Decision Trees for Classification

The Decision Tree algorithm is a powerful and versatile machine learning technique, it belongs to  supervised learning paradigm and is widely used for both Classification and Regression tasks, but is mainly used for classification problems. These algorithms work by creating a model that predicts the value of a target variable by learning simple decision rules based on the data features.

The importance of Decision Trees lies in their simplicity and interpretability. Unlike other machine learning algorithms, Decision Trees require little data preparation and can handle data which hasn't been normalized or encoded, as well as both categorical and numerical data. They also provide a clear indication of which fields are most important, making them a valuable tool for understanding complex datasets.

Decision Trees are fundamental components of random forests, which are among the most potent Machine Learning algorithms available today. In the real world, Decision Trees have a wide range of applications. They are used in the medical field for aiding in diagnosis, in finance to assess a potential investment risk, in retail to help predict a customer's likely purchases, and in many other fields.

Structure of Decision Trees

The structure of a Decision Tree is straightforward and intuitive. It consists of a root node, internal nodes, and leaf nodes. Each internal node represents a feature, each leaf node represents a class label, and each branch represents a rule. The topmost node in a Decision Tree is known as the root node. 

The tree starts at the root node and splits the data on the feature that results in the largest information gain (IG). In an iterative process, this splitting procedure is repeated at each child node until the leaves are pure. This means that all the samples at each node belong to the same class.

Metrics for Split Quality

There are many metrics used to evaluate the quality of a split in Decision Trees, some of which commonly used are Information Gain, Gini Impurity, and Entropy.

Entropy is a measure of the randomness or uncertainty, the goal of using entropy in a Decision Tree is to find and connect nodes with information that is most valuable for classification, which is done by minimizing the entropy, the formula of entropy is: which is the goal of the information gain criterion.

\begin{equation*} \mathbf{E} = \sum_{i=1}^{n}p_i \log_2 (p_i)  \end{equation*}

Information Gain is the reduction in entropy or impurity from a split. It measures how well a given attribute separates the training examples according to their target classification. It represents how much information a feature provides for the target variable, and this attribute with the highest information gain is chosen as the splitting feature at each node.

\begin{equation*} \mathbf{IG} = E_{parent} - E_{children}  \end{equation*}

Where $E_{parent}$ is the entropy of the parent node and $E_{children}$ represents the average entropy of the child nodes that follow this variable.

Gini Index is other alternative for splitting a decision tree. While the Entropy and Information Gain method focuses on purity and impurity in a node, the Gini Index measures the degree or probability for a particular variable being misclassified when it is randomly chosen. The formula for Gini Index is:

\begin{equation*} \mathbf{Gini} = 1 - \sum_{i=1}^{n}p_{i}{}^2  \end{equation*}

The lower the Gini Index, the lower the probability of misclassification, a Gini Impurity of 0 represents the best split.

Algorithms for Building Decision Trees

There are var popular algorithms used to build Decision Trees, including CART (Classification and Regression Trees), ID3 (Iterative Dichotomiser 3), and C4.5.

• CART: The CART algorithm is a binary tree algorithm, meaning that each parent node splits into exactly two child nodes. It can handle both numerical and categorical variables and also can be used for both classification and regression tasks. CART uses the Gini Index as a metric to create decision points for classification tasks, and mean squared error for regression tasks.

• ID3: It uses a top-down, greedy approach to construct a decision tree. It selects the best attribute to place at the root node based on the Information Gain metric. The ID3 algorithm can handle categorical input attributes well, but struggles with continuous attributes. It also doesn't handle missing values, so any missing data needs to be preprocessed before to be used in an ID3 algorithm.

• C4.5: This algorithm is an extension of ID3 that adjusts the Information Gain using a "Gain Ratio" to handle the bias towards attributes with many outcomes. It can handle both continuous and categorical attributes, and also allows for missing feature values. The C4.5 model includes a pruning step, where it uses a statistical test to estimate whether expanding or collapsing some branches can improve its predictions, which helps to avoid overfitting.

Implementing algorithm in Python

In this section, I explain an implementation of the Decision Tree algorithm in Python using the scikit-learn library. Here I use the Iris dataset, which is a multivariate dataset introduced by the British statistician and biologist Ronald Fisher in his 1936 paper. The task is building a Decision Tree model to classify the iris species based on their properties.

