Dịch đoạn văn sau sang tiếng việt trừ những từ ngữ chuyên ngành như trong kỹ thuật, lập trình, học máy,..., giữ nguyên định dạng và không đề cập thêm gì hết

All course materials will be made available online.



#### Assessment



Assessment for this course will be based on a combination of assignments, quizzes, a mid-term exam, and a final project. The final project will involve the application of the principles and methodologies learned throughout the course to a specific problem in Computational Cognitive Science.



#### Course Policies



Students are expected to adhere to all course policies, including those related to attendance, participation, and academic integrity. Any violations of these policies will be dealt with according to the university's disciplinary procedures.



In conclusion, this course aims to provide a comprehensive understanding of Computational Cognitive Science, its principles, methodologies, and applications. It is hoped that by the end of this course, students will not only have a deep understanding of the field but also be able to apply this knowledge in practical settings.



### Section: 1.2 Administrative details



#### Course Administration



The administration of this course is handled by the Administrative Division (AD) of the Department of Computer Science. The AD is responsible for personnel, property, and record administration. They ensure that all course materials are available online and that all administrative tasks related to the course are handled efficiently.



#### Course Schedule



The course is scheduled to run for a full academic semester. Specific dates and times for lectures, tutorials, and exams will be communicated to students via the course website and email. It is the responsibility of the students to keep themselves updated with the course schedule.



#### Course Instructors



The course is taught by a team of experienced instructors who are experts in the field of Computational Cognitive Science. The instructors are responsible for delivering lectures, conducting tutorials, and providing guidance to students throughout the course. Contact details of the instructors will be provided in the course syllabus.



#### Office Hours



Instructors will hold regular office hours to provide additional support to students. These sessions are an opportunity for students to ask questions, discuss course material, and seek clarification on any aspect of the course. The schedule for office hours will be communicated to students at the start of the course.



#### Course Communication



All communication related to the course will be conducted via email and the course website. Students are expected to check their email regularly and keep themselves updated with the latest course announcements.



#### Course Resources



The primary textbook for this course is "Computational Cognitive Science: A Comprehensive Guide". Additional readings will be assigned from various sources throughout the course. All course materials will be made available online.



#### Assessment



Assessment for this course will be based on a combination of assignments, quizzes, a mid-term exam, and a final project. The final project will involve the application of the principles and methodologies learned throughout the course to a specific problem in Computational Cognitive Science.



#### Course Policies



Students are expected to adhere to all course policies, including those related to academic integrity, attendance, and participation. Violation of these policies may result in penalties, including a reduction in grade or removal from the course. Detailed information about course policies will be provided in the course syllabus.



### Section: 1.3 Expectations



#### Course Expectations



This course is designed to provide a comprehensive understanding of Computational Cognitive Science. As such, it is expected that students will actively engage with the course material, participate in discussions, and complete all assignments and exams to the best of their ability.



#### Academic Integrity



Academic integrity is a fundamental value of this course. All work submitted for this course must be your own. Plagiarism, cheating, or any other form of academic dishonesty will not be tolerated. Any student found to be in violation of the academic integrity policy will face disciplinary action.



#### Attendance



Regular attendance is crucial for understanding the course material and for active participation in class discussions. While there is no strict attendance policy, students are strongly encouraged to attend all lectures and tutorials. If you must miss a class, it is your responsibility to catch up on the material covered.



#### Assignments



Assignments are designed to reinforce the concepts covered in the lectures and to provide practical experience with the computational tools and techniques used in cognitive science. All assignments must be submitted by the specified due date. Late submissions will be penalized unless a valid reason is provided.



#### Examinations



There will be two examinations for this course: a mid-term and a final exam. Both exams will cover the material presented in the lectures, tutorials, and assignments. The exams will test your understanding of the concepts and your ability to apply them to solve problems.



#### Participation



Active participation in class discussions and tutorials is strongly encouraged. This not only enhances your understanding of the course material but also helps to develop critical thinking and communication skills.



