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Gaussian Naive Bayes is the simplest probabilistic classifier in the FFD pipeline and acts as a fast, interpretable baseline. Given a transaction’s 29 PCA-derived features, GaussianNB estimates the probability that it belongs to Class 1 (fraud) by computing the posterior under Bayes’ theorem, modelling each feature’s class-conditional likelihood as a univariate Gaussian. Despite its strong independence assumption, it is competitive on the PCA-transformed FFD features because PCA explicitly decorrelates the input space.

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How Gaussian Naive Bayes Works

Bayes’ theorem relates the posterior probability of fraud to the likelihood and prior: 
P(fraud | x) ∝ P(x | fraud) × P(fraud)
The “naive” part is the conditional independence assumption: 
P(x | class) = ∏ P(xᵢ | class)
Each feature xᵢ is assumed to be drawn from a Gaussian distribution whose mean and variance are estimated from the training set for each class: 
P(xᵢ | class) = N(xᵢ; μᵢ_class, σ²ᵢ_class)
At inference time the class with the higher posterior wins. Training reduces to computing 2 × 29 = 58 means and 58 variances — extremely fast. Key properties:
  • Speed — parameter estimation is O(n · d); prediction is O(d) per sample.
  • No iterative optimisation — fits in a single pass over the training data.
  • Calibrated probabilities — outputs can be interpreted directly as fraud probabilities when the Gaussian assumption holds.

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Implementation

GaussianNB has no mandatory hyperparameters; the notebook uses all defaults: 
from sklearn.naive_bayes import GaussianNB

nb_model = GaussianNB()
nb_model.fit(x_train, y_train)
nb_predictions = nb_model.predict(x_test)

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Key Hyperparameters

ParameterDefaultGuidance
var_smoothing1e-9Adds a fraction of the largest feature variance to all feature variances, preventing division by zero for near-constant features. Increase (e.g. 1e-6) if the model is numerically unstable or if some PCA components have very low variance.
priorsNone (estimated from data)Supply priors=[p_genuine, p_fraud] to override the empirical class frequencies, for example when the training set has been augmented by GAN and no longer reflects real-world fraud rates.

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Evaluation

from sklearn.metrics import (
    classification_report,
    confusion_matrix,
    accuracy_score,
    precision_score,
    recall_score,
    f1_score,
)

print(classification_report(y_test, nb_predictions))
print("Confusion matrix:\n", confusion_matrix(y_test, nb_predictions))
print("Accuracy :", accuracy_score(y_test, nb_predictions))
print("Precision:", precision_score(y_test, nb_predictions))
print("Recall   :", recall_score(y_test, nb_predictions))
print("F1       :", f1_score(y_test, nb_predictions))
GaussianNB tends to have higher recall but lower precision compared to tree-based ensembles — it rarely misses a genuine fraud signal but may generate more false positives. The confusion matrix is the clearest tool for assessing this trade-off.
Why the independence assumption is reasonable here: The V1–V28 features are the result of Principal Component Analysis (PCA) applied to the original transaction features. By construction, PCA components are orthogonal and therefore linearly uncorrelated. Naive Bayes assumes conditional independence (a stronger claim), but decorrelated features violate this assumption less severely than correlated raw features would, making GaussianNB a natural fit for the FFD feature space.

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See Also

Training Pipeline

End-to-end pipeline: data loading, GAN augmentation, train/test split, and unified evaluation loop.

Metrics Reference

Definitions of accuracy, precision, recall, and F1 as used across all six FFD classifiers.

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