**Meet the editor**

Mohammad Saber Fallah Nezhad is an associate professor at Yazd University, Iran. He received his BS, MS, and PhD degrees, all in Industrial Engineering, from Sharif University of Technology, Tehran, Iran, under the supervision of Professor Akhavan Niaki. He was a visiting researcher at the Karlsruhe University under the supervision of Professor Juergen Branke. Also, he was award-

ed a silver medal in the 16th National Mathematics Olympiad in Iran. Dr. Fallah Nezhad was ranked first and eighth in the graduate national university comprehensive exam in System Management and Industrial Engineering in Iran, respectively. He was ranked 47th among all high school graduates in Iran. His areas of interest include dynamic programming, quality control, Bayesian inference, and operations research.

Contents

**Preface VII**

**RNA-seq Data 7**

**Analysis Models 21**

**System Analysis 41** Shigeru Kashiwaya

Douglas S. McNair

Bernd Panassiti

**Detection 109**

Chapter 1 **Introductory Chapter: Bayesian Thinking 1** Mohammad Saber Fallah Nezhad

Sunghee Oh and Seongho Song

Chapter 3 **Bayesian Analysis for Hidden Markov Factor**

Chapter 2 **Bayesian Modeling Approaches for Temporal Dynamics in**

Yemao Xia, Xiaoqian Zeng and Niansheng Tang

Chapter 4 **Dynamic Process Model Parameter Estimation by Global**

**Assessing Fairness in Machine-Learning**

Chapter 6 **Using Bayesian Inference to Investigate the Influence of**

Chapter 7 **A Bayesian Hau-Kashyap Approach for Hepatitis Disease**

Y.C. Tang and Moamin A. Mahmoud

**Environmental Factors on a Phytoplasma Disease 97**

Andino Maseleno, Rohmah Zahroh Hidayati, Marini Othman, Alicia

**Decision-Support Models 71**

Chapter 5 **Preventing Disparities: Bayesian and Frequentist Methods for**

## Contents

#### **Preface XI**

**Detection 109** Andino Maseleno, Rohmah Zahroh Hidayati, Marini Othman, Alicia Y.C. Tang and Moamin A. Mahmoud

Preface

in most real issues.

who have contributed to these materials.

This book is an introduction to the mathematical analysis of Bayesian decision-making when the state of the problem is unknown but further data about it can be obtained. The objective of such analysis is to determine the optimal decision or solution that is logically consistent with the preferences of the decision-maker, that can be analyzed using numerical utilities or criteria with the probabilities assigned to the possible state of the problem, such

It is seldom possible to take findings and methodologies from a basic conceptual framework and apply them directly to a real problem. Often, the findings of another field must be adapted to the problem. Thus, we need a methodology that can refine the results and data from different fields in an analytical framework, and lead these data to an appropriate sol‐ ution for the problem at hand. Since every kind of information, including qualitative or quantitative or fuzzy data, can be used in Bayesian thinking, this technique can also be used

The book focuses on the different models of Bayesian inference. It provides a description of methodologies along with the different approaches to Bayesian thinking. Throughout, the text concentrates on the results, rather than on the mathematical detail, but every effort has been made to elaborate the theoretical concepts and methodologies. A somewhat more de‐ tailed description of the topics covered in individual chapters may be needed for those who want to investigate about certain methodologies and basic formulations. The book arose from the materials contributed by scholars in the field of Bayesian inference. We thank all

**Mohammad Saber Fallah Nezhad**

Yazd University Yazd, Iran

that these probabilities are updated by gathering new information.

## Preface

This book is an introduction to the mathematical analysis of Bayesian decision-making when the state of the problem is unknown but further data about it can be obtained. The objective of such analysis is to determine the optimal decision or solution that is logically consistent with the preferences of the decision-maker, that can be analyzed using numerical utilities or criteria with the probabilities assigned to the possible state of the problem, such that these probabilities are updated by gathering new information.

It is seldom possible to take findings and methodologies from a basic conceptual framework and apply them directly to a real problem. Often, the findings of another field must be adapted to the problem. Thus, we need a methodology that can refine the results and data from different fields in an analytical framework, and lead these data to an appropriate sol‐ ution for the problem at hand. Since every kind of information, including qualitative or quantitative or fuzzy data, can be used in Bayesian thinking, this technique can also be used in most real issues.

The book focuses on the different models of Bayesian inference. It provides a description of methodologies along with the different approaches to Bayesian thinking. Throughout, the text concentrates on the results, rather than on the mathematical detail, but every effort has been made to elaborate the theoretical concepts and methodologies. A somewhat more de‐ tailed description of the topics covered in individual chapters may be needed for those who want to investigate about certain methodologies and basic formulations. The book arose from the materials contributed by scholars in the field of Bayesian inference. We thank all who have contributed to these materials.

> **Mohammad Saber Fallah Nezhad** Yazd University Yazd, Iran

**Chapter 1**

Provisional chapter

**Introductory Chapter: Bayesian Thinking**

Bayesian inference is developed based on the simple Bayesian rule in the probability theory but this method of thinking is one of the most important findings in the history of science. It means that we can modify our beliefs about the nature by gathering data from phenomenon or by analyzing the behavior of people around us or by investigating about historical events.

DOI: 10.5772/intechopen.75053

We start with an estimate of the probability that any claim, belief, hypothesis is true, then look at any new data and update the probability given the new data. Bayes' theorem is a method for analyzing the correctness of beliefs (hypotheses, claims and propositions) based on the best available evidence (observations, data, information). Here is the basic description: initial belief

We change our beliefs with objective information: initial beliefs + new objective data = posterior belief. Each time the system is updated, the posterior becomes the prior of the new stage. It was an evolving system; every bit of new information was closer to correct solution. This technique can be expressed both mathematically and philosophically about how we learn about the universe: that we learn about it through approximation, getting closer and closer to the fact as

Bayesian thinking is based on the idea that we can increase our information about a physical situation than is contained in the data from a single experiment. Bayesian methods can be applied for analyzing the data from different experiments, for example. In other situations, there may be sound reasons about the allowable values that can be assigned to a parameter. But often the data are scarce or noisy or biased or all of these. Experimental results are compared with predicted values, and the predictions are modified by arbitrarily subtracting off the discrepancy based on the difference. When new data are collected, once again the values disagree with predictions, and another "correction" is applied, leading to an aggregate of ad hoc tweaks, certainly not best practice, however common. But Bayesian methods can be

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Introductory Chapter: Bayesian Thinking

Mohammad Saber Fallah Nezhad

Mohammad Saber Fallah Nezhad

http://dx.doi.org/10.5772/intechopen.75053

plus new evidence = improved belief.

used here to avoid these heuristics [2].

we gather more evidence [1].

1. Introduction

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

#### **Chapter 1** Provisional chapter

#### **Introductory Chapter: Bayesian Thinking** Introductory Chapter: Bayesian Thinking

#### Mohammad Saber Fallah Nezhad Mohammad Saber Fallah Nezhad

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.75053

#### 1. Introduction

Bayesian inference is developed based on the simple Bayesian rule in the probability theory but this method of thinking is one of the most important findings in the history of science. It means that we can modify our beliefs about the nature by gathering data from phenomenon or by analyzing the behavior of people around us or by investigating about historical events.

DOI: 10.5772/intechopen.75053

We start with an estimate of the probability that any claim, belief, hypothesis is true, then look at any new data and update the probability given the new data. Bayes' theorem is a method for analyzing the correctness of beliefs (hypotheses, claims and propositions) based on the best available evidence (observations, data, information). Here is the basic description: initial belief plus new evidence = improved belief.

We change our beliefs with objective information: initial beliefs + new objective data = posterior belief. Each time the system is updated, the posterior becomes the prior of the new stage. It was an evolving system; every bit of new information was closer to correct solution. This technique can be expressed both mathematically and philosophically about how we learn about the universe: that we learn about it through approximation, getting closer and closer to the fact as we gather more evidence [1].

Bayesian thinking is based on the idea that we can increase our information about a physical situation than is contained in the data from a single experiment. Bayesian methods can be applied for analyzing the data from different experiments, for example. In other situations, there may be sound reasons about the allowable values that can be assigned to a parameter. But often the data are scarce or noisy or biased or all of these. Experimental results are compared with predicted values, and the predictions are modified by arbitrarily subtracting off the discrepancy based on the difference. When new data are collected, once again the values disagree with predictions, and another "correction" is applied, leading to an aggregate of ad hoc tweaks, certainly not best practice, however common. But Bayesian methods can be used here to avoid these heuristics [2].

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In fact, we all employ Bayesian inference in decision-making about the real problems in life. When we want to select a solution for a problem among several solutions, then we usually analyze the past data from each solution or we try to predict the outcome of each solution then the solution with higher success probability is selected. Thus, this process is very similar to Bayesian thinking when the decisions are selected based on posterior probabilities updated by Bayesian formula.

In fact, understanding the secret of mankind is still the most important challenge for humans, why the creature is so complex, and why some people come to a point where absolute destruction is called. Perhaps this is because learning the right way to think about is the most important duty of wise people who have been missing out on this training. Perhaps this should

Introductory Chapter: Bayesian Thinking http://dx.doi.org/10.5772/intechopen.75053 3

If we cannot instruct the right way of thinking to humans or we teach wrong beliefs because of the economic benefits and so on, then we have unknowingly created monsters that will once again come back to their sources. If scientists could teach the right thinking to all human beings, then there would never be imaginary thoughts and superstition that this illusion would be the root of the misery of the human race. The most dangerous illusion in the present world is to think that our beliefs and words are based on religion and divine law and it is in the interest of God and the law of nature, while even the foundations of our thinking are verged on

Even if one can analyze the events of human societies on the basis of Bayesian thinking, these methods cannot be implemented because the assumptions of human societies are based on concepts that cannot be explained by the use of classical science, and this may be the main reason for human disasters in the human history. The happiness of man depends on being able

But, we should note that if a wrong thought takes a lot of power, any wrong decision will cause great harm, and it is better to try to survive in this situation. If we look carefully at the present situation of the world and the leaders of the great powers, the equilibrium point of power warfare based on historical data will be probably in a difficult situation unless the wise men take over the world's affairs. If the wrong information is given to even a good person by the time gradually, then the soul of the individual will lose the ability to understand the truth at all because he becomes accustomed to the wrong information, and ultimately, it will be like a human with evil spirit. Usually, when a great mistake or sin is committed, most people will fall

Almost all ethnicities in the world with every religion have proven that their brutality is not limited; therefore, if these brutal thoughts are to be strengthened and combined simultaneously in all parts of the world, it will be a matter of concern. Maybe that is why the talk of the global village can have a reverse effect. When we cannot understand the alarming data in the present world, it may be the time to destroy all the ethical issues of the human race.

Thus, it is possible that many wars in the history of human have occurred due to wrong information while this information is generated by the winners of the wars before the wars.

When the human runs into a lot of trouble so that he cannot tolerate them then unconsciously he will think of supernatural power and ask for help. Perhaps understanding this vague relationship is the most important point in human intervention and fate, while God is not responsible for providing a life with prosperity and happiness to man, but he is surely respon-

Without ethical issues, there would be no reason for human interaction.

stay as secret because of unknown reasons.

actual sacrilege.

to think right.

asleep and false data are generated.

sible for ensuring a decent life for man.

It is seldom possible to take findings and methodologies from a basic conceptual framework and apply them directly to a real problem. Often, the findings of another field must be adapted to the problem. Thus, we need a methodology that can refine the results and data from different fields in an analytical framework such that these data lead to an appropriate solution for the problem at hand. Since every kind of information, including qualitative or quantitative or fuzzy data can be used in Bayesian thinking, so this technique can be used in most real issues.

