Bernoulli Naive Bayes, Explained: A Visual Guide with Code Examples for Beginners | by Samy Baladram | Aug, 2024

Unlocking Predictive Power Through Yes/No Probability

Unlike the baseline approach of dummy classifiers or the similarity-based reasoning of KNN, Naive Bayes leverages probability theory. It combines the individual probabilities of each “clue” (or feature) to make a final prediction. This straightforward yet powerful method has proven invaluable in various machine learning applications.

Naive Bayes is a machine learning algorithm that uses probability to classify data. It’s based on Bayes’ Theorem, a formula for calculating conditional probabilities. The “naive” part refers to its key assumption: it treats all features as independent of each other, even when they might not be in reality. This simplification, while often unrealistic, greatly reduces computational complexity and works well in many practical scenarios.

There are three main types of Naive Bayes classifiers. The key difference between these types lies in the assumption they make about the distribution of features:

1. Bernoulli Naive Bayes: Suited for binary/boolean features. It assumes each feature is a binary-valued (0/1) variable.
2. Multinomial Naive Bayes: Typically used for discrete counts. It’s often used in text classification, where features might be word counts.
3. Gaussian Naive Bayes: Assumes that continuous features follow a normal distribution.

It is a good start to focus on the simplest one which is Bernoulli NB. The “Bernoulli” in its name comes from the assumption that each feature is binary-valued.

Throughout this article, we’ll use this artificial golf dataset (inspired by [1]) as an example. This dataset predicts whether a person will play golf based on weather conditions.

# IMPORTING DATASET #
from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score
import pandas as pd
import numpy as np
dataset_dict = {
'Outlook': ['sunny', 'sunny', 'overcast', 'rain', 'rain', 'rain', 'overcast', 'sunny', 'sunny', 'rain', 'sunny', 'overcast', 'overcast', 'rain', 'sunny', 'overcast', 'rain', 'sunny', 'sunny', 'rain', 'overcast', 'rain', 'sunny', 'overcast', 'sunny', 'overcast', 'rain', 'overcast'],
'Temperature': [85.0, 80.0, 83.0, 70.0, 68.0, 65.0, 64.0, 72.0, 69.0, 75.0, 75.0, 72.0, 81.0, 71.0, 81.0, 74.0, 76.0, 78.0, 82.0, 67.0, 85.0, 73.0, 88.0, 77.0, 79.0, 80.0, 66.0, 84.0],
'Humidity': [85.0, 90.0, 78.0, 96.0, 80.0, 70.0, 65.0, 95.0, 70.0, 80.0, 70.0, 90.0, 75.0, 80.0, 88.0, 92.0, 85.0, 75.0, 92.0, 90.0, 85.0, 88.0, 65.0, 70.0, 60.0, 95.0, 70.0, 78.0],
'Wind': [False, True, False, False, False, True, True, False, False, False, True, True, False, True, True, False, False, True, False, True, True, False, True, False, False, True, False, False],
'Play': ['No', 'No', 'Yes', 'Yes', 'Yes', 'No', 'Yes', 'No', 'Yes', 'Yes', 'Yes', 'Yes', 'Yes', 'No', 'No', 'Yes', 'Yes', 'No', 'No', 'No', 'Yes', 'Yes', 'Yes', 'Yes', 'Yes', 'Yes', 'No', 'Yes']
}
df = pd.DataFrame(dataset_dict)
# ONE-HOT ENCODE 'Outlook' COLUMN
df = pd.get_dummies(df, columns=['Outlook'],  prefix='', prefix_sep='', dtype=int)
# CONVERT 'Windy' (bool) and 'Play' (binary) COLUMNS TO BINARY INDICATORS
df['Wind'] = df['Wind'].astype(int)
df['Play'] = (df['Play'] == 'Yes').astype(int)
# Set feature matrix X and target vector y
X, y = df.drop(columns='Play'), df['Play']
# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, train_size=0.5, shuffle=False)
print(pd.concat([X_train, y_train], axis=1), end='\n\n')
print(pd.concat([X_test, y_test], axis=1))

We’ll adapt it slightly for Bernoulli Naive Bayes by converting our features to binary.