Importing Libraries

First, we imported the libraries for this script, which includes:

# Importing libraries
import numpy as np
import pandas as pd
from sklearn.datasets import load_iris
from sklearn.model_selection import train_test_split
from sklearn.tree import DecisionTreeClassifier, export_graphviz
from sklearn.metrics import accuracy_score
import matplotlib.pyplot as plt
from graphviz import Source

• numpy: It adds support for multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays.

• pandas: Library for data manipulation and analysis. It offers data structures and operations for manipulating numerical tables and time series.

• sklearn: It is a machine learning library for Python. It features various machine learning algorithms and also supports Python's numerical and scientific libraries. We specifically imported the load_iris function for loading the Iris dataset, train_test_split for splitting our data into train and test sets, DecisionTreeClassifier for building the Decision Tree model, export_graphviz for exporting the tree in a Graphviz format, and accuracy_score for calculating the accuracy of the model.

• matplotlib: It is a visualization library in Python for creating static, animated, and interactive visualizations, as well as drawing attractive and informative statistical graphs.

• Graphviz: This is an open-source graph visualization software. It is used to represent structural information as diagrams of abstract graphs and networks. In this case, it is used to plot the tree diagram of the trained model.

Loading the Iris Dataset

This dataset is freely available and built into the scikit-learn library. It includes three iris species (setosa, versicolor and virginica) with 50 samples each, as well as some properties about each flower (sepal length, sepal width, petal length and petal width). We first loaded this dataset and split it into input features X and the target variable y. The input features are the measurements of each flower and the target variable is the species of the flower.

# Loading the Iris dataset
iris = load_iris()
# Splitting the data into input features (X) and target variable (y)
X = iris.data
y = iris.target

Next, we created a Pandas dataframe with the data and the column names. This allowed us to manipulate and explore the data more easily. 

# Creating a Pandas DataFrame with the data and the column names
df = pd.DataFrame(data=X, columns=iris.feature_names)
# Adding the target variable column to the DataFrame
df['target'] = y

To make the data more understandable, we mapped the integer values of the target variable to the names of the Iris species.

# Creating a dictionary to map the integer label values to the class names
class_mapping = {0: 'setosa', 1: 'versicolor', 2: 'virginica'}
# Map the integer label values to the class names
df['target'] = df['target'].map(class_mapping)

Finally, we printed out a random sample of 5 rows from our dataset. This is useful to get a quick overview of the data we are working with.

df.sample(n = 5, random_state = 0)

The random sample is shown below:

Applying the Decision Tree Model

Now that we have loaded the dataset, we can move on to applying the Decision Tree model. First, we split the data into train and test sets. This allowed us to evaluate the performance of our model on unseen data. We used the train_test_split function from scikit-learn library, resulting that 25% of the samples belong to test set and 75% to train set.

# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.25, random_state=0)

Next, we created an instance of the DecisionTreeClassifier and trained it using the training dataset.

# Creating an instance of DecisionTreeClassifier
decision_tree = DecisionTreeClassifier(criterion = 'gini', splitter = 'best', random_state = 0)
# Training the model using the train set
decision_tree.fit(X_train, y_train)

Once the model is trained, we applied it to make predictions on the test data. We also calculated the accuracy score to evaluate the performance of the model.

# Make predictions on the test set
y_pred = decision_tree.predict(X_test)
# Computing the accuracy of the model
accuracy = accuracy_score(y_test, y_pred)
print("The accuracy is:", accuracy)

The resulting accuracy of the models was 0.974, which indicates a good performance of the model.

Understanding the Structure of the Tree

One of the advantages of Decision Trees is that they are relatively easy to interpret. We can visualize the trained Decision Tree model using sklearn and the graphviz libraries.

# Generating a Graphviz representation of the Decision Tree
dot_data = export_graphviz(decision_tree, out_file=None, feature_names=iris.feature_names, class_names=iris.target_names, rounded=True)
# Creating a graph from the Graphviz representation
graph = Source(dot_data)
# Displaying the graph
graph

The export_graphviz function generated a Graphviz representation of the Decision Tree, which is a text language for describing graphs. Then, we used the Source function from the graphviz library to create a graph from the Graphviz representation of the Decision Tree. Finally, we displayed the graph, which is below:

Visualizing Feature Importance

Another advantage of Decision Trees is that they can compute the relative importance of features. This can be very useful in understanding which features are the most influential in making the predictions.

