test, y_test, max_estimators=50):
    """
    Estimate potential bagging benefit before full training.
    
    Trains progressively larger ensembles and measures improvement.
    """
    from sklearn.ensemble import BaggingClassifier
    
    estimator_counts = [1, 3, 5, 10, 20, 30, 50]
    estimator_counts = [e for e in estimator_counts if e <= max_estimators]
    
    scores = []
    
    for n_est in estimator_counts:
        if n_est == 1:
            # Single model baseline
            model = base_model_factory()
            model.fit(X_train, y_train)
            score = model.score(X_test, y_test)
        else:
            ensemble = BaggingClassifier(
                estimator=base_model_factory(),
                n_estimators=n_est,
                random_state=42,
                n_jobs=-1
            )
            ensemble.fit(X_train, y_train)
            score = ensemble.score(X_test, y_test)
        
        scores.append(score)
        print(f"n_estimators={n_est}: {score:.4f}")
    
    # Compute improvement
    single_score = scores[0]
    max_score = max(scores)
    improvement = max_score - single_score
    
    # Estimate saturation point
    improvements = [scores[i+1] - scores[i] for i in range(len(scores)-1)]
    if improvements and max(improvements) > 0:
        marginal_utility = improvements[-1] / improvements[0] if improvements[0] > 0 else 0
    else:
        marginal_utility = 0
    
    return {
        'estimator_counts': estimator_counts,
        'scores': scores,
        'single_model_score': single_score,
        'best_ensemble_score': max_score,
        'improvement': improvement,
        'marginal_utility': marginal_utility,
        'recommendation': (
            "Strong bagging benefit" if improvement > 0.03 else
            "Moderate bagging benefit" if improvement > 0.01 else
            "Limited bagging benefit"
        )
    }
 
 
def compare_variance_across_models(X, y):
    """
    Compare variance characteristics of different model types.
    """
    models = [
        ("Deep Tree (max_depth=None)", 
         lambda: DecisionTreeClassifier(max_depth=None)),
        ("Shallow Tree (max_depth=3)", 
         lambda: DecisionTreeClassifier(max_depth=3)),
        ("Logistic Regression", 
         lambda: LogisticRegression(max_iter=1000)),
    ]
    
    results = {}
    for name, factory in models:
        result = variance_diagnostic(factory, X, y, n_iterations=20)
        results[name] = result
        print(f"{name}:")
        print(f"  Mean: {result['mean_score']:.4f}")
        print(f"  Std:  {result['std_score']:.4f}")
        print(f"  Level: {result['variance_level']}")
        print(f"  {result['recommendation']}")
        print()
    
    return results

Rule of Thumb

When Bagging Doesn't Help

Recognizing when bagging won't help is as important as knowing when it will. Here are the key scenarios where bagging provides minimal benefit:

Bagging Provides Minimal Benefit When:

•Base model has low variance — Linear regression, Naive Bayes, heavily regularized models. Averaging similar predictions doesn't improve them.
•Base model has high bias — Bagging cannot fix underfitting. Shallow trees need boosting, not bagging.
•Dataset is very large — Bootstrap samples overlap almost entirely, producing nearly identical models. Diversity vanishes.
•Single feature dominates — All trees split on the same feature first, making them correlated. Consider feature randomization.
•Problem is inherently noisy — If irreducible noise dominates, variance reduction helps little. Focus on noise reduction.
•Already at optimal complexity — If your model is well-calibrated (good bias-variance balance), ensemble methods have limited upside.

Quantitative Thresholds:

Diagnostic	Bagging Unlikely to Help If
CV standard deviation	< 1% of mean
Improvement from 1→10 trees	< 0.5%
Model correlation $\rho$	> 0.8
Training-test gap	< 2% (underfit)
Feature importance concentration	Top feature > 70%

Case Study: Why Bagging a Linear Model Fails

Consider bagging logistic regression for a linearly separable problem:

Each bootstrap sample produces nearly identical coefficients
Decision boundaries differ by tiny amounts
Averaged predictions essentially equal any individual prediction
Computational cost multiplied by $B$ with no benefit

In this case, invest computation in better features or more data, not bagging.

