#### Overview of this book

Machine learning applications are highly automated and self-modifying, and continue to improve over time with minimal human intervention, as they learn from the trained data. To address the complex nature of various real-world data problems, specialized machine learning algorithms have been developed. Through algorithmic and statistical analysis, these models can be leveraged to gain new knowledge from existing data as well. Data Science Algorithms in a Week addresses all problems related to accurate and efficient data classification and prediction. Over the course of seven days, you will be introduced to seven algorithms, along with exercises that will help you understand different aspects of machine learning. You will see how to pre-cluster your data to optimize and classify it for large datasets. This book also guides you in predicting data based on existing trends in your dataset. This book covers algorithms such as k-nearest neighbors, Naive Bayes, decision trees, random forest, k-means, regression, and time-series analysis. By the end of this book, you will understand how to choose machine learning algorithms for clustering, classification, and regression and know which is best suited for your problem
Title Page
Packt Upsell
Contributors
Preface
Free Chapter
Classification Using K-Nearest Neighbors
Time Series Analysis
Python Reference
Statistics
Glossary of Algorithms and Methods in Data Science
Other Books You May Enjoy
Index

## Bayes' theorem and its extension

In this section, we will state and prove both Bayes' theorem and its extension.

### Bayes' theorem

Bayes' theorem states the following:

#### Proof

We can prove this theorem by using elementary set theory on the probability spaces of the events A and B. In other words, here, a probability event will be defined as a set of the possible outcomes in the probability space:

Figure 2.1: Probability space for the two events

As you can see in the preceding diagram, we can state the following relationships:

Rearranging these relationships, we get the following:

This is, in fact, Bayes' theorem:

This concludes the proof.

### Extended Bayes' theorem

We can extend Bayes' theorem by taking more probability events into consideration. Suppose that the events B1,…,Bn are conditionally independent given A. Let ~A denote the complement of A. Then, we have the following:

#### Proof

Since the events B1,…,Bn are conditionally independent given A (and also given ~A), we get the following:

Applying the simple...