When choosing the right choice from several available options, we need to analyze the observations from each of the options. Now, if there is some kind of competition between options, we need to balance them somehow so that each of the options can produce observations close to their actual performance. This is the reason for combining game theory and Bayesian inference in real cases.

Also, mathematical methodologies such as game theory and dynamic programming and others should be redefined using Bayesian thinking. For example, in dynamic planning, since decisions are selected stage by stage, after observing information in each stage, we can update the probability functions of random variables and solve the problem for the number of remaining stages that may lead to a change in optimal policy. In the game theory, it can be assumed that after one stage of game and observing the rewards of each decision, it is possible to replay the game, assuming that rewards are random variable such that their probability function is determined based on past data and the optimal decision after each stage may change. The author thinks that three concepts of dynamic programming and game theory and Bayesian inference should be considered in the formulation of each real problem because the interests of decision-makers in each problem are different and the decisions are effective on the future and the next stages and the new data will be available in each new stage thus a new mathematical formulation is needed regarding these issues.

The main disadvantage of Bayesian thinking is the existence of false information that cripples the mechanism of this method and results in disastrous issues in real terms. It should be noted that our view of life is based on the information we receive and, if there is a deviation in the part of the information received, will lead to wrong decisions. Then, we have to treat the human as mad people or animals where his only sin is to receive false information. Maybe, this is the reason for thinking about the paradise for the poor people.

When there is a lot of false information around us, we can only hope for God to provide small light in this darkness. In the case of sacred books, it should be noted that the prophets knew that these books would also be read by their enemies, so the facts are usually in the veil of insight and understanding of the facts that require consideration in many complications.

In fact, understanding the secret of mankind is still the most important challenge for humans, why the creature is so complex, and why some people come to a point where absolute destruction is called. Perhaps this is because learning the right way to think about is the most important duty of wise people who have been missing out on this training. Perhaps this should stay as secret because of unknown reasons.

In fact, we all employ Bayesian inference in decision-making about the real problems in life. When we want to select a solution for a problem among several solutions, then we usually analyze the past data from each solution or we try to predict the outcome of each solution then the solution with higher success probability is selected. Thus, this process is very similar to Bayesian thinking when the decisions are selected based on posterior probabilities updated by

It is seldom possible to take findings and methodologies from a basic conceptual framework and apply them directly to a real problem. Often, the findings of another field must be adapted to the problem. Thus, we need a methodology that can refine the results and data from different fields in an analytical framework such that these data lead to an appropriate solution for the problem at hand. Since every kind of information, including qualitative or quantitative or fuzzy data can be used in Bayesian thinking, so this technique can be used in

When choosing the right choice from several available options, we need to analyze the observations from each of the options. Now, if there is some kind of competition between options, we need to balance them somehow so that each of the options can produce observations close to their actual performance. This is the reason for combining game theory and Bayesian

Also, mathematical methodologies such as game theory and dynamic programming and others should be redefined using Bayesian thinking. For example, in dynamic planning, since decisions are selected stage by stage, after observing information in each stage, we can update the probability functions of random variables and solve the problem for the number of remaining stages that may lead to a change in optimal policy. In the game theory, it can be assumed that after one stage of game and observing the rewards of each decision, it is possible to replay the game, assuming that rewards are random variable such that their probability function is determined based on past data and the optimal decision after each stage may change. The author thinks that three concepts of dynamic programming and game theory and Bayesian inference should be considered in the formulation of each real problem because the interests of decision-makers in each problem are different and the decisions are effective on the future and the next stages and the new data will be available in each new stage thus a new

The main disadvantage of Bayesian thinking is the existence of false information that cripples the mechanism of this method and results in disastrous issues in real terms. It should be noted that our view of life is based on the information we receive and, if there is a deviation in the part of the information received, will lead to wrong decisions. Then, we have to treat the human as mad people or animals where his only sin is to receive false information. Maybe,

When there is a lot of false information around us, we can only hope for God to provide small light in this darkness. In the case of sacred books, it should be noted that the prophets knew that these books would also be read by their enemies, so the facts are usually in the veil of insight and understanding of the facts that require consideration in many complications.

mathematical formulation is needed regarding these issues.

this is the reason for thinking about the paradise for the poor people.

Bayesian formula.

2 New Insights into Bayesian Inference

most real issues.

inference in real cases.

If we cannot instruct the right way of thinking to humans or we teach wrong beliefs because of the economic benefits and so on, then we have unknowingly created monsters that will once again come back to their sources. If scientists could teach the right thinking to all human beings, then there would never be imaginary thoughts and superstition that this illusion would be the root of the misery of the human race. The most dangerous illusion in the present world is to think that our beliefs and words are based on religion and divine law and it is in the interest of God and the law of nature, while even the foundations of our thinking are verged on actual sacrilege.

Even if one can analyze the events of human societies on the basis of Bayesian thinking, these methods cannot be implemented because the assumptions of human societies are based on concepts that cannot be explained by the use of classical science, and this may be the main reason for human disasters in the human history. The happiness of man depends on being able to think right.

But, we should note that if a wrong thought takes a lot of power, any wrong decision will cause great harm, and it is better to try to survive in this situation. If we look carefully at the present situation of the world and the leaders of the great powers, the equilibrium point of power warfare based on historical data will be probably in a difficult situation unless the wise men take over the world's affairs. If the wrong information is given to even a good person by the time gradually, then the soul of the individual will lose the ability to understand the truth at all because he becomes accustomed to the wrong information, and ultimately, it will be like a human with evil spirit. Usually, when a great mistake or sin is committed, most people will fall asleep and false data are generated.

Almost all ethnicities in the world with every religion have proven that their brutality is not limited; therefore, if these brutal thoughts are to be strengthened and combined simultaneously in all parts of the world, it will be a matter of concern. Maybe that is why the talk of the global village can have a reverse effect. When we cannot understand the alarming data in the present world, it may be the time to destroy all the ethical issues of the human race. Without ethical issues, there would be no reason for human interaction.

Thus, it is possible that many wars in the history of human have occurred due to wrong information while this information is generated by the winners of the wars before the wars.

When the human runs into a lot of trouble so that he cannot tolerate them then unconsciously he will think of supernatural power and ask for help. Perhaps understanding this vague relationship is the most important point in human intervention and fate, while God is not responsible for providing a life with prosperity and happiness to man, but he is surely responsible for ensuring a decent life for man.

### 2. Bayesian thinking at the time of Chaos in society

In common usage, "chaos" means "a state of disorder." However, in chaos theory, the term is defined more precisely. Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic systems are predictable for a while and then appear to become random. Chaos can occur in societies that can effect on the human history and these effects must be analyzed in order to prevent any harm in the future.

simply, which is recognized as an advantage of the current world, but false and malicious

Introductory Chapter: Bayesian Thinking http://dx.doi.org/10.5772/intechopen.75053 5

Perhaps the main reason for the misconduct of human beliefs and thoughts is that the received information is wrong, leading to wrong decisions and behaviors. Repeating these mistakes results in the stress that it will slowly lead to a mental problem, so these people will also be victimized by the complex order prevailing in the world. The major problem is that these people usually do not know about their own deterioration, and the behaviors that are legiti-

Therefore, living in a time of chaos in the community will be accompanied by a lot of troubles and problems, because most people will be away from humanity and there will be plenty of psychological harm to the people of the society, whose effects will remain for the next generations. But any chaos ultimately leads to the order. If we can analyze the data of the chaos effects on human, and analyze the cause of the destruction of the social, religious, and cultural foundations, then the principles of the new order can be effectively designed so that better people can be trained. People who remain loyal to ethical issues in the most difficult conditions. Perhaps this ideal goal cannot be achieved, but it seems to be the only way for human in the near future.

Given the current state of the world, it can be said that the current situation in the world is rapidly jumping to an intensive chaos. In fact, what is worrying is that this disorder is being systematically created by world powers, and they certainly are aware of the disadvantages and effects of this chaos and there is no justifiable reason to create this chaos at this very high level. It seems that key and critical problems are solved based on instantaneous and emotional decisions. Mistaken decisions under butterfly effect may lead to global disruptions, such that no wise

What is mentioned in the foregoing is only my own perceptions of Bayesian thinking that is

[1] https://gainweightjournal.com/bayesian-thinking-if-you-want-to-be-a-critical-thinker-you-

information is also simply exchanged.

person can solve them.

Author details

References

based on limited experiences and studies.

need-to-understand-this-concept/

Address all correspondence to<EMAIL_ADDRESS>
Department of Industrial Engineering, Yazd University, Yazd, Iran

[2] https://www.statisticalengineering.com/bayes\_thinking.htm

Mohammad Saber Fallah Nezhad

mate for them may be very hideous on the basis of divine law.

The basis of all the sciences and knowledge is the order of the universe that the purpose of the knowledge is the discovery of this order. One of the important methods for discovering how the world order is ruling is to employ empirical method that, based on the data collected, we can identify the pattern in the behavior of phenomena in order to anticipate patterns and future relationships, or even to design and optimize, according to these detected relations.

But we know that there will be a chaos in some conditions, which, based on the concept of chaos theory, it will lead to disorder in order to create a new order. The chaos theory is a complicated and disputed mathematical theory that seeks to explain the effect of seemingly insignificant factors. The chaos theory name originates from the idea that the theory can give an explanation for chaotic or random occurrences. When the data are analyzed at the time of the disorder, no specific conception and result can be observed, since the data are not produced based on a specific pattern and they do not have a specific behavior.

But it can be said that the more the disorder is, the new order will be more important. Also, when a disorder occurs in a community, most people in the community will experience a psychological breakdown in a variety of ways, which can be stressful. Anyway, the greater the degree of disorder, the greater the damage done to these individuals.

Such people usually have very contradictory behaviors and are quite isolated and alone, but it can be said that they have a stronger understanding of their surroundings and, when they cannot adapt the conditions of society to their logical criteria, they experience a kind of paradox and contradiction. Thus these Attitudes with respect to the conditions of society and their stress will lead to some kind of psychological breakdown.

Such people usually have to live in a fantasy world, and this imagination will lead to a departure from the real situation. These people are somehow the victim of this new order, but the data that they produce will be quite close to the features of the new order because their damaged psyche grows and reconstructs in accordance with the new order, and the insinuation of these people takes form in the new order.

Therefore, it can be said that the existential essence of these people actually represents the new order that is being created. So, if the psychosocial vulnerability of these people is analyzed accurately, according to the data produced by these individuals, one can identify problems of the new order.

But the feature of new orders in the present world is their complexity. In fact, understanding the behavior of such people and analysis is not simply possible, because the truth is never told and what is seen in public is more of a show. Perhaps the root cause of the mental problems in the current societies is generation of false beliefs. Because every information is transmitted simply, which is recognized as an advantage of the current world, but false and malicious information is also simply exchanged.

Perhaps the main reason for the misconduct of human beliefs and thoughts is that the received information is wrong, leading to wrong decisions and behaviors. Repeating these mistakes results in the stress that it will slowly lead to a mental problem, so these people will also be victimized by the complex order prevailing in the world. The major problem is that these people usually do not know about their own deterioration, and the behaviors that are legitimate for them may be very hideous on the basis of divine law.

Therefore, living in a time of chaos in the community will be accompanied by a lot of troubles and problems, because most people will be away from humanity and there will be plenty of psychological harm to the people of the society, whose effects will remain for the next generations. But any chaos ultimately leads to the order. If we can analyze the data of the chaos effects on human, and analyze the cause of the destruction of the social, religious, and cultural foundations, then the principles of the new order can be effectively designed so that better people can be trained. People who remain loyal to ethical issues in the most difficult conditions. Perhaps this ideal goal cannot be achieved, but it seems to be the only way for human in the near future.