#### Course Feedback



Feedback is an essential part of the learning process. Students are encouraged to provide feedback on the course content, teaching methods, and their overall learning experience. This feedback will be used to improve the course in future semesters.



In conclusion, this course requires a significant commitment of time and effort. However, with active participation and diligent study, you will gain a deep understanding of Computational Cognitive Science and its applications.



### Conclusion



In this introductory chapter, we have laid the groundwork for our exploration into the fascinating field of computational cognitive science. We have set the stage for a comprehensive understanding of how computational models can be used to simulate and understand cognitive processes. While we have not delved into the specifics yet, we have established the importance of computational cognitive science in the broader context of cognitive science and artificial intelligence.



Computational cognitive science is a multidisciplinary field that draws from various areas such as psychology, computer science, neuroscience, and linguistics. It is a field that is constantly evolving and expanding, with new models and theories being developed to explain complex cognitive processes. As we move forward in this book, we will delve deeper into these models and theories, exploring how they can be used to simulate and understand cognitive processes.



In the upcoming chapters, we will explore various computational models, their applications, and their implications for our understanding of cognition. We will also discuss the challenges and limitations of these models, and how they can be addressed. By the end of this book, you should have a comprehensive understanding of computational cognitive science and its role in advancing our understanding of cognition.



### Exercises



#### Exercise 1

Research and write a brief summary about the history of computational cognitive science. How has it evolved over the years?



#### Exercise 2

Identify and describe three key disciplines that contribute to computational cognitive science. How do these disciplines intersect in this field?



#### Exercise 3

Why is computational cognitive science important in the broader context of cognitive science and artificial intelligence? Provide at least two reasons.



#### Exercise 4

What are some of the challenges and limitations of computational cognitive science? How can these be addressed?



#### Exercise 5

Choose a cognitive process (e.g., memory, attention, perception) and discuss how computational cognitive science might help us understand this process better.



### Conclusion



In this introductory chapter, we have laid the groundwork for our exploration into the fascinating field of computational cognitive science. We have set the stage for a comprehensive understanding of how computational models can be used to simulate and understand cognitive processes. While we have not delved into the specifics yet, we have established the importance of computational cognitive science in the broader context of cognitive science and artificial intelligence.



Computational cognitive science is a multidisciplinary field that draws from various areas such as psychology, computer science, neuroscience, and linguistics. It is a field that is constantly evolving and expanding, with new models and theories being developed to explain complex cognitive processes. As we move forward in this book, we will delve deeper into these models and theories, exploring how they can be used to simulate and understand cognitive processes.



In the upcoming chapters, we will explore various computational models, their applications, and their implications for our understanding of cognition. We will also discuss the challenges and limitations of these models, and how they can be addressed. By the end of this book, you should have a comprehensive understanding of computational cognitive science and its role in advancing our understanding of cognition.



### Exercises



#### Exercise 1

Research and write a brief summary about the history of computational cognitive science. How has it evolved over the years?



#### Exercise 2

Identify and describe three key disciplines that contribute to computational cognitive science. How do these disciplines intersect in this field?



#### Exercise 3

Why is computational cognitive science important in the broader context of cognitive science and artificial intelligence? Provide at least two reasons.



#### Exercise 4

What are some of the challenges and limitations of computational cognitive science? How can these be addressed?



#### Exercise 5

Choose a cognitive process (e.g., memory, attention, perception) and discuss how computational cognitive science might help us understand this process better.



## Chapter: Tutorial on Probability Theory, Bayesian Inference, Bayes Nets



### Introduction



In this chapter, we delve into the fascinating world of Probability Theory, Bayesian Inference, and Bayes Nets, three fundamental pillars of Computational Cognitive Science. These concepts form the bedrock of many algorithms and models used in this field, and understanding them is crucial to gaining a comprehensive understanding of Computational Cognitive Science.