# Get the feature importances
importances = decision_tree.feature_importances_

# Creating a bar plot to visualize the feature importances
plt.rcParams['font.size'] = '12'
fig, ax = plt.subplots(figsize=(8,6), facecolor='#F5F5F5')
ax.set_facecolor('#F5F5F5')
bars = ax.bar(range(len(importances)), importances, tick_label=iris.feature_names)
ax.set_ylabel('Importance')
ax.set_title('Importance of features in decision tree model')
ax.set_xticklabels(iris.feature_names, rotation=45)

# Adding the value above each bar
for bar in bars:
    yval = bar.get_height()
    ax.text(bar.get_x() + bar.get_width()/2, yval, round(yval, 4), ha='center', va='bottom')

plt.show()

The feature_importances_ attribute of the model gave us the importance of each feature. These importances were computed as the total reduction of the criterion (Gini) brought by that feature. Then, we plotted these feature importances on a bar plot for easy visualization.

This graph indicates that sepal length was not used in the decision-making process of the model, as it had an importance of 0.0. Sepal width, with an importance of 0.0201, played a minor role. But, the petal length and petal width features were the most influential in the decision-making process, with importances of 0.3993 and 0.5806, respectively. This suggests that these two features contributed the most to the reduction of impurity in the nodes, in other words, they were the most important features for the Decision Tree model.

Training a New Decision Tree Model with less Features

Given the importance scores that we obtained from the initial Decision Tree model, we saw that petal length and petal width were the most influential features in classifying the Iris species. To illustrate the impact of these features, we trained a new Decision Tree model using only these two features.

We created new training and testing sets that only include the petal length and petal width features.

# Selecting only the 'petal length' and 'petal width' features for training and testing
X_train_reduced = X_train[:, 2:4]
X_test_reduced = X_test[:, 2:4]

Then, we defined a new instance of the DecisionTreeClassifier and trained it using the reduced-features training data.

# Creating a new DecisionTreeClassifier model
decision_tree_2 = DecisionTreeClassifier(criterion = 'gini', splitter = 'best', random_state = 0)
# Train the new model using the reduced training set
decision_tree_2.fit(X_train_reduced, y_train)

Finally, we computed the accuracy of the new model.

# Make predictions on the reduced testing set
y_pred_2 = decision_tree_2.predict(X_test_reduced)
# Computing the accuracy of the new model
accuracy_2 = accuracy_score(y_test, y_pred_2)
print("The accuracy of the new model is:", accuracy_2)

The accuracy of this new model, which only uses the petal length and petal width features, was 0.947. This is slightly lower than the accuracy of the initial model, which was 0.973. But, it's important to note that this reduction is minimal.

Moreover, the fact that we were able to achieve nearly the same level of accuracy while reducing the number of features by half is quite significant. In larger, more complex datasets, reducing the dimensionality of the data can lead to substantial computational savings and can also help to mitigate the risk of overfitting.

Conclusion

In this post, we have explored the Decision Tree algorithm and its application in classification tasks. Through our hands-on implementation with the Iris dataset, we have seen how the model can be trained, how predictions can be made, and how the model's performance can be evaluated. We have also discovered the power of feature selection, by focusing on the most influential features, we were able to create a more efficient model that maintained nearly the same level of accuracy. This highlights the balance between model complexity and performance, a key consideration in machine learning.

In conclusion, we have learned about the power and importance of Decision Tree models, their simplicity, interpretability, and versatility make them a valuable tool in any data scientist's toolkit.

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Naive Bayes Algorithm

The Naive Bayes is fundamental algorithm in the world of machine learning and data science. This probabilistic model, rooted in the principles of Bayesian statistics, is famous and useful for its efficiency, simplicity, and surprisingly robust performance across a wide array of applications. Naive Bayes classifier is based on the Bayes' theorem, which is a principle in probability theory and statistics that describes the relationship of conditional probabilities of statistical quantities, in other words, it provides a way to calculate the probability of an event (variable) based on prior knowledge of another variable. 