Given the current state of the world, it can be said that the current situation in the world is rapidly jumping to an intensive chaos. In fact, what is worrying is that this disorder is being systematically created by world powers, and they certainly are aware of the disadvantages and effects of this chaos and there is no justifiable reason to create this chaos at this very high level. It seems that key and critical problems are solved based on instantaneous and emotional decisions. Mistaken decisions under butterfly effect may lead to global disruptions, such that no wise person can solve them.

What is mentioned in the foregoing is only my own perceptions of Bayesian thinking that is based on limited experiences and studies.

### Author details

2. Bayesian thinking at the time of Chaos in society

4 New Insights into Bayesian Inference

must be analyzed in order to prevent any harm in the future.

In common usage, "chaos" means "a state of disorder." However, in chaos theory, the term is defined more precisely. Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic systems are predictable for a while and then appear to become random. Chaos can occur in societies that can effect on the human history and these effects

The basis of all the sciences and knowledge is the order of the universe that the purpose of the knowledge is the discovery of this order. One of the important methods for discovering how the world order is ruling is to employ empirical method that, based on the data collected, we can identify the pattern in the behavior of phenomena in order to anticipate patterns and future relationships, or even to design and optimize, according to these detected relations.

But we know that there will be a chaos in some conditions, which, based on the concept of chaos theory, it will lead to disorder in order to create a new order. The chaos theory is a complicated and disputed mathematical theory that seeks to explain the effect of seemingly insignificant factors. The chaos theory name originates from the idea that the theory can give an explanation for chaotic or random occurrences. When the data are analyzed at the time of the disorder, no specific conception and result can be observed, since the data are not pro-

But it can be said that the more the disorder is, the new order will be more important. Also, when a disorder occurs in a community, most people in the community will experience a psychological breakdown in a variety of ways, which can be stressful. Anyway, the greater the degree of

Such people usually have very contradictory behaviors and are quite isolated and alone, but it can be said that they have a stronger understanding of their surroundings and, when they cannot adapt the conditions of society to their logical criteria, they experience a kind of paradox and contradiction. Thus these Attitudes with respect to the conditions of society and their stress will lead to some

Such people usually have to live in a fantasy world, and this imagination will lead to a departure from the real situation. These people are somehow the victim of this new order, but the data that they produce will be quite close to the features of the new order because their damaged psyche grows and reconstructs in accordance with the new order, and the insinua-

Therefore, it can be said that the existential essence of these people actually represents the new order that is being created. So, if the psychosocial vulnerability of these people is analyzed accurately, according to the data produced by these individuals, one can identify problems of

But the feature of new orders in the present world is their complexity. In fact, understanding the behavior of such people and analysis is not simply possible, because the truth is never told and what is seen in public is more of a show. Perhaps the root cause of the mental problems in the current societies is generation of false beliefs. Because every information is transmitted

duced based on a specific pattern and they do not have a specific behavior.

disorder, the greater the damage done to these individuals.

kind of psychological breakdown.

the new order.

tion of these people takes form in the new order.

Mohammad Saber Fallah Nezhad

Address all correspondence to<EMAIL_ADDRESS>
Department of Industrial Engineering, Yazd University, Yazd, Iran

#### References

**Chapter 2**

**Provisional chapter**

**Bayesian Modeling Approaches for Temporal Dynamics**

Analysis of differential expression has been a central role to address the variety of biological questions in the manner to characterize abnormal patterns of cellular and molecular functions for last decades. To date, identification of differentially expressed genes and isoforms has been more intensively focused on temporal dynamics over a series of time points. Bayesian strategies have been successfully employed to uncover the complexity of biological interest with the methodological and analytical perspectives for the various platforms of high-throughput data, for instance, methods in differential expression analysis and network modules in transcriptome data, peak-callers in ChipSeq data, target prediction in microRNA data and meta-methods between different platforms. In this chapter, we will discuss how our methodological works based on Bayesian models address important questions to arise in the architecture of temporal dynamics in RNA-

**Keywords:** hierarchical Dirichlet Bayesian mixture model, Poisson Gamma auto

**Bayesian Modeling Approaches for Temporal Dynamics**

DOI: 10.5772/intechopen.73062

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

The differential expression analysis across external conditions (e.g. drug treatments, or between- or within-cell/tissue types) in stimuli-response data has long been crucial part on clinical applications [1–11]. The primary goal of these studies is to target therapeutic effects on genes and their pathways that are highly associated with the alterations between different conditions, corresponding underlying biological mechanisms, and condition-specific molecular processes from microarray until recent RNA-seq platform [1–3, 5, 6, 9–43]. Such substantial effects on transcriptome data involved by various types of human diseases have

**in RNA-seq Data**

**Abstract**

seq data.

**1. Introduction**

**in RNA-seq Data**

Sunghee Oh and Seongho Song

Sunghee Oh and Seongho Song

http://dx.doi.org/10.5772/intechopen.73062

Additional information is available at the end of the chapter

regressive model, temporal dynamics, RNA-seq

Additional information is available at the end of the chapter

**Provisional chapter**

#### **Bayesian Modeling Approaches for Temporal Dynamics in RNA-seq Data in RNA-seq Data**

**Bayesian Modeling Approaches for Temporal Dynamics**

DOI: 10.5772/intechopen.73062

Sunghee Oh and Seongho Song

Sunghee Oh and Seongho Song Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.73062

#### **Abstract**

Analysis of differential expression has been a central role to address the variety of biological questions in the manner to characterize abnormal patterns of cellular and molecular functions for last decades. To date, identification of differentially expressed genes and isoforms has been more intensively focused on temporal dynamics over a series of time points. Bayesian strategies have been successfully employed to uncover the complexity of biological interest with the methodological and analytical perspectives for the various platforms of high-throughput data, for instance, methods in differential expression analysis and network modules in transcriptome data, peak-callers in ChipSeq data, target prediction in microRNA data and meta-methods between different platforms. In this chapter, we will discuss how our methodological works based on Bayesian models address important questions to arise in the architecture of temporal dynamics in RNAseq data.

**Keywords:** hierarchical Dirichlet Bayesian mixture model, Poisson Gamma auto regressive model, temporal dynamics, RNA-seq

#### **1. Introduction**

The differential expression analysis across external conditions (e.g. drug treatments, or between- or within-cell/tissue types) in stimuli-response data has long been crucial part on clinical applications [1–11]. The primary goal of these studies is to target therapeutic effects on genes and their pathways that are highly associated with the alterations between different conditions, corresponding underlying biological mechanisms, and condition-specific molecular processes from microarray until recent RNA-seq platform [1–3, 5, 6, 9–43]. Such substantial effects on transcriptome data involved by various types of human diseases have

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

significantly addressed fundamental issues characterized by biological phenomena in transcriptional regulation. For instance, examination of classification of subtypes on hereditary breast and ovary cancer [44–46], reciprocal phylogenetic conservation and heterogeneity between closest animal models of aging and depression in brain tissues [47–49], identification of differential expression on enzyme effect in Gaucher's disease across distinct three tissues [14, 50], developmental transient patterns in mouse embryonic stem cells in pre-frontal cortex and limb tissues [13, 51], and et cetera.

developmental stages; multi-series of factorial time course with multiple external conditions at each time point; cell-cyclic periodical data with (or without) external stimuli [7, 8, 22, 23,

Bayesian Modeling Approaches for Temporal Dynamics in RNA-seq Data

http://dx.doi.org/10.5772/intechopen.73062

9

To date, despite the substantial potential and significance to explore temporal dynamics at genes and other genomic features in various human disease progressive models and therapeutic effects, the lack of analytical methodologies in order to precisely characterize temporal dynamics has been an important challenging issue to better understand biological mechanisms relevant with both time specific and responsively altered changes by given external

In this chapter, we propose Bayesian approaches to better infer differential expression in temporal and spatial dynamic regulation that can be widely adopted in the community of biomedical research such as pediatric disease progressive models, age-related neurodegen-

Dynamic gene regulation within a time window in transcriptome data is generally subcategorized into (1) within-subject longitudinally repeatedly measured stimuli-response data in a single series of time course experiment, (2) between-subject factorial multi-series of time course data with different conditions at each time point, and (3) periodical data in cell-cycle or circadian rhythmic patterns with or without external conditions. For the first type of temporal dynamics, we initially proposed Bayesian Poisson Gamma (negative binomial model) strategy to identify temporally differentially expressed genes in the previous study [23]. In this model, each gene is statistically tested whether it is equal or differential expression by auto-regressive (AR) model. The detailed description for notations is given in the following,

**Model I**: supposedly, a gene expression profile across a series of multiple time points, *ygrt* is independently distributed as Poisson Gamma (negative binomial model to account for variability of biological replicates within a group), *g* = 1, 2, …,*G* (gene or other genomic feature),

> <sup>~</sup>*N*(0, *<sup>σ</sup>*<sup>2</sup> \_\_\_\_\_ 1 − *ϕgr*<sup>1</sup> 2 ),

~*N*(*ϕgr*<sup>1</sup> ∗ *wgrt*−1, *σ*<sup>2</sup>

+ *β<sup>g</sup>* ,

), *t* > 2

*t* = 1, 2, …,*T* (time point), and *r* = 1, 2, …,*R* (biological replicate at each time point),

*ygrt*~*POI*(*μgrt*),

log(*μgrt*) = *wgrt*

*wgr*<sup>1</sup>

*wgrt* ∣ *wgr*,1,2,…,*t*−<sup>1</sup>

erative diseases, other types of longitudinal and multi-series of time course data.

29, 30, 60].

stimulus.

**2. Methods**

**2.1. Data types of time course experiments**

The characterization based on Bayesian strategies for whole-genome wide transcriptome data and other types of Seq data has been successfully addressed on the variety of questions to arise in biomedical community [8, 9, 52]. As a naïve Bayesian method, baySeq framework has been proposed in differential expression analysis between different groups with replicates [52]. ShrinkBayes has been developed to identify differential expression analysis at static data on the basis of zero-inflated Poisson Gamma model and Integrated Nested Laplace Approximation to estimate shrunken parameters [53]. And another Bayesian technique in ChipSeq, BayesPeak has been developed to detect significantly enriched regions in transcription factor binding sites and histone modification datasets. It is on the basis of Bayesian hidden Markov model strategy and MCMC simulations with Poisson Gamma distribution to take into account over-dispersion in the abundance of read counts in different regions by comparing to the existing methods of peak callers in ChipSeq data, MACS, PeakSeq, and ChipSeq Peak Finder [54]. Additionally, Bayesian approaches to define differential expression of alternative splicing in RNA-seq have been proposed by MISO (as known as mixture of isoforms) model and MATS (as known as multivariate analysis of transcript splicing) [9, 18, 55]. MISO method is to estimate differential expression of alternatively spliced exons and isoforms based on Bayes factor to quantify the odds of differential regulation of given isoform for the ratio of inclusion and exclusion levels. Similarly, MATS (and rMATS) is implemented to test differential alternative splicing patterns by estimating exon inclusion levels between two samples without (and with replicates), respectively. In addition, microRNAs target prediction methods that play a key regulatory role in gene regulation on the variety of biological processes in human diseases, especially, cancer development, have been proposed by Bayesian methods [56, 57]. And also, Bayesian network methods have been proposed to predict the important functions of long non-coding RNAs as well as coding genes in RNA-seq [58, 59]. Thus, RNA-seq has become the alternative in transcriptome studies with advantageous features over arrays, dynamic range of expression signals, higher reproducibility and quality on samples and upgraded annotation without any known priori [3, 16, 19, 20, 24, 26–28, 32].

With the strength on improvements on technology and continuously declining cost to sequencing, RNA-seq enables to facilitate to perform more dense experimental designs such as time course data with abundant resources of dynamic gene regulation. Furthermore, transcriptome data and other types of meta-framed data across different platforms will be more popularly investigated in years to come. Indeed, next generation sequencing technologies have been steadily improved with higher throughput, longer reads, deeper sequencings, larger samples size of replicates, and less biases on data. Such advances allow investigators to conduct more complex experimental studies, various types of time course experiments, such as single series of longitudinally measured time course and transient dynamic patterns in developmental stages; multi-series of factorial time course with multiple external conditions at each time point; cell-cyclic periodical data with (or without) external stimuli [7, 8, 22, 23, 29, 30, 60].