Probability Theory is the branch of mathematics that deals with the analysis of random phenomena. It is a framework for quantifying our uncertainty about the world and making predictions based on observed data. We will explore the basic principles of Probability Theory, including the concepts of random variables, probability distributions, and conditional probability.



Next, we will introduce Bayesian Inference, a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian Inference is a powerful tool in Computational Cognitive Science, allowing us to make predictions and decisions under uncertainty.



Finally, we will discuss Bayes Nets, also known as Bayesian Networks. These are graphical models that represent the probabilistic relationships among a set of variables. Bayes Nets are a powerful tool for modeling complex systems, allowing us to capture the dependencies between variables and make predictions based on these dependencies.



Throughout this chapter, we will use the popular Markdown format for writing and the MathJax library for rendering mathematical expressions. This will allow us to present complex mathematical concepts in a clear and understandable way. For example, we will use the `$y_j(n)$` format for inline math and the `$$\Delta w = ...$$` format for equations.



By the end of this chapter, you will have a solid understanding of Probability Theory, Bayesian Inference, and Bayes Nets, and you will be well-equipped to apply these concepts in your own work in Computational Cognitive Science.



### Section: 2.1 Basic probability theory



Probability theory is a mathematical framework for quantifying our uncertainty about various phenomena. It provides a means to calculate the likelihood of an event given the occurrence of other events. In this section, we will introduce the basic concepts of probability theory, including the definitions of probability, conditional probability, and the chain rule of probability.



#### Definition of Probability



The probability of an event $A$ is a measure of the likelihood that $A$ will occur. It is denoted by $P(A)$ and satisfies the following properties:



1. Non-negativity: For any event $A$, $P(A) \geq 0$.

2. Normalization: The probability of the sample space $S$ is 1, i.e., $P(S) = 1$.

3. Additivity: For any two mutually exclusive events $A$ and $B$, $P(A \cup B) = P(A) + P(B)$.



#### Conditional Probability



Conditional probability is a measure of the probability of an event given that another event has occurred. If the event of interest is $A$ and event $B$ is known or assumed to have occurred, the conditional probability of $A$ given $B$ is usually written as $P(A|B)$.



#### Chain Rule of Probability



The chain rule, also known as the general product rule, allows the calculation of any member of the joint distribution of a set of random variables using only conditional probabilities. The chain rule is derived from the definition of conditional probability.



For events $A_1,\ldots,A_n$ whose intersection has not probability zero, the chain rule states:



$$

P(A_1 \cap A_2 \cap \ldots \cap A_n) = \prod_{k=1}^n P(A_k \mid A_1 \cap \dots \cap A_{k-1})

$$



This rule can be illustrated with the following examples:



##### Example 1



For four events, the chain rule reads:



$$

P(A_1 \cap A_2 \cap A_3 \cap A_4) = P(A_4 \mid A_3 \cap A_2 \cap A_1)P(A_3 \mid A_2 \cap A_1)P(A_2 \mid A_1)P(A_1)

$$



##### Example 2



We randomly draw 4 cards without replacement from a deck of 52 cards. What is the probability that we have picked 4 aces?



First, we set $A_n := \left\{ \text{draw an ace in the } n^{\text{th}} \text{ try} \right\}$. Obviously, we get the following probabilities:



$$

P(A_2 \mid A_1) = \frac 3{51}, 

P(A_3 \mid A_1 \cap A_2) = \frac 2{50}, 

P(A_4 \mid A_1 \cap A_2 \cap A_3) = \frac 1{49}

$$



Applying the chain rule, we can calculate the probability of drawing 4 aces.



In the next section, we will delve deeper into the concept of Bayesian Inference, a powerful tool for updating our beliefs based on new evidence.



### Section: 2.2 Bayesian inference



Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics.



#### Bayes' Theorem



Bayes' theorem is a fundamental theorem in probability theory and statistics that describes how to update the probabilities of hypotheses when given evidence. It is named after Thomas Bayes, who provided the first mathematical treatment of a non-trivial problem of statistical data analysis using what is now known as Bayesian inference.



