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

Here:

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

By using explicit integration of the expression in exercise 2, show that the marginalization of bivariate normal distribution will result in univariate normal distribution.

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:

What is the probability of finding flowers with a sepal length more than 5 cm and a sepal width less than 3 cm?

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?

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?

#### Learning Bayesian Models with R

##### By :

#### Learning Bayesian Models with R

##### By:

#### Overview of this book

Table of Contents (16 chapters)

Learning Bayesian Models with R

Credits

About the Author

About the Reviewers

www.PacktPub.com

Preface

Free Chapter

Introducing the Probability Theory

The R Environment

Introducing Bayesian Inference

Machine Learning Using Bayesian Inference

Bayesian Regression Models

Bayesian Classification Models

Bayesian Models for Unsupervised Learning

Bayesian Neural Networks

Bayesian Modeling at Big Data Scale

Index

Customer Reviews