To date, despite the substantial potential and significance to explore temporal dynamics at genes and other genomic features in various human disease progressive models and therapeutic effects, the lack of analytical methodologies in order to precisely characterize temporal dynamics has been an important challenging issue to better understand biological mechanisms relevant with both time specific and responsively altered changes by given external stimulus.

In this chapter, we propose Bayesian approaches to better infer differential expression in temporal and spatial dynamic regulation that can be widely adopted in the community of biomedical research such as pediatric disease progressive models, age-related neurodegenerative diseases, other types of longitudinal and multi-series of time course data.

#### **2. Methods**

significantly addressed fundamental issues characterized by biological phenomena in transcriptional regulation. For instance, examination of classification of subtypes on hereditary breast and ovary cancer [44–46], reciprocal phylogenetic conservation and heterogeneity between closest animal models of aging and depression in brain tissues [47–49], identification of differential expression on enzyme effect in Gaucher's disease across distinct three tissues [14, 50], developmental transient patterns in mouse embryonic stem cells in pre-frontal cortex

The characterization based on Bayesian strategies for whole-genome wide transcriptome data and other types of Seq data has been successfully addressed on the variety of questions to arise in biomedical community [8, 9, 52]. As a naïve Bayesian method, baySeq framework has been proposed in differential expression analysis between different groups with replicates [52]. ShrinkBayes has been developed to identify differential expression analysis at static data on the basis of zero-inflated Poisson Gamma model and Integrated Nested Laplace Approximation to estimate shrunken parameters [53]. And another Bayesian technique in ChipSeq, BayesPeak has been developed to detect significantly enriched regions in transcription factor binding sites and histone modification datasets. It is on the basis of Bayesian hidden Markov model strategy and MCMC simulations with Poisson Gamma distribution to take into account over-dispersion in the abundance of read counts in different regions by comparing to the existing methods of peak callers in ChipSeq data, MACS, PeakSeq, and ChipSeq Peak Finder [54]. Additionally, Bayesian approaches to define differential expression of alternative splicing in RNA-seq have been proposed by MISO (as known as mixture of isoforms) model and MATS (as known as multivariate analysis of transcript splicing) [9, 18, 55]. MISO method is to estimate differential expression of alternatively spliced exons and isoforms based on Bayes factor to quantify the odds of differential regulation of given isoform for the ratio of inclusion and exclusion levels. Similarly, MATS (and rMATS) is implemented to test differential alternative splicing patterns by estimating exon inclusion levels between two samples without (and with replicates), respectively. In addition, microRNAs target prediction methods that play a key regulatory role in gene regulation on the variety of biological processes in human diseases, especially, cancer development, have been proposed by Bayesian methods [56, 57]. And also, Bayesian network methods have been proposed to predict the important functions of long non-coding RNAs as well as coding genes in RNA-seq [58, 59]. Thus, RNA-seq has become the alternative in transcriptome studies with advantageous features over arrays, dynamic range of expression signals, higher reproducibility and quality on samples and upgraded annotation without any known priori [3, 16, 19, 20, 24, 26–28, 32]. With the strength on improvements on technology and continuously declining cost to sequencing, RNA-seq enables to facilitate to perform more dense experimental designs such as time course data with abundant resources of dynamic gene regulation. Furthermore, transcriptome data and other types of meta-framed data across different platforms will be more popularly investigated in years to come. Indeed, next generation sequencing technologies have been steadily improved with higher throughput, longer reads, deeper sequencings, larger samples size of replicates, and less biases on data. Such advances allow investigators to conduct more complex experimental studies, various types of time course experiments, such as single series of longitudinally measured time course and transient dynamic patterns in

and limb tissues [13, 51], and et cetera.

8 New Insights into Bayesian Inference

#### **2.1. Data types of time course experiments**

Dynamic gene regulation within a time window in transcriptome data is generally subcategorized into (1) within-subject longitudinally repeatedly measured stimuli-response data in a single series of time course experiment, (2) between-subject factorial multi-series of time course data with different conditions at each time point, and (3) periodical data in cell-cycle or circadian rhythmic patterns with or without external conditions. For the first type of temporal dynamics, we initially proposed Bayesian Poisson Gamma (negative binomial model) strategy to identify temporally differentially expressed genes in the previous study [23]. In this model, each gene is statistically tested whether it is equal or differential expression by auto-regressive (AR) model. The detailed description for notations is given in the following,

**Model I**: supposedly, a gene expression profile across a series of multiple time points, *ygrt* is independently distributed as Poisson Gamma (negative binomial model to account for variability of biological replicates within a group), *g* = 1, 2, …,*G* (gene or other genomic feature), *t* = 1, 2, …,*T* (time point), and *r* = 1, 2, …,*R* (biological replicate at each time point),

$$\begin{aligned} \mathcal{Y}\_{\mathcal{Y}^t} & \sim \text{POI}\{\mu\_{\mathcal{Y}^t}\} \\\\ \log\left(\mu\_{\mathcal{Y}^t}\right) &= \left. w\_{\mathcal{Y}^t} + \beta\_{\mathcal{Y}^t} \right. \\\\ w\_{\mathcal{Y}^t} & \sim \text{N}\left(0, \frac{\sigma^2}{1 - \rho\_{\mathcal{Y}^t}^2}\right) \\\\ \left. w\_{\mathcal{Y}^t} \right| & \left. w\_{\mathcal{Y}^t, 1, 2, \dots, t-1} \sim \text{N}\left(\rho\_{\mathcal{Y}^t} \* w\_{\mathcal{Y}^{t-1}}, \sigma^2\right), t \ge 2 \right. \end{aligned}$$

In this proposed model, *β<sup>g</sup>* is assumed to have non-informative priors and time series random effects model for sequentially measured single series of time course data is assumed. To update the defined auto-regressive model and estimate the posterior probabilities of given parameter sets, we employ Markov Chain Monte Carlo simulations, *N* = 10,000 iterations and 8000 burn-ins. To define temporal dynamics whether or not a given gene is temporally differentially expressed, we further examine the most interesting parameter of auto-coefficient representing time series sequential random effects in this proposed model. To classify between equal and differential expression, we implemented to compute Bayesian credible interval estimates. This proposed model is implemented by OpenBUGS (WinBUGS) in R (submitted paper). Basically, it allows us to include major factor of time and variability of replicates at each time point and this simple linear auto-regressive (AR) model is straightforwardly extended to identify temporally differentially expressed other genomic features, for instance, quantified abundance of transcripts. Despite the much better improved quality of samples in RNA-seq data when compared to the beginning of technology, the preprocessing and normalization procedures are still required to precisely infer temporally differentially expressed genes and isoforms and to reduce the misleading results in subsequent analyses. And some samples are discarded in the preprocessing step such as due to sample preparation in experiments and the rest of corresponding samples for the given missing sample should be also deleted as it is the method for longitudinal data with repeated measurements across the series of time points. More importantly, RNA-seq data is highly skewed expression data toward to zero and low expression levels than high expression level. Collectively, in the format of longitudinally measured experimental designs in temporal dynamics of RNA-seq data, as further improved strategy, we are currently developing a zero-inflated Poisson Gamma model with missing observations in longitudinal data to improve the capability of detection of temporal changes in highly skewed count data [61, 62]. The detailed descriptions and notations are given in the following, we consider the conditional distribution of *ygrt* <sup>∣</sup> *Egrt*,

code for different forms of a protein (isoform) as a result of alternative splicing that increases

Bayesian Modeling Approaches for Temporal Dynamics in RNA-seq Data

http://dx.doi.org/10.5772/intechopen.73062

11

In the previous literatures, deep sequencing based transcriptome data predicts that more than ~95% of human genes typically contain multi-exons and transcript-variants to undergo alternative spliced events [8, 9, 25, 28, 33, 35, 63–65, 67]. The aberrant alternative splicing events occurring in post-transcriptional and translational procedures are highly associated with different tissues- or developmental stages- or environmental condition-specific manner. It has been investigated that malformation and dysfunctional mechanisms by the majority of abnormal alternative splicings in human brain diseases. Aberrant patterns of splicings in neurodegenerative Alzheimer's patients and other types of pediatric cancer progression could be significant contributors to targeted therapies on disease progressive models and developmental evolutionary processes in transcriptional activity in temporal dynamics [28, 63–66]. Despite the importance of alternative splicing events in recent technology, characterization of dynamic processes has been merely limited to gene level and static data

In the methodological point of view, for quantification and identification of isoforms, a couple of bioinformatics tools including IQSeq, rSeq, MapSplice have been recently developed [15, 17, 31], and identification of differentially expressed isoforms at static data types have been pursued by MATS (focused on a specified experimental design on a sample versus another single sample comparison at a fixed time point) [9], DEXSeq (flexibly to allow various types of experimental and biological conditions in generalized linear model from the basis of multiple comparisons at exon levels) [5], and cufflinks and cuffdiff (as known as on the of most popular versatile tools for quantification and identification of differential expression at isoform levels,

To our best knowledge, none of current static and dynamic methods can identify temporally differentially expression at alternative splicing by explicitly accounting for data-driven nature

For the first type of longitudinal time course experiments, quantified expression levels at isoforms and other types of genomic features can be directly applied for our proposed dynamic

For the second type of between-subject factorial multi-series of time course data, another our previous study [8] proposed a hierarchical Bayesian modeling approach to define differential expression analysis at isoforms when having multiple conditions at each time point, such as different tissues, drug treatments, stress, and trauma in temporal dynamics (see **Figure 1**).

**Model II**: the detailed description for notations in the proposed model is given in the following, for the sake of consistency in notations of the proposed models, we pursue Bayesian hierarchical Dirichlet Bayesian mixture model of temporal dynamics in multi-series of time course data by making use of the identical notations on the transcriptomic expression levels as shown in the Model I. Suppose that a gene (or other genomic feature) expression profile across a series of multiple time points is tested and each time point contains different external conditions, such as different types of tissues, cell lines, drug treatments, trauma, stress, and et

but restricted to simple pairwise comparison with replicates) [10, 11, 30, 68].

cetera, where we denote a condition factor, *c* = 1, 2, …,*C* at each time point.

the complexity of mammalian transcriptome.

of various time course experimental settings.

approaches.

AR model.

$$\mathcal{Y}\_{\mathcal{g}^{rt}} \mid E\_{\mathcal{g}^{rt}} \sim \text{POL} \{ \mu\_{\mathcal{g}^{rt}} \} \text{ if } E\_{\mathcal{g}^{rt}} = 1$$

and

$$\mathbf{P}\left(\left.y\_{grt}=\mathbf{0}\right|E\_{grt}=\mathbf{0}\right)=\mathbf{1}\text{ if }E\_{grt}=\mathbf{0}$$

where *Egrt* is the binary indicator of whether a gene expression profile across a series of multiple time points is present for a gene *g*, time *t* and replication *r*. We also assume that conditioned on *pgrt*, the *Egrt*'s are independent Bernoulli random variables with *P*(*Egrt* <sup>=</sup> <sup>1</sup>) <sup>=</sup> *pgrt* for *<sup>g</sup>* <sup>=</sup> <sup>1</sup>, <sup>2</sup>, …,*G*, *<sup>t</sup>* <sup>=</sup> <sup>1</sup>, <sup>2</sup>, …,*T*, and *<sup>r</sup>* <sup>=</sup> <sup>1</sup>, <sup>2</sup>, …,*R*. Here, given *Egrt* <sup>=</sup> <sup>1</sup>, assume that the *ygrt*'s are conditionally independent.