The theorem is stated mathematically as the following equation:



$$

P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)}

$$



where:



- $P(H|E)$ is the probability of hypothesis $H$ given the evidence $E$. This is called the posterior probability.

- $P(E|H)$ is the probability of the evidence given that the hypothesis is true.

- $P(H)$ and $P(E)$ are the probabilities of the hypothesis and the evidence, respectively.



#### Bayesian Inference in Practice



In practice, Bayesian inference is done by specifying a prior probability distribution for the parameters of interest, collecting data, and then updating the prior distribution using Bayes' theorem to get a posterior distribution. The posterior distribution can then be used to make statistical inferences about the parameters.



For example, let's say we have a prior belief about the probability of a coin landing heads. We then flip the coin a number of times and record the results. Using Bayes' theorem, we can update our prior belief with the evidence from the coin flips to get a posterior belief about the probability of the coin landing heads.



#### Bayesian Networks



A Bayesian network, also known as a Bayes network, belief network, or decision network, is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor.



For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. Efficient algorithms exist that perform inference and learning in Bayesian networks. Bayesian networks that model sequences of variables, like speech signals or protein sequences, are called dynamic Bayesian networks.



In the next section, we will delve deeper into the concept of Bayesian networks and how they can be used in computational cognitive science.



### Section: 2.3 Bayes nets



Bayesian networks, also known as Bayes nets, belief networks, or decision networks, are a type of probabilistic graphical model that represent a set of variables and their conditional dependencies via a directed acyclic graph (DAG). 



#### Structure of a Bayes Net



A Bayes net is composed of nodes and edges. Each node in the network represents a random variable, which can be either discrete or continuous. The edges, on the other hand, represent the probabilistic dependencies among the variables. An edge from node A to node B indicates that B is conditionally dependent on A. The absence of an edge indicates conditional independence.



The structure of a Bayes net encodes the joint distribution of its variables. Specifically, the joint distribution is the product of the conditional probabilities specified by the network. Mathematically, this is expressed as:



$$

P(X_1, X_2, ..., X_n) = \prod_{i=1}^{n} P(X_i | Parents(X_i))

$$



where $Parents(X_i)$ denotes the parents of node $X_i$ in the network.



#### Inference in Bayes Nets



Inference in Bayes nets involves computing the posterior distribution of a set of query variables given some observed variables. This is typically done using algorithms like variable elimination, belief propagation, or sampling methods.



For example, consider a Bayes net with variables $A$, $B$, and $C$, where $A$ and $B$ are parents of $C$. If we observe that $C$ is true, we might want to infer the posterior distribution of $A$ given this observation. This involves summing over all possible values of $B$:



$$

P(A | C=true) = \sum_{b \in \{true, false\}} P(A, B=b | C=true)

$$



This computation can be expensive, especially for large networks, but various algorithms have been developed to perform it efficiently.



#### Learning Bayes Nets



Learning a Bayes net from data involves two tasks: learning the structure (i.e., the DAG) and learning the parameters (i.e., the conditional probability distributions). 



Structure learning can be done using search-and-score methods, which search over the space of possible DAGs and score each one according to how well it fits the data. Common scoring metrics include the Bayesian Information Criterion (BIC) and the Akaike Information Criterion (AIC).



Parameter learning, on the other hand, can be done using maximum likelihood estimation or Bayesian estimation. In the maximum likelihood approach, the parameters are chosen to maximize the likelihood of the observed data given the parameters. In the Bayesian approach, a prior distribution is specified over the parameters, and this is updated to a posterior distribution given the observed data.



In the next section, we will delve deeper into the algorithms used for inference and learning in Bayes nets.