The term naive in Naive Bayes refers to the assumption that all features in the dataset used for classification are mutually independent, this assumption simplifies the calculations for the posterior probabilities and makes the algorithm computationally efficient. Although this assumption is not common in real-world data, the Naive Bayes classifier often delivers robust performance, even when there is no independence among features.

There are several types of Naive Bayes models, including Gaussian Naive Bayes, Multinomial Naive Bayes, and Bernoulli Naive Bayes. Each type is suited to different kinds of data, In this post, I focus on the Gaussian Naive Bayes classifier, which assumes that the features follow a normal distribution. This variant is particularly useful when dealing with continuous data. 

How does the Gaussian Naive Bayes classifier work?

The Gaussian Naive Bayes classifier operates under the framework of Bayes' theorem, which in its simplest form can be expressed as:

\begin{equation*} \mathbf{P} \left(Y | X \right)= \frac {\mathbf{P} \left(X | Y \right) \mathbf{P} \left( Y \right)} {\mathbf{P} \left( X \right)} \end{equation*}

In the context of the Gaussian Naive Bayes classifier, Y and X are events where Y is the hypothesis (class label) and X is the evidence (features). $\mathbf{P} \left(Y | X \right)$ is the posterior probability, $\mathbf{P} \left(X | Y \right)$ is the likelihood, $\mathbf{P} \left(Y \right)$ is the prior probability, and $\mathbf{P} \left(X \right)$ is the evidence.

• Posterior probability $P \left(Y | X \right)$: It can be understood as the probability that an instance X (data point with specific features values) belongs to class Y.

• Likelihood $P \left(X | Y \right)$: It can be understood as the probability that, given a certain class label Y, the observed features have been X. In the Gaussian Naive Bayes classifier, this is calculated using the Gaussian (normal) distribution, hence the name.

• Prior probability $P \left(Y \right)$: It is the initial probability of an specific class Y, calculated as the proportion of samples of that class in the training set.

• Evidence $P \left(X \right)$: It is the total probability of the features X. In practice, this term is often ignored during the calculation, because it doesn't affect which class has the highest probability.

The pseudocode is described in the next image, which provides an outline of the basic Naive Bayes algorithm for classification.

Implementing algorithm in Python

According to each dataset, the selection process for the number of neighbors K may be different. In this post the number K will be taken as arbitrary, but in a future post I'll talk about how to correctly select the number K.

Importing Libraries

First, we imported the necessary libraries for the code, which includes:

import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
from matplotlib.colors import ListedColormap
from sklearn.naive_bayes import GaussianNB
from sklearn.metrics import confusion_matrix
from sklearn import datasets
from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score

• numpy: Provides mathematical functions to operate on arrays.

• sklearn: One of the most popular libraries for machine learning in Python. It features many machine learning algorithms, including the Naive Bayes algorithm (imported as GaussianNB). It also provides tools for model fitting, data preprocessing, model selection, model evaluation and other utilities.

• seaborn and matplotlib: These are visualization libraries in Python for creating static, animated, and interactive visualizations, as well as drawing attractive and informative statistical graphs.

Defining Functions

Now that we have imported the necessary libraries, we can proceed to implement the Naive Bayes algorithm. We defined two functions: calculate_parameters and naive_bayes.

# This function calculates the means, variances and prior probabilities
def calculate_parameters(X, y):
    unique_labels = np.unique(y)
    prior_probs = []
    means = []
    variances = []
    for label in unique_labels:
        # Calculating the prior probability of the label
        prob = np.count_nonzero(y == label)/len(y)
        prior_probs.append(prob)
        # Calculating the mean of the features
        mean = X[y==label].mean(axis = 0)
        means.append(mean)
        # Calculating the variance of the features
        variance = X[y==label].var(axis = 0)
        variances.append(variance)
    return np.array(means), np.array(variances), np.array(prior_probs)


# This function apply the Gaussian Naive Bayes algorithm to a test dataset
def naive_bayes(Test_Dataset, Train_Dataset, Train_Labels):
    means, variances, prior_probs = calculate_parameters(Train_Dataset, Train_Labels)
    Predicted_Labels = []