Compared to arrays, the major strength of RNA-seq transcriptome data enables to quantify and identify spliced isoforms as well as individual exon-level expression which had not been previously done due to low resolution [8–11, 13, 14, 25, 28, 30, 33, 35, 60, 63–68]. In addition to gene level analyses, it is well established that alternative splicing is a prevalent mechanism on the variety of organisms. It involves multiple selective schemes of splice sites to construct diverse functional pathways and protein structures in gene regulation. A single mRNA may code for different forms of a protein (isoform) as a result of alternative splicing that increases the complexity of mammalian transcriptome.

In this proposed model, *β<sup>g</sup>*

10 New Insights into Bayesian Inference

and

ally independent.

is assumed to have non-informative priors and time series ran-

dom effects model for sequentially measured single series of time course data is assumed. To update the defined auto-regressive model and estimate the posterior probabilities of given parameter sets, we employ Markov Chain Monte Carlo simulations, *N* = 10,000 iterations and 8000 burn-ins. To define temporal dynamics whether or not a given gene is temporally differentially expressed, we further examine the most interesting parameter of auto-coefficient representing time series sequential random effects in this proposed model. To classify between equal and differential expression, we implemented to compute Bayesian credible interval estimates. This proposed model is implemented by OpenBUGS (WinBUGS) in R (submitted paper). Basically, it allows us to include major factor of time and variability of replicates at each time point and this simple linear auto-regressive (AR) model is straightforwardly extended to identify temporally differentially expressed other genomic features, for instance, quantified abundance of transcripts. Despite the much better improved quality of samples in RNA-seq data when compared to the beginning of technology, the preprocessing and normalization procedures are still required to precisely infer temporally differentially expressed genes and isoforms and to reduce the misleading results in subsequent analyses. And some samples are discarded in the preprocessing step such as due to sample preparation in experiments and the rest of corresponding samples for the given missing sample should be also deleted as it is the method for longitudinal data with repeated measurements across the series of time points. More importantly, RNA-seq data is highly skewed expression data toward to zero and low expression levels than high expression level. Collectively, in the format of longitudinally measured experimental designs in temporal dynamics of RNA-seq data, as further improved strategy, we are currently developing a zero-inflated Poisson Gamma model with missing observations in longitudinal data to improve the capability of detection of temporal changes in highly skewed count data [61, 62]. The detailed descriptions and notations are

given in the following, we consider the conditional distribution of *ygrt* <sup>∣</sup> *Egrt*,

<sup>|</sup>*Egrt* <sup>=</sup> <sup>0</sup>) <sup>=</sup> <sup>1</sup> if *Egrt* <sup>=</sup> <sup>0</sup>

where *Egrt* is the binary indicator of whether a gene expression profile across a series of multiple time points is present for a gene *g*, time *t* and replication *r*. We also assume that conditioned on *pgrt*, the *Egrt*'s are independent Bernoulli random variables with *P*(*Egrt* <sup>=</sup> <sup>1</sup>) <sup>=</sup> *pgrt* for *<sup>g</sup>* <sup>=</sup> <sup>1</sup>, <sup>2</sup>, …,*G*, *<sup>t</sup>* <sup>=</sup> <sup>1</sup>, <sup>2</sup>, …,*T*, and *<sup>r</sup>* <sup>=</sup> <sup>1</sup>, <sup>2</sup>, …,*R*. Here, given *Egrt* <sup>=</sup> <sup>1</sup>, assume that the *ygrt*'s are condition-

Compared to arrays, the major strength of RNA-seq transcriptome data enables to quantify and identify spliced isoforms as well as individual exon-level expression which had not been previously done due to low resolution [8–11, 13, 14, 25, 28, 30, 33, 35, 60, 63–68]. In addition to gene level analyses, it is well established that alternative splicing is a prevalent mechanism on the variety of organisms. It involves multiple selective schemes of splice sites to construct diverse functional pathways and protein structures in gene regulation. A single mRNA may

*ygrt* ∣ *Egrt* ∼ *POI*(*μgrt*), if *Egrt* = 1

P( *ygrt* = 0

In the previous literatures, deep sequencing based transcriptome data predicts that more than ~95% of human genes typically contain multi-exons and transcript-variants to undergo alternative spliced events [8, 9, 25, 28, 33, 35, 63–65, 67]. The aberrant alternative splicing events occurring in post-transcriptional and translational procedures are highly associated with different tissues- or developmental stages- or environmental condition-specific manner. It has been investigated that malformation and dysfunctional mechanisms by the majority of abnormal alternative splicings in human brain diseases. Aberrant patterns of splicings in neurodegenerative Alzheimer's patients and other types of pediatric cancer progression could be significant contributors to targeted therapies on disease progressive models and developmental evolutionary processes in transcriptional activity in temporal dynamics [28, 63–66]. Despite the importance of alternative splicing events in recent technology, characterization of dynamic processes has been merely limited to gene level and static data approaches.

In the methodological point of view, for quantification and identification of isoforms, a couple of bioinformatics tools including IQSeq, rSeq, MapSplice have been recently developed [15, 17, 31], and identification of differentially expressed isoforms at static data types have been pursued by MATS (focused on a specified experimental design on a sample versus another single sample comparison at a fixed time point) [9], DEXSeq (flexibly to allow various types of experimental and biological conditions in generalized linear model from the basis of multiple comparisons at exon levels) [5], and cufflinks and cuffdiff (as known as on the of most popular versatile tools for quantification and identification of differential expression at isoform levels, but restricted to simple pairwise comparison with replicates) [10, 11, 30, 68].

To our best knowledge, none of current static and dynamic methods can identify temporally differentially expression at alternative splicing by explicitly accounting for data-driven nature of various time course experimental settings.

For the first type of longitudinal time course experiments, quantified expression levels at isoforms and other types of genomic features can be directly applied for our proposed dynamic AR model.

For the second type of between-subject factorial multi-series of time course data, another our previous study [8] proposed a hierarchical Bayesian modeling approach to define differential expression analysis at isoforms when having multiple conditions at each time point, such as different tissues, drug treatments, stress, and trauma in temporal dynamics (see **Figure 1**).

**Model II**: the detailed description for notations in the proposed model is given in the following, for the sake of consistency in notations of the proposed models, we pursue Bayesian hierarchical Dirichlet Bayesian mixture model of temporal dynamics in multi-series of time course data by making use of the identical notations on the transcriptomic expression levels as shown in the Model I. Suppose that a gene (or other genomic feature) expression profile across a series of multiple time points is tested and each time point contains different external conditions, such as different types of tissues, cell lines, drug treatments, trauma, stress, and et cetera, where we denote a condition factor, *c* = 1, 2, …,*C* at each time point.

*ng* <sup>=</sup> <sup>|</sup>{*g*; *<sup>l</sup>*

for equal expression (EEX) and *G*<sup>1</sup>

rior probability is given by, P(*γ<sup>g</sup>* <sup>=</sup> *<sup>d</sup>* <sup>∣</sup> *<sup>l</sup>*

**3. Closing remarks in future directions**

of stimuli-response data in this chapter.

genomic levels such as transcripts and exon levels.

parameter *α*. And, *F*<sup>0</sup>

expressed gene.

data.

*<sup>g</sup>* = *l*

∗ ~ \_\_1 <sup>2</sup> *<sup>N</sup>*(−*δ*, *<sup>σ</sup>*<sup>2</sup>

*F*(*β*)~*DP*(*F*0(*F*(*β*)); *η*0),

*wgrtc* ∣ (*rg* = *d*, *l*

}<sup>|</sup> denote the size of the *<sup>l</sup>*th cluster,

)~*DP*(*Gd* ∗ ; *M*),

represents the mixture format of *β* with p(*β*) = *NG*−1(0, *<sup>ψ</sup>*0) such that *G*<sup>0</sup>

∗ ~*N*(0, *σ*<sup>2</sup> )

> , *ψ*<sup>0</sup> ,

13

) for differential expression (DEX) and *η*<sup>0</sup>

*<sup>g</sup>* <sup>=</sup> *<sup>l</sup>*, *yg*111, …,*yg*,*r*=*R*,*t*=*T*,*c*=*<sup>C</sup>*) for the temporally differentially

Bayesian Modeling Approaches for Temporal Dynamics in RNA-seq Data

http://dx.doi.org/10.5772/intechopen.73062

*<sup>g</sup>* = *l*

where *DP*(H;*α*) stands for the Dirichlet process having its baseline distribution H(∙) and mass

and *M* are fixed hyper-parameters based on prior information [6, 69]. Therefore, the poste-

Based on our proposed Bayesian approach for multi-series of factorial time course data, we are currently implementing OpenBUGS(WinBUGS) in R to perform differential expression analysis. In order to validate our proposed model, we need to compare to other maSigPro for RNA-seq data and Gaussian Process modeling approach in terms of temporally differentially expressed genes in the multiple datasets after transformation of stabilizing variance on counts

In earlier sections, we have discussed Bayesian techniques to address different types of experimental (clinical) settings in temporal dynamics by focusing on Poisson Gamma autoregressive model for longitudinally measured single-series of time course data and hierarchical Dirichlet Bayesian mixture model framework for multi-series of factorial time course data, respectively. Thus, as the continuous efforts to modeling approaches, we propose differential expression analytical frameworks that precisely characterize temporal dynamics for each type

The novel features in the proposed hierarchical Dirichlet Bayesian mixture model enables to identify significant temporal changes of expression levels between at least two external conditional factors over a series of time points based on Bayesian strategy grouping clusters by the patterns of differential or equal expression [6, 69]. The identified temporal changes are determined as the putative biomarkers that could be relevantly linked with various dynamic genetic mechanisms of molecular and physiological processes. Additionally, our proposed method enables to allow more than two genetic and environmental factors within a time point and to address how the intra-factor of multiple conditions and time factor affect altered expression patterns as significant contributors independently and interactively. Furthermore, this proposed model is straightforwardly extended to detect temporal changes at other

) <sup>+</sup> \_\_1 <sup>2</sup> *<sup>N</sup>*(*δ*, *<sup>σ</sup>*<sup>2</sup>

**Figure 1.** It depicts the latent variable to be estimated in hierarchical Dirichlet Bayesian mixture model.

For a particular time point at *T* = *t*, and condition at *C* = *c*, the given expression level is given by *ygrtc*, *<sup>g</sup>* <sup>=</sup> <sup>1</sup>, <sup>2</sup>, …,*<sup>G</sup>* (gene), *<sup>r</sup>* <sup>=</sup> <sup>1</sup>, <sup>2</sup>, …,*R* (replicate at each time point and a condition), *<sup>t</sup>* <sup>=</sup> <sup>1</sup>, <sup>2</sup>, …,*<sup>T</sup>* (time point), and *<sup>c</sup>* <sup>=</sup> <sup>1</sup>, <sup>2</sup>, …,*C* (external condition at each time point). Let *γ<sup>g</sup>* be a latent variable to be estimated in a hierarchical mixture model whether equal expression (EEX) or temporally differentially expression (TDEX) is at a given gene of interest. At a given time point *T* = *t*, *τg* denotes the *g*th gene effect, *β<sup>c</sup>* denotes the effect of the *c*th condition. The latent variable *γg* represents an indicator variable such that the given *g*th gene is temporally differentially expressed or not, that is, *γ<sup>g</sup>* <sup>=</sup> { <sup>1</sup>, when TDEX 0, when EEX .

$$\begin{aligned} \mathcal{Y}\_{grtc} \mid \mathcal{Y}\_{\mathcal{S}} &= d \sim \text{POI}\{\mu\_{grtc}\} \\\\ \log\left(\mu\_{grtc}\right) &= \tau\_{\mathcal{S}} + \beta\_c + F(\beta) + w\_{grtc} \\\\ \beta\_c \mid l\_{\mathcal{S}} &= l \sim F(\beta\_l) \end{aligned}$$