# One-hot encode the categorized columns and drop them after, but do it separately for training and test sets
# Define categories for 'Temperature' and 'Humidity' for training set
X_train['Temperature'] = pd.cut(X_train['Temperature'], bins=[0, 80, 100], labels=['Warm', 'Hot'])
X_train['Humidity'] = pd.cut(X_train['Humidity'], bins=[0, 75, 100], labels=['Dry', 'Humid'])
# Similarly, define for the test set
X_test['Temperature'] = pd.cut(X_test['Temperature'], bins=[0, 80, 100], labels=['Warm', 'Hot'])
X_test['Humidity'] = pd.cut(X_test['Humidity'], bins=[0, 75, 100], labels=['Dry', 'Humid'])
# One-hot encode the categorized columns
one_hot_columns_train = pd.get_dummies(X_train[['Temperature', 'Humidity']], drop_first=True, dtype=int)
one_hot_columns_test = pd.get_dummies(X_test[['Temperature', 'Humidity']], drop_first=True, dtype=int)
# Drop the categorized columns from training and test sets
X_train = X_train.drop(['Temperature', 'Humidity'], axis=1)
X_test = X_test.drop(['Temperature', 'Humidity'], axis=1)
# Concatenate the one-hot encoded columns with the original DataFrames
X_train = pd.concat([one_hot_columns_train, X_train], axis=1)
X_test = pd.concat([one_hot_columns_test, X_test], axis=1)
print(pd.concat([X_train, y_train], axis=1), '\n')
print(pd.concat([X_test, y_test], axis=1))

Bernoulli Naive Bayes operates on data where each feature is either 0 or 1.

1. Calculate the probability of each class in the training data.
2. For each feature and class, calculate the probability of the feature being 1 and 0 given the class.
3. For a new instance: For each class, multiply its probability by the probability of each feature value (0 or 1) for that class.
4. Predict the class with the highest resulting probability.

The training process for Bernoulli Naive Bayes involves calculating probabilities from the training data:

1. Class Probability Calculation: For each class, calculate its probability: (Number of instances in this class) / (Total number of instances)

from fractions import Fraction
def calc_target_prob(attr):
total_counts = attr.value_counts().sum()
prob_series = attr.value_counts().apply(lambda x: Fraction(x, total_counts).limit_denominator())
return prob_series
print(calc_target_prob(y_train))

2.Feature Probability Calculation: For each feature and each class, calculate:

• (Number of instances where feature is 0 in this class) / (Number of instances in this class)
• (Number of instances where feature is 1 in this class) / (Number of instances in this class)

from fractions import Fraction
def sort_attr_label(attr, lbl):
return (pd.concat([attr, lbl], axis=1)
.sort_values([attr.name, lbl.name])
.reset_index()
.rename(columns={'index': 'ID'})
.set_index('ID'))
def calc_feature_prob(attr, lbl):
total_classes = lbl.value_counts()
counts = pd.crosstab(attr, lbl)
prob_df = counts.apply(lambda x: [Fraction(c, total_classes[x.name]).limit_denominator() for c in x])
return prob_df
print(sort_attr_label(y_train, X_train['sunny']))
print(calc_feature_prob(X_train['sunny'], y_train))

for col in X_train.columns:
print(calc_feature_prob(X_train[col], y_train), "\n")

3. Smoothing (Optional): Add a small value (usually 1) to the numerator and denominator of each probability calculation to avoid zero probabilities

# In sklearn, all processes above is summarized in this 'fit' method:
from sklearn.naive_bayes import BernoulliNB
nb_clf = BernoulliNB(alpha=1)
nb_clf.fit(X_train, y_train)

4. Store Results: Save all calculated probabilities for use during classification.

Given a new instance with features that are either 0 or 1:

1. Probability Collection: For each possible class:

• Start with the probability of this class occurring (class probability).
• For each feature in the new instance, collect the probability of this feature being 0/1 for this class.

2. Score Calculation & Prediction: For each class:

• Multiply all the collected probabilities together
• The result is the score for this class
• The class with the highest score is the prediction

y_pred = nb_clf.predict(X_test)
print(y_pred)

# Evaluate the classifier
print(f"Accuracy: {accuracy_score(y_test, y_pred)}")

Bernoulli Naive Bayes has a few important parameters:

1. Alpha (α): This is the smoothing parameter. It adds a small count to each feature to prevent zero probabilities. Default is usually 1.0 (Laplace smoothing) as what was shown before.
2. Binarize: If your features aren’t already binary, this threshold converts them. Any value above this threshold becomes 1, and any value below becomes 0.