    # Classifying the test dataset
    for x in Test_Dataset:
        post_probs = []
        for i in range(len(prior_probs)):
            # Calculating P(xi | y) for each feature xi
            numerator = np.exp( -(x-means[i])**2 / (2 * variances[i]) )
            denominator = np.sqrt(2 * np.pi * variances[i])
            # Calculating P(x | y)
            p_x_given_y = np.prod(numerator/denominator)
            # Calculating P(y | X) for the class y
            prob_class_i = p_x_given_y*prior_probs[i]
            post_probs.append(prob_class_i)

        # Assigning the class with the highest posterior probability
        probable_class = np.argmax(post_probs)
        Predicted_Labels.append(probable_class)

    return np.array(Predicted_Labels)

The calculate_parameters function takes the training dataset and the corresponding labels as inputs. It calculates the mean and variance for each feature for each class, as well as the prior probability of each class. These parameters are used later when applying the Naive Bayes algorithm.

Next, we define the naive_bayes function, which applies the Naive Bayes algorithm to a test dataset. It takes the test dataset, the training dataset, and the training labels as inputs, and returns the predicted labels for the test dataset. This function calculates the posterior probability for each class for each data point in the test set, based on the parameters calculated by the calculate_parameters function. It then assigns the class with the highest posterior probability to each data point.

Generating and Visualizing Dataset

Now that we have the Naive Bayes classifier ready, we need a dataset to apply it. For this post, we generated a synthetic dataset using the make_circles function from the sklearn library. This function generates a large circle containing a smaller circle in two dimensions. We can control the number of data points and the amount of noise in the data.

# Setting parameters for the dataset
n_points = 3500 # Number of data points
noise = 0.12 # Standard deviation of Gaussian noise added to the data
color_map = ListedColormap(['mediumseagreen','mediumblue']) # Color map for the two classes

# Creating dataset
X,y = datasets.make_circles(n_samples=n_points, noise=noise, factor = 0.6, random_state=0)

After generating the dataset, we used matplotlib to create a scatter plot of the data points. The two classes are represented by different colors.

# Plotting generated dataset
plt.rcParams['font.size'] = '12'
fig, ax = plt.subplots(figsize=(8,6), facecolor='#F5F5F5')
ax.set_facecolor('#F5F5F5')
# Creating a scatter plot
scatter = ax.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap=color_map, alpha = 0.4)
# Adding a legend to the plot
ax.legend(handles=scatter.legend_elements()[0], labels=['0', '1'], title = 'Classes')
ax.set_title("Generated Dataset", fontsize=15)

Splitting the Dataset for Train and Test

After generating the dataset, the next step is to split it into a train set and a test set. The train set was used to train the Naive Bayes classifier (dataset used for calculate the means, variances and prior probabilities), while the test set was used to evaluate the classifier's performance on new data. A common practice is to allocate 75% of the dataset to the train set and 25% to the test set, which is what we did in this case.

# Splitting the dataset into a train set and a test set
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.25, random_state=0)

# Creating a figure with two subplots
fig, ax = plt.subplots(1,2, figsize=(14,5), facecolor='#F5F5F5')
# Plotting the train set
ax[0].set_facecolor('#F5F5F5')
scatter = ax[0].scatter(X_train[:, 0], X_train[:, 1], c=y_train, s=50, cmap=color_map, alpha = 0.4)
ax[0].legend(handles=scatter.legend_elements()[0], labels=['0', '1'], title = 'Classes')
ax[0].set_title("Train Dataset", fontsize=14)
# Plotting the test set
ax[1].set_facecolor('#F5F5F5')
scatter = ax[1].scatter(X_test[:, 0], X_test[:, 1], c=y_test, s=50, cmap=color_map, alpha = 0.4)
ax[1].legend(handles=scatter.legend_elements()[0], labels=['0', '1'], title = 'Classes')
ax[1].set_title("Test Dataset", fontsize=14)

We used the train_test_split function from the sklearn library to perform this split. This function shuffles the dataset and then splits it into train and test sets. Then, we created scatter plots to visualize both sets using the matplotlib library. These plots are shown below:

Applying Naive Bayes Algorithm

We already have the dataset and the Naive Bayes classifier, so we can apply this classifier to the dataset and evaluate its performance.

The naive_bayes function takes the test set, the train set and the train labels as inputs, and returns the predicted labels for the test set as output. We then calculate the accuracy of the predictions using the accuracy_score function from sklearn.