The Diminishing Returns Trap

Bagging Benefit by Problem Type

Different problem types present different opportunities for bagging. Here's what to expect based on your domain:

Expected Bagging Benefit by Problem Domain
Problem Type	Typical Benefit	Key Considerations
Tabular classification (small)	High (3-8%)	Trees have high variance; bagging works well
Tabular classification (large)	Moderate (1-3%)	Less variance due to data quantity; still helps
Tabular regression	High (5-15%)	Regression targets amplify variance; bagging very effective
Text classification	Moderate (2-5%)	Feature sparsity increases diversity
Image classification (CNN)	Moderate (1-3%)	Initialization variance enables ensemble benefit
Time series forecasting	Moderate (3-8%)	Care needed: temporal structure in bootstrap
Ranking problems	Moderate (2-4%)	Ensemble rankings more stable than individual
Anomaly detection	High (5-10%)	Boundary estimation highly variable

Domain-Specific Considerations:

Medical Diagnosis:

Bagging improves both accuracy AND calibration
Multiple models provide uncertainty estimates critical for clinical decisions
Regulatory requirements often favor interpretable ensembles
Strong benefit: 3-8% improvement typical

Financial Prediction:

High noise levels can limit bagging's benefit
But uncertainty quantification is valuable for risk management
Non-stationarity means different trees capture different regimes
Moderate benefit: 2-5% improvement typical

Computer Vision:

CNN ensembles show consistent improvements
Test-time augmentation is a form of implicit bagging
Main benefit is robustness to adversarial examples
Moderate benefit: 1-3% improvement, but reliability boost is significant

problem_type_bagging.py
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import numpy as np
from sklearn.datasets import (
    make_classification, make_regression, 
    fetch_20newsgroups_vectorized
)
from sklearn.ensemble import RandomForestClassifier, RandomForestRegressor
from sklearn.tree import DecisionTreeClassifier, DecisionTreeRegressor
from sklearn.model_selection import cross_val_score
 
def estimate_bagging_benefit_for_problem_type():
    """
    Demonstrate bagging benefit varies by problem type.
    """
    results = {}
    
    # Binary classification - balanced
    X, y = make_classification(
        n_samples=1000, n_features=20, n_informative=10,
        random_state=42
    )
    
    single_scores = cross_val_score(
        DecisionTreeClassifier(random_state=42), X, y, cv=5
    )
    ensemble_scores = cross_val_score(
        RandomForestClassifier(n_estimators=50, random_state=42), X, y, cv=5
    )
    
    results['binary_classification'] = {
        'single': single_scores.mean(),
        'ensemble': ensemble_scores.mean(),
        'improvement': ensemble_scores.mean() - single_scores.mean()
    }
    
    # Regression
    X, y = make_regression(
        n_samples=1000, n_features=20, n_informative=10,
        noise=10, random_state=42
    )
    
    single_scores = cross_val_score(
        DecisionTreeRegressor(random_state=42), X, y, cv=5,
        scoring='neg_mean_squared_error'
    )
    ensemble_scores = cross_val_score(
        RandomForestRegressor(n_estimators=50, random_state=42), X, y, cv=5,
        scoring='neg_mean_squared_error'
    )
    
    # Convert to RMSE improvement
    single_rmse = np.sqrt(-single_scores.mean())
    ensemble_rmse = np.sqrt(-ensemble_scores.mean())
    
    results['regression'] = {
        'single_rmse': single_rmse,
        'ensemble_rmse': ensemble_rmse,
        'improvement_percent': (single_rmse - ensemble_rmse) / single_rmse * 100
    }
    
    # Imbalanced classification
    X, y = make_classification(
        n_samples=1000, n_features=20, n_informative=10,
        weights=[0.9, 0.1], random_state=42  # 90-10 imbalance
    )
    
    single_scores = cross_val_score(
        DecisionTreeClassifier(random_state=42), X, y, cv=5,
        scoring='f1'
    )
    ensemble_scores = cross_val_score(
        RandomForestClassifier(n_estimators=50, random_state=42), X, y, cv=5,
        scoring='f1'
    )
    
    results['imbalanced'] = {
        'single_f1': single_scores.mean(),
        'ensemble_f1': ensemble_scores.mean(),
        'improvement': ensemble_scores.mean() - single_scores.mean()
    }
    