*l* represents distinct patterns between conditions across time points,

 Namely, *l <sup>g</sup>* = *l* iff *g*th gene belongs to cluster *l*,

Let *L* denote the number of clusters on expression data,

 *ng* <sup>=</sup> <sup>|</sup>{*g*; *<sup>l</sup> <sup>g</sup>* = *l* }<sup>|</sup> denote the size of the *<sup>l</sup>*th cluster,

$$F(\beta) \sim DP\left(F\_o(F(\beta)); \eta\_o\right),$$

$$w\_{g\_{\mathcal{I}^{\text{tot}}}} \mid \ (r\_{\mathcal{I}} = d, l\_{\mathcal{I}} = l) \sim DP(G\_{d'}^\* M)\_{\mathcal{I}^{\text{tot}}}$$

where *DP*(H;*α*) stands for the Dirichlet process having its baseline distribution H(∙) and mass parameter *α*. And, *F*<sup>0</sup> represents the mixture format of *β* with p(*β*) = *NG*−1(0, *<sup>ψ</sup>*0) such that *G*<sup>0</sup> ∗ ~*N*(0, *σ*<sup>2</sup> ) for equal expression (EEX) and *G*<sup>1</sup> ∗ ~ \_\_1 <sup>2</sup> *<sup>N</sup>*(−*δ*, *<sup>σ</sup>*<sup>2</sup> ) <sup>+</sup> \_\_1 <sup>2</sup> *<sup>N</sup>*(*δ*, *<sup>σ</sup>*<sup>2</sup> ) for differential expression (DEX) and *η*<sup>0</sup> , *ψ*<sup>0</sup> , and *M* are fixed hyper-parameters based on prior information [6, 69]. Therefore, the posterior probability is given by, P(*γ<sup>g</sup>* <sup>=</sup> *<sup>d</sup>* <sup>∣</sup> *<sup>l</sup> <sup>g</sup>* <sup>=</sup> *<sup>l</sup>*, *yg*111, …,*yg*,*r*=*R*,*t*=*T*,*c*=*<sup>C</sup>*) for the temporally differentially expressed gene.

Based on our proposed Bayesian approach for multi-series of factorial time course data, we are currently implementing OpenBUGS(WinBUGS) in R to perform differential expression analysis. In order to validate our proposed model, we need to compare to other maSigPro for RNA-seq data and Gaussian Process modeling approach in terms of temporally differentially expressed genes in the multiple datasets after transformation of stabilizing variance on counts data.

#### **3. Closing remarks in future directions**

For a particular time point at *T* = *t*, and condition at *C* = *c*, the given expression level is given by *ygrtc*, *<sup>g</sup>* <sup>=</sup> <sup>1</sup>, <sup>2</sup>, …,*<sup>G</sup>* (gene), *<sup>r</sup>* <sup>=</sup> <sup>1</sup>, <sup>2</sup>, …,*R* (replicate at each time point and a condition), *<sup>t</sup>* <sup>=</sup> <sup>1</sup>, <sup>2</sup>, …,*<sup>T</sup>*

to be estimated in a hierarchical mixture model whether equal expression (EEX) or temporally differentially expression (TDEX) is at a given gene of interest. At a given time point *T* = *t*,

represents an indicator variable such that the given *g*th gene is temporally differentially

*<sup>g</sup>* = *l*~*F*(*β<sup>l</sup>*

),

*<sup>g</sup>* = *l* iff *g*th gene belongs to cluster *l*,

denotes the effect of the *c*th condition. The latent variable

be a latent variable

(time point), and *<sup>c</sup>* <sup>=</sup> <sup>1</sup>, <sup>2</sup>, …,*C* (external condition at each time point). Let *γ<sup>g</sup>*

**Figure 1.** It depicts the latent variable to be estimated in hierarchical Dirichlet Bayesian mixture model.

<sup>1</sup>, when TDEX 0, when EEX .

*l* represents distinct patterns between conditions across time points,

Let *L* denote the number of clusters on expression data,

*ygrtc* ∣ *γ<sup>g</sup>* = *d*~*POI*(*μgrtc*),

log(*μgrtc*) = *τ<sup>g</sup>* + *β<sup>c</sup>* + *F*(*β*) + *wgrtc*,

*τg*

*γg*

denotes the *g*th gene effect, *β<sup>c</sup>*

*β<sup>c</sup>* ∣ *l*

Namely, *l*

expressed or not, that is, *γ<sup>g</sup>* <sup>=</sup> {

12 New Insights into Bayesian Inference

In earlier sections, we have discussed Bayesian techniques to address different types of experimental (clinical) settings in temporal dynamics by focusing on Poisson Gamma autoregressive model for longitudinally measured single-series of time course data and hierarchical Dirichlet Bayesian mixture model framework for multi-series of factorial time course data, respectively. Thus, as the continuous efforts to modeling approaches, we propose differential expression analytical frameworks that precisely characterize temporal dynamics for each type of stimuli-response data in this chapter.

The novel features in the proposed hierarchical Dirichlet Bayesian mixture model enables to identify significant temporal changes of expression levels between at least two external conditional factors over a series of time points based on Bayesian strategy grouping clusters by the patterns of differential or equal expression [6, 69]. The identified temporal changes are determined as the putative biomarkers that could be relevantly linked with various dynamic genetic mechanisms of molecular and physiological processes. Additionally, our proposed method enables to allow more than two genetic and environmental factors within a time point and to address how the intra-factor of multiple conditions and time factor affect altered expression patterns as significant contributors independently and interactively. Furthermore, this proposed model is straightforwardly extended to detect temporal changes at other genomic levels such as transcripts and exon levels.

As the extension of this study, we are currently developing how to measure the relationship of a parent gene-to-multiple child isoforms in temporal dynamic patterns. For the task of this procedure, we carry out directional comparison, gene-to-isoform in differential expression based on similarity and discrepancy on magnitude and pattern of expression. This proposed model enables to define connectivity visualization of splicing maps on the variety of structural formations by switchable exon usage. Moreover, we are currently developing differential expression method for cell-cyclic periodical data with or without external conditions in stimuli-response data [70]. Thus, this proposed study is timely crucial to define temporal dynamics at alternative splicing diversity related with disease progression by discovering which splicing events are condition and time specifically observed and how eventually their spliced abnormal patterns and splicing maps are associated with biological functions. And it is essential to develop strategies to correct aberrant splicing as well as gene approaches in temporal and spatial dynamics on the variety of disease progression and evolutionary comparative studies between human diseases and other closely related species.

**Author details**

\* and Seongho Song<sup>2</sup>

\*Address all correspondence to<EMAIL_ADDRESS>
Biology. 2010;**11**:R106. DOI: 10.1186/gb-2010-11-10-r106

2010;**11**:94. DOI: 10.1186/1471-2105-11-94

10.1093/bioinformatics/btp616

6753-7\_12

gkr1291

1 Department of Computer Science and Statistics, Jeju National University, Jeju City,

Bayesian Modeling Approaches for Temporal Dynamics in RNA-seq Data

http://dx.doi.org/10.5772/intechopen.73062

15

2 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, USA

[1] Anders S, Huber W. Differential expression analysis for sequence count data. Genome

[2] Bullard JH, Purdom E, Hansen KD, Dudoit S. Evaluation of statistical methods for normalization and differential expression in mRNA-Seq experiments. BMC Bioinformatics.

[3] Ritchie ME et al. Limma powers differential expression analyses for RNA-sequencing and microarray studies. Nucleic Acids Research. 2015;**43**:e47. DOI: 10.1093/nar/gkv007

[4] Robinson MD, DJ MC, Smyth GK. edgeR: A bioconductor package for differential expression analysis of digital gene expression data. Bioinformatics. 2010;**26**:139-140. DOI:

[5] Anders S, Reyes A, Huber W. Detecting differential usage of exons from RNA-seq data.

[6] Guindani M, Sepulveda N, Paulino CD, Muller P. A Bayesian semi-parametric approach for the differential analysis of sequence counts data. Journal of the Royal Statistical

[7] Heinonen M et al. Detecting time periods of differential gene expression using Gaussian processes: An application to endothelial cells exposed to radiotherapy dose fraction.

[8] Oh S, Song S. Differential gene expression (DEX) and alternative splicing events (ASE) for temporal dynamic processes using HMMs and hierarchical Bayesian modeling approaches. Methods in Molecular Biology. 2017;**1552**:165-176. DOI: 10.1007/978-1-4939-

[9] Shen S et al. MATS: A Bayesian framework for flexible detection of differential alternative splicing from RNA-Seq data. Nucleic Acids Research. 2012;**40**:e61. DOI: 10.1093/nar/

[10] Trapnell C et al. Differential analysis of gene regulation at transcript resolution with

RNA-seq. Nature Biotechnology. 2013;**31**:46-53. DOI: 10.1038/nbt.2450

Society. Series C, Applied Statistics. 2014;**63**:385-404. DOI: 10.1111/rssc.12041

Genome Research. 2012;**22**:2008-2017. DOI: 10.1101/gr.133744.111

Bioinformatics. 2015;**31**:728-735. DOI: 10.1093/bioinformatics/btu699

Sunghee Oh<sup>1</sup>

South Korea

**References**

#### **Conflict of interest**

The authors have no conflicts of interest to disclose.

#### **Contributors' statements**

SO wrote manuscript and SO and SS conceived this study.

For discloser of any prior publications or submission with any overlapping information including studies and patients, there are no prior publications or submissions with any overlapping information including studies and patients.

The manuscript has not been and will not be submitted to any other journal while it is under consideration by this book chapter in Bayesian inference.

All authors approved the final manuscript as submitted and agree to be accountable for all aspects of the work.

#### **Funding source**

This study is supported by an internal grant from Jeju National University to Dr. Sunghee Oh.

#### **Financial disclosure**

The authors have no financial relationships relevant to this article to disclose.

### **Author details**

As the extension of this study, we are currently developing how to measure the relationship of a parent gene-to-multiple child isoforms in temporal dynamic patterns. For the task of this procedure, we carry out directional comparison, gene-to-isoform in differential expression based on similarity and discrepancy on magnitude and pattern of expression. This proposed model enables to define connectivity visualization of splicing maps on the variety of structural formations by switchable exon usage. Moreover, we are currently developing differential expression method for cell-cyclic periodical data with or without external conditions in stimuli-response data [70]. Thus, this proposed study is timely crucial to define temporal dynamics at alternative splicing diversity related with disease progression by discovering which splicing events are condition and time specifically observed and how eventually their spliced abnormal patterns and splicing maps are associated with biological functions. And it is essential to develop strategies to correct aberrant splicing as well as gene approaches in temporal and spatial dynamics on the variety of disease progression and evolutionary comparative studies between human diseases and other closely

For discloser of any prior publications or submission with any overlapping information including studies and patients, there are no prior publications or submissions with any over-

The manuscript has not been and will not be submitted to any other journal while it is under

All authors approved the final manuscript as submitted and agree to be accountable for all

This study is supported by an internal grant from Jeju National University to Dr. Sunghee Oh.

The authors have no financial relationships relevant to this article to disclose.

related species.

**Conflict of interest**

14 New Insights into Bayesian Inference

aspects of the work.

**Funding source**

**Financial disclosure**

**Contributors' statements**

The authors have no conflicts of interest to disclose.

SO wrote manuscript and SO and SS conceived this study.

lapping information including studies and patients.

consideration by this book chapter in Bayesian inference.