# Applying Naive Bayes algorithm
y_pred = naive_bayes(X_test, X_train, y_train)
print("The accuracy is:", accuracy_score(y_test, y_pred))

We can also use the GaussianNB model from the library sklearn. For this, we created a classifier object, then we fitted it to the train set and finally predicted the labels for the test set. The accuracy of these predictions was also calculated using the accuracy_score function.

# Applying Naive Bayes algorithm using model from Sklearn
bayes_sklearn = GaussianNB()
bayes_sklearn.fit(X_train, y_train)
y_pred = bayes_sklearn.predict(X_test)
print("The accuracy is:", accuracy_score(y_test, y_pred))

For both alternatives, the accuracy of the models was 0.95. 

After making the predictions, we created a confusion matrix using the confusion_matrix function from sklearn. The confusion matrix is a table that is often used to describe the performance of a classification model on a test set for which the true values are known. Then, we visualized the confusion matrix using a heatmap from the seaborn library.

# Creating a confusion matrix
cm = confusion_matrix(y_test, y_pred)
# Visualizing the confusion matrix using a heatmap
fig, ax = plt.subplots(figsize=(7,5), facecolor='#F5F5F5')
ax = sns.heatmap(cm, annot=True, fmt="d", cmap="YlOrBr", annot_kws={"size": 16})
ax.set_xlabel('Predicted labels', fontsize=14)
ax.set_ylabel('True labels', fontsize=14)
ax.set_xticklabels(['Class 0', 'Class 1'], fontsize=13)
ax.set_yticklabels(['Class 0', 'Class 1'], fontsize=13)
plt.show()

The confusion matrix showed that for the class 0, 423 points were classified correctly and 10 points were classified incorrectly. For the class 1, 410 points were classified correctly and 32 points were classified incorrectly. This indicates that the Naive Bayes classifier had a high accuracy. The confusion matrix is shown below:

Bonus: Visualizing the Predicted Labels

After applying the Naive Bayes classifier and evaluating its performance, we plotted the test set with the correct labels and the predicted labels. It helps for a visual understanding of how well the classifier performed.

We created two scatter plots: one for the test set with the correct labels, and another one for the test set with the predicted labels.

# Creating a figure and a set of subplots
fig, ax = plt.subplots(1,2, figsize=(14,5), facecolor='#F5F5F5')

# Plotting the test set with the correct labels
ax[0].set_facecolor('#F5F5F5')
scatter = ax[0].scatter(X_test[:, 0], X_test[:, 1], c=y_test, s=50, cmap=color_map, alpha = 0.4)
ax[0].legend(handles=scatter.legend_elements()[0], labels=['0', '1'], title = 'Classes')
ax[0].set_title("Correct Classes", fontsize=14)

# Creating a new color map for the predicted labels
color_map2 = ListedColormap(['mediumseagreen','mediumblue', 'firebrick'])
# Finding the indices of the points that were classified incorrectly
incorrect_indices = np.where(y_pred != y_test)[0]
# The labels of the points that were classified incorrectly are set to 2
y_plot = y_pred.copy()
y_plot[incorrect_indices] = 2

# Plotting the test set with the predicted labels
ax[1].set_facecolor('#F5F5F5')
scatter = ax[1].scatter(X_test[:, 0], X_test[:, 1], c=y_plot, s=50, cmap=color_map2, alpha = 0.4)
ax[1].legend(handles=scatter.legend_elements()[0], labels=['0', '1', 'Incorrect'], title = 'Classes')
ax[1].set_title("Predicted Classes", fontsize=14)

These graphs were as follows:

In the second plot, the red points are those that were incorrectly classified by the model.

As we have shown, the Naive Bayes algorithm is a powerful and efficient tool for classification tasks. Despite its simplicity and the "naive" assumption of feature independence, it often performs surprisingly well in practice, even when the independence assumption is violated. Its efficiency and scalability make it particularly suitable for large datasets and applications where speed is crucial.

Through this post, we have seen how the Naive Bayes model can be implemented from scratch and applied to a synthetic dataset. We have also compared its performance with the implementation provided by the sklearn library. This exploration has demonstrated the practicality and effectiveness of the Naive Bayes algorithm. As with any machine learning model, it's important to understand its strengths and limitations, in order to consider them when we select an algorithm for a specific task.

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