3. Fit Prior: Whether to learn class prior probabilities or assume uniform priors (50/50).

Like any algorithm in machine learning, Bernoulli Naive Bayes has its strengths and limitations.

1. Simplicity: Easy to implement and understand.
2. Efficiency: Fast to train and predict, works well with large feature spaces.
3. Performance with Small Datasets: Can perform well even with limited training data.
4. Handles High-Dimensional Data: Works well with many features, especially in text classification.

1. Independence Assumption: Assumes all features are independent, which is often not true in real-world data.
2. Limited to Binary Features: In its pure form, only works with binary data.
3. Sensitivity to Input Data: Can be sensitive to how the features are binarized.
4. Zero Frequency Problem: Without smoothing, zero probabilities can strongly affect predictions.

The Bernoulli Naive Bayes classifier is a simple yet powerful machine learning algorithm for binary classification. It excels in text analysis and spam detection, where features are typically binary. Known for its speed and efficiency, this probabilistic model performs well with small datasets and high-dimensional spaces.

Despite its naive assumption of feature independence, it often rivals more complex models in accuracy. Bernoulli Naive Bayes serves as an excellent baseline and real-time classification tool.

# Import needed libraries
import pandas as pd
from sklearn.naive_bayes import BernoulliNB
from sklearn.preprocessing import StandardScaler
from sklearn.metrics import accuracy_score
from sklearn.model_selection import train_test_split
# Load the dataset
dataset_dict = {
'Outlook': ['sunny', 'sunny', 'overcast', 'rainy', 'rainy', 'rainy', 'overcast', 'sunny', 'sunny', 'rainy', 'sunny', 'overcast', 'overcast', 'rainy', 'sunny', 'overcast', 'rainy', 'sunny', 'sunny', 'rainy', 'overcast', 'rainy', 'sunny', 'overcast', 'sunny', 'overcast', 'rainy', 'overcast'],
'Temperature': [85.0, 80.0, 83.0, 70.0, 68.0, 65.0, 64.0, 72.0, 69.0, 75.0, 75.0, 72.0, 81.0, 71.0, 81.0, 74.0, 76.0, 78.0, 82.0, 67.0, 85.0, 73.0, 88.0, 77.0, 79.0, 80.0, 66.0, 84.0],
'Humidity': [85.0, 90.0, 78.0, 96.0, 80.0, 70.0, 65.0, 95.0, 70.0, 80.0, 70.0, 90.0, 75.0, 80.0, 88.0, 92.0, 85.0, 75.0, 92.0, 90.0, 85.0, 88.0, 65.0, 70.0, 60.0, 95.0, 70.0, 78.0],
'Wind': [False, True, False, False, False, True, True, False, False, False, True, True, False, True, True, False, False, True, False, True, True, False, True, False, False, True, False, False],
'Play': ['No', 'No', 'Yes', 'Yes', 'Yes', 'No', 'Yes', 'No', 'Yes', 'Yes', 'Yes', 'Yes', 'Yes', 'No', 'No', 'Yes', 'Yes', 'No', 'No', 'No', 'Yes', 'Yes', 'Yes', 'Yes', 'Yes', 'Yes', 'No', 'Yes']
}
df = pd.DataFrame(dataset_dict)
# Prepare data for model
df = pd.get_dummies(df, columns=['Outlook'],  prefix='', prefix_sep='', dtype=int)
df['Wind'] = df['Wind'].astype(int)
df['Play'] = (df['Play'] == 'Yes').astype(int)
# Split data into training and testing sets
X, y = df.drop(columns='Play'), df['Play']
X_train, X_test, y_train, y_test = train_test_split(X, y, train_size=0.5, shuffle=False)
# Scale numerical features (for automatic binarization)
scaler = StandardScaler()
float_cols = X_train.select_dtypes(include=['float64']).columns
X_train[float_cols] = scaler.fit_transform(X_train[float_cols])
X_test[float_cols] = scaler.transform(X_test[float_cols])
# Train the model
nb_clf = BernoulliNB()
nb_clf.fit(X_train, y_train)
# Make predictions
y_pred = nb_clf.predict(X_test)
# Check accuracy
print(f"Accuracy: {accuracy_score(y_test, y_pred)}")