Sunghee Oh<sup>1</sup> \* and Seongho Song<sup>2</sup>

\*Address all correspondence to<EMAIL_ADDRESS>
1 Department of Computer Science and Statistics, Jeju National University, Jeju City, South Korea

2 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, USA

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**Chapter 3**

Provisional chapter

**Bayesian Analysis for Hidden Markov Factor Analysis**

DOI: 10.5772/intechopen.72837

The purpose of this chapter is to provide an introduction to Bayesian approach within a general framework and develop a Bayesian procedure for analyzing multivariate

The Bayesian approach is now well recognized in the statistics literature as an attractive approach to analyzing a wide variety of models [1], and there is rich literature on this issue. Here, we are not going to present a full coverage on the general Bayesian theory, and readers may refer to excellent books, for example [2, 3], for more details for this general statistical method. This chapter provides an introduction to the Bayesian approach within a general framework and develops a specific Bayesian procedure for analyzing multivariate longitudinal data within the hidden Markov factor analysis framework. We begin with the basic ideas of the Bayesian approach and then describe the model under consideration in the second section. The following section considers Bayesian inferences including parameter estimation, model selection, and posterior density estimates. The final section demonstrates the practical value of proposed methodology to cocaine use data to get some Bayesian results. Some technical details

Consider a data set Y with the probability model p Yð Þ jθ where θ is a univariate or multivariate population parameters vector, which quantifies the uncertainty of data. In the statistical literature,

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

longitudinal data within the hidden Markov factor analysis framework.

Keywords: hidden Markov factor analysis model, Markov chain Monte Carlo

Bayesian Analysis for Hidden Markov Factor Analysis

Yemao Xia, Xiaoqian Zeng and Niansheng Tang

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72837

Yemao Xia, Xiaoqian Zeng and

**Models**

Models

Niansheng Tang

Abstract

1. Introduction

are given in the Appendix.

sampling, cocaine use

#### **Bayesian Analysis for Hidden Markov Factor Analysis Models** Bayesian Analysis for Hidden Markov Factor Analysis Models

DOI: 10.5772/intechopen.72837

Yemao Xia, Xiaoqian Zeng and Niansheng Tang Yemao Xia, Xiaoqian Zeng and

Additional information is available at the end of the chapter Niansheng Tang

http://dx.doi.org/10.5772/intechopen.72837 Additional information is available at the end of the chapter

Abstract

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10.1038/nrg3310

20 New Insights into Bayesian Inference

Cell. 1998;**9**:3273-3297

2008;**456**:470-476. DOI: 10.1038/nature07509

expression profiles. Bioinformatics. 2002;**18**:1194-1206

The purpose of this chapter is to provide an introduction to Bayesian approach within a general framework and develop a Bayesian procedure for analyzing multivariate longitudinal data within the hidden Markov factor analysis framework.

Keywords: hidden Markov factor analysis model, Markov chain Monte Carlo sampling, cocaine use

#### 1. Introduction

The Bayesian approach is now well recognized in the statistics literature as an attractive approach to analyzing a wide variety of models [1], and there is rich literature on this issue. Here, we are not going to present a full coverage on the general Bayesian theory, and readers may refer to excellent books, for example [2, 3], for more details for this general statistical method. This chapter provides an introduction to the Bayesian approach within a general framework and develops a specific Bayesian procedure for analyzing multivariate longitudinal data within the hidden Markov factor analysis framework. We begin with the basic ideas of the Bayesian approach and then describe the model under consideration in the second section. The following section considers Bayesian inferences including parameter estimation, model selection, and posterior density estimates. The final section demonstrates the practical value of proposed methodology to cocaine use data to get some Bayesian results. Some technical details are given in the Appendix.

Consider a data set Y with the probability model p Yð Þ jθ where θ is a univariate or multivariate population parameters vector, which quantifies the uncertainty of data. In the statistical literature,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

p Yð Þ jθ is called likelihood or sampling distribution and often represented as Lð Þ θ . From the frequency statistics point of view, statistical inferences are carried out based on Lð Þ θ . In this case, θ, though unknown, is treated as fixed. Unlike the frequency statistical inferences, the Bayesian approach for data analysis assumes that θ is random and has a distribution π θð Þ. This distribution, which represents the knowledge about θ, is referred to as prior distribution or prior. When data are available, the information on θ is summarized within the posterior distribution or posterior, a conditional distribution θ given data, i.e.,

$$p(\theta|Y) = \frac{p(Y|\theta)\pi(\theta)}{p(Y)} \propto p(Y|\theta)p(\theta) \tag{1}$$

investigating its transition pattern over time. In these cases, hidden Markov latent variable model (HMLVM) [10–13] provides a feasible and unified framework to address these issues. HMLVM assumes that the overall model constitutes the observed process and the underlying hidden state process. The state process, as the convention in the classic HMM (see for example, [14–17]), is an univariate discrete process, which follows a first-order Markov chain, while the observed process, conditional on the state sequence, is an independent process with emission distribution specified via LVMs [18]. Hence, in this regard, HMLVM provides a unified way of describing the correlation of multiple items, temporal dependence, and heterogeneity among the data simultaneously. However, the current existing developments cited beforehand focus on the maximum likelihood analysis in which statistical inferences heavily depend on the asymptotic properties. As an illustration of Bayesian inferences on practical problems, in this chapter, we develop a Bayesian procedure to analyze cocaine use data within the hidden Markov factor analysis model framework. Compared to ML, a basic nice feature of a Bayesian approach is its flexibility to utilize useful prior information for achieving better results. Additionally, simulation-based Bayesian methods depend less on asymptotic theory and hence have

Bayesian Analysis for Hidden Markov Factor Analysis Models

http://dx.doi.org/10.5772/intechopen.72837

23

Consider a set of multivariate longitudinal observations formed by p-dimensional observed

across N subjects: i ¼ 1, ⋯, N. In the field of multivariate analysis, interest mainly focuses on exploring item dependence since measurements may be highly correlated arising from the multicollinearity problem. But more often, interest also concentrates on the heterogeneity resulting from the situation where the population of yit constitutes more than one component. This is particularly true in the situation where the data illustrate extreme behaviors such as multimodal and/or skewed characteristics. In these cases, a finite mixture factor analysis model (FMFAM) can provide a powerful tool to address these issues. Typically, FMFAM assumes that conditioning on an univariate discrete value state variable zit and an m-dimensional (m < p) continuous latent factor vector ωit, yit are independent and distributed with a p-dimensional multivariate normal distribution, and meanwhile, given zit, ωit also follows an m-dimensional

<sup>y</sup>itjωit; zit <sup>¼</sup> <sup>r</sup> � <sup>N</sup> <sup>p</sup> <sup>μ</sup><sup>r</sup> <sup>þ</sup> <sup>Λ</sup><sup>r</sup> <sup>ω</sup>it; <sup>Ψ</sup>ε<sup>r</sup>

a p � p diagonal matrix with the jth diagonal element Ψ<sup>e</sup>krj > 0, and Φ<sup>r</sup> is an m � m positive

ωit ð Þ� jzit ¼ r N <sup>m</sup>ð Þ 0; Φ<sup>r</sup>

, which are recorded on p items over periods of length T: t ¼ 1, ⋯, T

is a p-dimensional intercept vector, which represents the baseline

is a <sup>p</sup> � <sup>m</sup> factor loading matrix, <sup>Ψ</sup><sup>e</sup><sup>r</sup> <sup>¼</sup> diag <sup>Ψ</sup><sup>e</sup>kr1; <sup>⋯</sup>; <sup>Ψ</sup><sup>e</sup>krp is

(2)

the potential to produce more reliable results even with small samples.

2. Model description

vectors yit <sup>¼</sup> yit1;…; yitp <sup>⊺</sup>

normal distribution, that is,

where <sup>μ</sup><sup>r</sup> <sup>¼</sup> <sup>μ</sup>r1;…; <sup>μ</sup>rp <sup>⊺</sup>

level of <sup>y</sup>it, <sup>Λ</sup><sup>r</sup> <sup>¼</sup> <sup>Λ</sup><sup>⊺</sup>

definite matrix.

<sup>r</sup>1; …; Λ<sup>⊺</sup> rp <sup>⊺</sup>

2.1. Hidden Markov factor analysis model

where p Yð Þ¼ <sup>Ð</sup> p Yð Þ jθ π θð Þdθ is the marginal distribution of Y. The right-hand-side term in (1) omits the factor p Yð Þ since given Y it is a known constant. In Bayes literature, p Yð Þ jθ pð Þ θ is also termed the unnormalized posterior. Analogous to the role of likelihood in frequency statistical inferences, posterior is the starting point of Bayesian inferences.

Selecting proper priors for parameters is fundamental to Bayesian analysis. Basically, there are two kinds of prior distributions, namely, the noninformative prior distributions and the informative prior distributions. Noninformative prior distributions associate with situations when the prior distributions have no population basis. They are used when we have little prior information on θ and desire that the prior distributions play a minimal role in the posterior distribution distribution. Informative prior distribution represents the distribution of possible parameter values, from which the parameter θ has been drawn. We may have prior knowledge about this distribution, either from closed related data or from the subjective knowledge of experts. A commonly used informative prior distribution in the general Bayesian approach to statistical problems is the conjugate prior distribution, a prior ensuring that the posterior distribution follows the same parametric form as the prior distribution [1, 3].

A potential difficulty underlying Bayesian inferences is the statistical computation when posterior distribution takes on the complicated form. This is particularly true in the situation where latent variables or other unobservable quantities are involved in the model, as discussed in this chapter. In such cases, statistical inferences usually recur to simulation-based methods. Among various sampling methods, Markov chains Monte Carlo methods (MCMC) provide powerful tools for simulating observations from posterior. The key to Markov chain simulation is to create a Markov sequence whose stationary distribution is a specified posterior pð Þ θjY . Posterior inferences are carried out based on these simulated observations. There are many ways of constructing these Markov chains, but all of them, including the Gibbs sampler [4, 5], are special cases of the general framework of Metropolis et al. [6] and Hastings [7]. However, we do not intend to pursue this issue here, and details on simulation-based methods can be referenced to [2, 3, 8, 9].

In what follows, as an illustration, we will develop a Bayesian analysis procedure for multivariate data under longitudinal setting. Multivariate longitudinal or clustered data occur when multiple items are measured repeatedly over periods of time or across occasions. Under such setting, the primary interest is inference about the dependence of the multiple measurements and the temporal correlation resulting from the repeated measures on the same items. But more often, particular interest also focuses on exploring the potential heterogeneity of data and investigating its transition pattern over time. In these cases, hidden Markov latent variable model (HMLVM) [10–13] provides a feasible and unified framework to address these issues. HMLVM assumes that the overall model constitutes the observed process and the underlying hidden state process. The state process, as the convention in the classic HMM (see for example, [14–17]), is an univariate discrete process, which follows a first-order Markov chain, while the observed process, conditional on the state sequence, is an independent process with emission distribution specified via LVMs [18]. Hence, in this regard, HMLVM provides a unified way of describing the correlation of multiple items, temporal dependence, and heterogeneity among the data simultaneously. However, the current existing developments cited beforehand focus on the maximum likelihood analysis in which statistical inferences heavily depend on the asymptotic properties. As an illustration of Bayesian inferences on practical problems, in this chapter, we develop a Bayesian procedure to analyze cocaine use data within the hidden Markov factor analysis model framework. Compared to ML, a basic nice feature of a Bayesian approach is its flexibility to utilize useful prior information for achieving better results. Additionally, simulation-based Bayesian methods depend less on asymptotic theory and hence have the potential to produce more reliable results even with small samples.