    # High-dimensional sparse (text-like)
    X, y = make_classification(
        n_samples=1000, n_features=1000, n_informative=100,
        random_state=42
    )
    
    single_scores = cross_val_score(
        DecisionTreeClassifier(random_state=42), X, y, cv=5
    )
    ensemble_scores = cross_val_score(
        RandomForestClassifier(n_estimators=50, random_state=42), X, y, cv=5
    )
    
    results['high_dimensional'] = {
        'single': single_scores.mean(),
        'ensemble': ensemble_scores.mean(),
        'improvement': ensemble_scores.mean() - single_scores.mean()
    }
    
    # Print summary
    print("Bagging Benefit by Problem Type:")
    print("=" * 50)
    for problem, metrics in results.items():
        print(f"\n{problem}:")
        for k, v in metrics.items():
            if isinstance(v, float):
                print(f"  {k}: {v:.4f}")
            else:
                print(f"  {k}: {v}")
    
    return results
 
 
def time_series_bagging_considerations():
    """
    Special considerations for time series bagging.
    
    Standard bagging violates temporal structure.
    Use block bootstrap or moving block bootstrap instead.
    """
    def block_bootstrap(data, block_size, n_samples):
        """
        Block bootstrap for time series.
        
        Rather than sampling individual points,
        sample contiguous blocks to preserve temporal structure.
        """
        n = len(data)
        n_blocks = n_samples // block_size
        
        bootstrapped = []
        for _ in range(n_blocks):
            # Random start position for block
            start = np.random.randint(0, n - block_size + 1)
            block = data[start:start + block_size]
            bootstrapped.extend(block)
        
        return np.array(bootstrapped[:n_samples])
    
    # Example usage
    np.random.seed(42)
    time_series = np.sin(np.linspace(0, 4*np.pi, 100)) + np.random.randn(100) * 0.1
    
    # Standard bootstrap (breaks temporal structure)
    standard_boot = np.random.choice(time_series, size=100, replace=True)
    
    # Block bootstrap (preserves local structure)
    block_boot = block_bootstrap(time_series, block_size=10, n_samples=100)
    
    print("Standard bootstrap loses temporal structure")
    print("Block bootstrap maintains local patterns")
    
    return {
        'original_autocorr': np.corrcoef(time_series[:-1], time_series[1:])[0,1],
        'block_boot_autocorr': np.corrcoef(block_boot[:-1], block_boot[1:])[0,1],
    }

Practical Decision Framework

Use this decision framework to determine if bagging is right for your problem:

Step 1: Characterize Your Base Model

Ask: Does my base model have high variance?

Train 5 models with different seeds
Compute prediction agreement (% samples with same prediction)
If agreement < 85%: HIGH variance → Bagging opportunity
If agreement > 95%: LOW variance → Bagging unlikely to help

Step 2: Check for Bias Issues

Ask: Is my base model underfitting?

Compute training accuracy
If training accuracy < 95% (for classification) or R² < 0.9 (regression): Likely underfitting
If underfitting: Fix with more complex model or boosting instead

Step 3: Estimate Diversity Potential

Ask: Will bootstrap samples produce diverse models?

Check feature importance concentration
If top 3 features > 80% importance: LOW diversity → Consider feature randomization
If importance spread across 10+ features: HIGH diversity → Standard bagging sufficient

Step 4: Run Quick Experiment

# 5-minute bagging test
from sklearn.ensemble import BaggingClassifier

baseline = base_model.fit(X_train, y_train).score(X_test, y_test)

bagged = BaggingClassifier(base_model, n_estimators=20).fit(X_train, y_train)
bagged_score = bagged.score(X_test, y_test)

improvement = bagged_score - baseline
print(f"Improvement: {improvement:.3f}")
# > 0.02: Significant benefit, proceed with full bagging
# 0.01-0.02: Moderate benefit, consider if compute allows
# < 0.01: Limited benefit, likely not worth it

The 5-Minute Test

Decision Tree Summary:

Start
│
├── Is training accuracy high (>95%)?
│   ├── No → Model underfits. Use boosting or more complex model.
│   └── Yes → Continue
│
├── Is there a large train-test gap (>5%)?
│   ├── Yes → High variance! Bagging will likely help.
│   └── No → May be optimal. Continue.
│
├── 5 random seeds → prediction agreement?
│   ├── < 85% → High variance. Bagging will help!
│   └── > 95% → Low variance. Bagging minimal benefit.
│
├── Top-3 feature importance > 80%?
│   ├── Yes → Models will be correlated. Use Random Forest (feature sampling)
│   └── No → Standard bagging sufficient.
│
└── Quick 20-tree experiment → improvement?
    ├── > 2% → Proceed with full bagging
    ├── 0.5-2% → Marginal benefit. Consider if compute allows.
    └── < 0.5% → Not worth it. Try other approaches.

