Book Image

Learning Bayesian Models with R

By : Hari Manassery Koduvely
Book Image

Learning Bayesian Models with R

By: Hari Manassery Koduvely

Overview of this book

Bayesian Inference provides a unified framework to deal with all sorts of uncertainties when learning patterns form data using machine learning models and use it for predicting future observations. However, learning and implementing Bayesian models is not easy for data science practitioners due to the level of mathematical treatment involved. Also, applying Bayesian methods to real-world problems requires high computational resources. With the recent advances in computation and several open sources packages available in R, Bayesian modeling has become more feasible to use for practical applications today. Therefore, it would be advantageous for all data scientists and engineers to understand Bayesian methods and apply them in their projects to achieve better results. Learning Bayesian Models with R starts by giving you a comprehensive coverage of the Bayesian Machine Learning models and the R packages that implement them. It begins with an introduction to the fundamentals of probability theory and R programming for those who are new to the subject. Then the book covers some of the important machine learning methods, both supervised and unsupervised learning, implemented using Bayesian Inference and R. Every chapter begins with a theoretical description of the method explained in a very simple manner. Then, relevant R packages are discussed and some illustrations using data sets from the UCI Machine Learning repository are given. Each chapter ends with some simple exercises for you to get hands-on experience of the concepts and R packages discussed in the chapter. The last chapters are devoted to the latest development in the field, specifically Deep Learning, which uses a class of Neural Network models that are currently at the frontier of Artificial Intelligence. The book concludes with the application of Bayesian methods on Big Data using the Hadoop and Spark frameworks.
Table of Contents (11 chapters)
10
Index

Exercises

  1. By using the definition of conditional probability, show that any multivariate joint distribution of N random variables Exercises has the following trivial factorization:
    Exercises
  2. The bivariate normal distribution is given by:
    Exercises

    Here:

    Exercises

    By using the definition of conditional probability, show that the conditional distribution Exercises can be written as a normal distribution of the form Exercises where Exercises and Exercises.

  3. By using explicit integration of the expression in exercise 2, show that the marginalization of bivariate normal distribution will result in univariate normal distribution.
  4. In the following table, a dataset containing the measurements of petal and sepal sizes of 15 different Iris flowers are shown (taken from the Iris dataset, UCI machine learning dataset repository). All units are in cms:

    Sepal Length

    Sepal Width

    Petal Length

    Petal Width

    Class of Flower

    5.1

    3.5

    1.4

    0.2

    Iris-setosa

    4.9

    3

    1.4

    0.2

    Iris-setosa

    4.7

    3.2

    1.3

    0.2

    Iris-setosa

    4.6

    3.1

    1.5

    0.2

    Iris-setosa

    5

    3.6

    1.4

    0.2

    Iris-setosa

    7

    3.2

    4.7

    1.4

    Iris-versicolor

    6.4

    3.2

    4.5

    1.5

    Iris-versicolor

    6.9

    3.1

    4.9

    1.5

    Iris-versicolor

    5.5

    2.3

    4

    1.3

    Iris-versicolor

    6.5

    2.8

    4.6

    1.5

    Iris-versicolor

    6.3

    3.3

    6

    2.5

    Iris-virginica

    5.8

    2.7

    5.1

    1.9

    Iris-virginica

    7.1

    3

    5.9

    2.1

    Iris-virginica

    6.3

    2.9

    5.6

    1.8

    Iris-virginica

    6.5

    3

    5.8

    2.2

    Iris-virginica

    Answer the following questions:

    1. What is the probability of finding flowers with a sepal length more than 5 cm and a sepal width less than 3 cm?
    2. What is the probability of finding flowers with a petal length less than 1.5 cm; given that petal width is equal to 0.2 cm?
    3. What is the probability of finding flowers with a sepal length less than 6 cm and a petal width less than 1.5 cm; given that the class of the flower is Iris-versicolor?