#### 2. Model description

p Yð Þ jθ is called likelihood or sampling distribution and often represented as Lð Þ θ . From the frequency statistics point of view, statistical inferences are carried out based on Lð Þ θ . In this case, θ, though unknown, is treated as fixed. Unlike the frequency statistical inferences, the Bayesian approach for data analysis assumes that θ is random and has a distribution π θð Þ. This distribution, which represents the knowledge about θ, is referred to as prior distribution or prior. When data are available, the information on θ is summarized within the posterior

omits the factor p Yð Þ since given Y it is a known constant. In Bayes literature, p Yð Þ jθ pð Þ θ is also termed the unnormalized posterior. Analogous to the role of likelihood in frequency statistical

Selecting proper priors for parameters is fundamental to Bayesian analysis. Basically, there are two kinds of prior distributions, namely, the noninformative prior distributions and the informative prior distributions. Noninformative prior distributions associate with situations when the prior distributions have no population basis. They are used when we have little prior information on θ and desire that the prior distributions play a minimal role in the posterior distribution distribution. Informative prior distribution represents the distribution of possible parameter values, from which the parameter θ has been drawn. We may have prior knowledge about this distribution, either from closed related data or from the subjective knowledge of experts. A commonly used informative prior distribution in the general Bayesian approach to statistical problems is the conjugate prior distribution, a prior ensuring that the posterior

A potential difficulty underlying Bayesian inferences is the statistical computation when posterior distribution takes on the complicated form. This is particularly true in the situation where latent variables or other unobservable quantities are involved in the model, as discussed in this chapter. In such cases, statistical inferences usually recur to simulation-based methods. Among various sampling methods, Markov chains Monte Carlo methods (MCMC) provide powerful tools for simulating observations from posterior. The key to Markov chain simulation is to create a Markov sequence whose stationary distribution is a specified posterior pð Þ θjY . Posterior inferences are carried out based on these simulated observations. There are many ways of constructing these Markov chains, but all of them, including the Gibbs sampler [4, 5], are special cases of the general framework of Metropolis et al. [6] and Hastings [7]. However, we do not intend to pursue this issue here, and details on simulation-based methods can be

In what follows, as an illustration, we will develop a Bayesian analysis procedure for multivariate data under longitudinal setting. Multivariate longitudinal or clustered data occur when multiple items are measured repeatedly over periods of time or across occasions. Under such setting, the primary interest is inference about the dependence of the multiple measurements and the temporal correlation resulting from the repeated measures on the same items. But more often, particular interest also focuses on exploring the potential heterogeneity of data and

p Yð Þ jθ π θð Þdθ is the marginal distribution of Y. The right-hand-side term in (1)

p Yð Þ <sup>∝</sup>p Yð Þ <sup>j</sup><sup>θ</sup> <sup>p</sup>ð Þ <sup>θ</sup> (1)

distribution or posterior, a conditional distribution θ given data, i.e.,

inferences, posterior is the starting point of Bayesian inferences.

where p Yð Þ¼ <sup>Ð</sup>

22 New Insights into Bayesian Inference

referenced to [2, 3, 8, 9].

<sup>p</sup>ð Þ¼ <sup>θ</sup>j<sup>Y</sup> p Yð Þ <sup>j</sup><sup>θ</sup> π θð Þ

distribution follows the same parametric form as the prior distribution [1, 3].

#### 2.1. Hidden Markov factor analysis model

Consider a set of multivariate longitudinal observations formed by p-dimensional observed vectors yit <sup>¼</sup> yit1;…; yitp <sup>⊺</sup> , which are recorded on p items over periods of length T: t ¼ 1, ⋯, T across N subjects: i ¼ 1, ⋯, N. In the field of multivariate analysis, interest mainly focuses on exploring item dependence since measurements may be highly correlated arising from the multicollinearity problem. But more often, interest also concentrates on the heterogeneity resulting from the situation where the population of yit constitutes more than one component. This is particularly true in the situation where the data illustrate extreme behaviors such as multimodal and/or skewed characteristics. In these cases, a finite mixture factor analysis model (FMFAM) can provide a powerful tool to address these issues. Typically, FMFAM assumes that conditioning on an univariate discrete value state variable zit and an m-dimensional (m < p) continuous latent factor vector ωit, yit are independent and distributed with a p-dimensional multivariate normal distribution, and meanwhile, given zit, ωit also follows an m-dimensional normal distribution, that is,

$$\begin{cases} \left( \mathbf{y}\_{it} | \boldsymbol{\omega}\_{it}, \boldsymbol{z}\_{it} = \boldsymbol{r} \right) \sim \mathcal{N}\_p \left( \boldsymbol{\mu}\_r + \boldsymbol{\Lambda}\_r \boldsymbol{\omega}\_{it}, \ \boldsymbol{\Psi}\_{\varepsilon r} \right) \\\ \left( \boldsymbol{\omega}\_{it} | \boldsymbol{z}\_{it} = \boldsymbol{r} \right) \sim \mathcal{N}\_m (\mathbf{0}, \boldsymbol{\Phi}\_r) \end{cases} \tag{2}$$

where <sup>μ</sup><sup>r</sup> <sup>¼</sup> <sup>μ</sup>r1;…; <sup>μ</sup>rp <sup>⊺</sup> is a p-dimensional intercept vector, which represents the baseline level of <sup>y</sup>it, <sup>Λ</sup><sup>r</sup> <sup>¼</sup> <sup>Λ</sup><sup>⊺</sup> <sup>r</sup>1; …; Λ<sup>⊺</sup> rp <sup>⊺</sup> is a <sup>p</sup> � <sup>m</sup> factor loading matrix, <sup>Ψ</sup><sup>e</sup><sup>r</sup> <sup>¼</sup> diag <sup>Ψ</sup><sup>e</sup>kr1; <sup>⋯</sup>; <sup>Ψ</sup><sup>e</sup>krp is a p � p diagonal matrix with the jth diagonal element Ψ<sup>e</sup>krj > 0, and Φ<sup>r</sup> is an m � m positive definite matrix.

Formulation given in (2) has two basic features: one is to characterize heterogeneity of population of yit at the occasion level and the other is to establish the dependence among the multiple measurements. The heterogeneous population is specified via state-specific parameters contained in the model while the dependence between different measurements is identified via sharing the common factors in the manner of liner combinations. In particular, apart from explaining the idiosyncratic part of measurements, latent factors also characterize the association between any two measurements. As a matter of fact, one can show that the correlation coefficient between yitj and yitk at state zit is given by

P<sup>S</sup>

<sup>s</sup>¼<sup>1</sup> Qrs <sup>¼</sup> <sup>1</sup>:0 for <sup>r</sup> <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, S. Modeling state sequences into (5) allows us to explore the transition pattern of individuals across occasions exactly. For example, in the cocaine use data analysis, zit is often identified with the latent state of patient i at time t, then Qrs specifies how individual i being in state r transfers to state s on two successive occasions. Surely, we can relax the time-homogeneous assumption of transition probabilities by including relevant covariates to interpret the inhomogeneous transition behavior among observation data (see, for example,

The current model defined in (2)–(5) provides a comprehensive framework for modeling the multivariate longitudinal data with the latent variables. It accommodates the dynamic behavior of observed sequences, heterogeneity of observed data at the occasion level, and dependence among the multiple items simultaneously. In particular, it makes sense to measure

Let Y be the collection of all observations, and Ω be the set of corresponding factors. Denote Z ¼ zit f g : 1 ≤ i ≤ N; 1 ≤ t ≤ T be set of state variables. It follows from Eqs. (2), (4) and (5) that the

t¼2

� �

trΨ<sup>e</sup>zit <sup>y</sup>it � <sup>μ</sup>zit � <sup>Λ</sup><sup>⊺</sup>

� <sup>Y</sup> N

i¼1

YT t¼1

2

where <sup>θ</sup> is formed by free parameters in <sup>μ</sup>r, <sup>Λ</sup>r, <sup>Ψ</sup>r, and <sup>Φ</sup>r. Here, we write <sup>a</sup> <sup>⊗</sup> <sup>2</sup> <sup>¼</sup> aa<sup>⊺</sup> and denote I Að Þ the indicator function of a set A. The observed data likelihood is then achieved by taking integration of pð Þ Y; Ω;Zjθ; δ; Q over Ω and Z, which involves high-dimensional inte-

� �, <sup>Λ</sup> <sup>¼</sup> f g <sup>Λ</sup><sup>r</sup> , <sup>Ψ</sup><sup>e</sup> <sup>¼</sup> f g <sup>Ψ</sup><sup>e</sup><sup>r</sup> , and <sup>Φ</sup> <sup>¼</sup> f g <sup>Φ</sup>kr . For the Bayesian analysis, we need to

pð Þ¼ θ; δ; Q pð Þ μ pð Þ Λ; Ψ<sup>e</sup> pð Þ Φ pð Þ δ pð Þ Q : (7)

assign priors to the unknown parameters involved for completing model specification. Since θ, δ, and Q are involved in different submodels, it is natural to assume that θ, δ, and Q are mutually independent and the components contained in θ are also mutually independent, that is,

<sup>p</sup> <sup>y</sup>it; <sup>ω</sup>itjzit; <sup>θ</sup> � �p zit ð Þ <sup>j</sup>zi,t�<sup>1</sup>; <sup>Q</sup>

zit <sup>ω</sup>it � � <sup>⊗</sup> <sup>2</sup>

Y S

δI zf g <sup>i</sup>1¼<sup>r</sup> r

Y S

Bayesian Analysis for Hidden Markov Factor Analysis Models

http://dx.doi.org/10.5772/intechopen.72837

rs !

<sup>Q</sup>I z f g i,t�1¼r;zit¼<sup>s</sup>

(6)

25

s¼1

r¼1

[12, 13, 16]) but at the expense of computational burden.

effects of latent factors on the manifest variables quantitatively.

<sup>p</sup> <sup>y</sup>i1; <sup>ω</sup>itjzi1; <sup>θ</sup> � �p zð Þ <sup>i</sup>1j<sup>δ</sup> <sup>Y</sup><sup>T</sup>

�<sup>1</sup>=<sup>2</sup> exp � <sup>1</sup>

2 trΦ�<sup>1</sup> zit <sup>ω</sup> <sup>⊗</sup> <sup>2</sup> it � �!

1 Ψ<sup>e</sup>zit � � �

<sup>1</sup>=<sup>2</sup> exp � <sup>1</sup>

joint sampling distribution of Y, Ω, and Z is given by

N

i¼1

YT t¼1

i¼1

� <sup>1</sup> Φzit � � � �

∝ Y N

<sup>p</sup>ðY; <sup>Ω</sup>; <sup>Z</sup>jθ; <sup>δ</sup>; <sup>Q</sup>Þ ¼ <sup>Y</sup>

3. Posterior inferences

3.1. Prior specifications

grations.

Let μ ¼ μ<sup>r</sup>

$$\text{Corr}\left(y\_{t\circ j}, y\_{tk}|z\_{lt} = r\right) = \frac{\sum\_{t=1}^{m} \sum\_{h=1}^{m} \lambda\_{r\circ l} \lambda\_{rkh} \Phi\_{rlh}}{\sqrt{\sum\_{t=1}^{m} \sum\_{h=1}^{m} \lambda\_{r\circ l} \lambda\_{r\circ h} \Phi\_{rlh} + \Psi\_{erj}} \sqrt{\sum\_{t=1}^{m} \sum\_{h=1}^{m} \lambda\_{rkt} \lambda\_{rkh} \Phi\_{rlh} + \Psi\_{erk}}} \tag{3}$$

in which λrjk is the ð Þ j; k th element of Λ<sup>r</sup> and Φr, hk is the ð Þ h; k th element in Φ, respectively. The strength of correlation is identified by the factor loadings and covariance of factors together. In the case when ωit degenerates to zero (i.e., Φ ¼ 0) or Λ ¼ 0, the association among items disappears and model (2) reduces to p-independent mean-variance models within cluster r. Hence, latent factors play a dominant role in characterizing association of multiple items. Note that, in actual applications, latent factors, though unobservable, often have their own physical interpretations. In psychology, for example, latent factors are often used to identify concepts such as treatment, temper, and anxiety, which are important within the framework of theoretical models. The measurements are just proxies for these unobserved concepts of interest. We will provide further interpretations in the real example.