Real-World Case Studies

Let's examine real scenarios where bagging's impact was measured:

Case Study 1: Medical Image Classification (Skin Cancer Detection)

Setup: 25,000 dermoscopy images, 7-class classification, ResNet-50 base model

Results:

Single model: 87.2% accuracy
5-model ensemble: 89.8% accuracy (+2.6%)
10-model ensemble: 90.4% accuracy (+3.2%)

Analysis:

High improvement due to initialization variance in CNNs
Critical for clinical deployment: reduced false negative rate by 15%
Uncertainty estimates from ensemble enabled "refer to specialist" workflow

Lesson: In high-stakes applications, even 2-3% improvement matters, and uncertainty quantification is invaluable.

Case Study 2: Credit Scoring (Financial Services)

Setup: 100,000 loan applications, 50 features, binary default prediction

Results:

Logistic Regression: 78.4% AUC (no bagging benefit)
Single Decision Tree: 72.1% AUC
Random Forest (100 trees): 81.2% AUC
XGBoost: 82.5% AUC

Analysis:

Bagging logistic regression provided < 0.1% improvement
Random Forest massively improved over single tree
But boosting ultimately outperformed bagging

Lesson: Bagging helps high-variance models; low-variance models need different approaches.

Case Study 3: E-commerce Product Categorization

Setup: 2 million products, 500 categories, text + numeric features

Results:

Single gradient boosted tree: 89.3% accuracy
Bagged GBT (5 models): 90.1% accuracy (+0.8%)
Single Random Forest: 88.7% accuracy
Time: RF trained in 20 min (parallel), GBT ensemble took 3 hours

Analysis:

Bagging boosted models provided modest improvement
Training time trade-off significant at scale
For this use case, single well-tuned model was preferred

Lesson: At large scale, the compute cost of ensembles must be weighed against accuracy gains.

Case Study Summary
Domain	Base Model	Improvement from Bagging	Verdict
Medical imaging	Deep CNN	+3.2%	Essential for deployment
Credit scoring (LogReg)	Logistic Regression	<0.1%	Not worth it
Credit scoring (Trees)	Decision Tree	+9.1%	Highly effective
E-commerce (large scale)	Gradient Boosting	+0.8%	Marginal, high compute cost
Fraud detection	Random Forest	+4.2%	Effective for rare event
Time series forecasting	ARIMA ensemble	+12%	Very effective (block bootstrap)

Key Insight from Case Studies

Summary: When Bagging Helps

Knowing when bagging will help—and when it won't—separates ensemble practitioners from ensemble experts. Let's consolidate the key insights:

Key Takeaways

•Three conditions — Bagging needs high base variance, low model correlation, and acceptable bias. Missing any one limits benefit.
•Diagnose before committing — Use repeated CV, prediction stability, and quick experiments to estimate potential benefit.
•Low variance = no benefit — Bagging linear models or heavily regularized models is typically wasted computation.
•High bias needs boosting — If the model underfits, averaging more underfitting models doesn't help.
•Problem type matters — Regression often benefits more than classification; high-dimensional often benefits more than low.
•Always validate empirically — Theoretical expectations should be confirmed with quick experiments on your specific data.

Module Complete!

With this page, you've completed Module 6: Bagging for Other Models. You've learned:

Bagged Decision Trees — The canonical bagging application
Bagged Neural Networks — Extending bagging to deep learning
Model Aggregation Strategies — How to combine predictions effectively
Bagging vs Boosting — When each paradigm excels
When Bagging Helps — Practical decision frameworks

You're now equipped to apply bagging intelligently across model families, diagnose when it will help, and implement effective ensemble strategies for real-world problems.

Module Complete

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