### Conclusion



In this chapter, we have explored the fundamental concepts of Probability Theory, Bayesian Inference, and Bayes Nets, which are crucial in the field of Computational Cognitive Science. We began by understanding the basics of Probability Theory, which is the mathematical framework that allows us to quantify uncertainty. We then delved into Bayesian Inference, a method of statistical inference that updates the probability for a hypothesis as more evidence or information becomes available. Lastly, we discussed Bayes Nets, a graphical model that represents the probabilistic relationships among a set of variables.



These concepts are not just theoretical constructs, but they have practical applications in various fields such as artificial intelligence, machine learning, and cognitive modeling. Understanding these concepts is essential for anyone who wishes to delve deeper into the field of Computational Cognitive Science. They provide the mathematical and conceptual foundation upon which more complex models and theories are built.



As we move forward, we will see how these concepts are applied and extended in various ways to model and understand cognitive processes. The journey of learning Computational Cognitive Science is a challenging one, but with a solid understanding of these foundational concepts, you are well-equipped to tackle the complexities that lie ahead.



### Exercises



#### Exercise 1

Given a coin that you know is biased but you do not know the bias, how would you use Bayesian Inference to estimate the bias?



#### Exercise 2

Draw a Bayes Net that represents the probabilistic relationships among the following variables: Rain, Traffic, Late for Work. Explain the relationships you have drawn.



#### Exercise 3

Using the Bayes Net you drew in Exercise 2, if you know that it is raining, how would you update the probabilities of Traffic and Late for Work?



#### Exercise 4

Explain the difference between frequentist and Bayesian interpretations of probability. Give an example of a situation where the Bayesian interpretation would be more appropriate.



#### Exercise 5

Given the following probabilities: P(A) = 0.6, P(B) = 0.4, P(A|B) = 0.3, calculate P(B|A) using Bayes' theorem.



### Conclusion



In this chapter, we have explored the fundamental concepts of Probability Theory, Bayesian Inference, and Bayes Nets, which are crucial in the field of Computational Cognitive Science. We began by understanding the basics of Probability Theory, which is the mathematical framework that allows us to quantify uncertainty. We then delved into Bayesian Inference, a method of statistical inference that updates the probability for a hypothesis as more evidence or information becomes available. Lastly, we discussed Bayes Nets, a graphical model that represents the probabilistic relationships among a set of variables.



These concepts are not just theoretical constructs, but they have practical applications in various fields such as artificial intelligence, machine learning, and cognitive modeling. Understanding these concepts is essential for anyone who wishes to delve deeper into the field of Computational Cognitive Science. They provide the mathematical and conceptual foundation upon which more complex models and theories are built.



As we move forward, we will see how these concepts are applied and extended in various ways to model and understand cognitive processes. The journey of learning Computational Cognitive Science is a challenging one, but with a solid understanding of these foundational concepts, you are well-equipped to tackle the complexities that lie ahead.



### Exercises



#### Exercise 1

Given a coin that you know is biased but you do not know the bias, how would you use Bayesian Inference to estimate the bias?



#### Exercise 2

Draw a Bayes Net that represents the probabilistic relationships among the following variables: Rain, Traffic, Late for Work. Explain the relationships you have drawn.



#### Exercise 3

Using the Bayes Net you drew in Exercise 2, if you know that it is raining, how would you update the probabilities of Traffic and Late for Work?



#### Exercise 4

Explain the difference between frequentist and Bayesian interpretations of probability. Give an example of a situation where the Bayesian interpretation would be more appropriate.



#### Exercise 5

Given the following probabilities: P(A) = 0.6, P(B) = 0.4, P(A|B) = 0.3, calculate P(B|A) using Bayes' theorem.



## Chapter 3: Induction



### Introduction



Induction, as a fundamental concept in cognitive science, is the process by which we form generalizations based on specific observations. This chapter delves into the computational aspects of induction, exploring how it is modeled and understood within the realm of cognitive science.



Induction is a cornerstone of human cognition, enabling us to navigate our world by making predictions and assumptions based on our experiences. It is the mechanism that allows us to learn from our past, anticipate the future, and make sense of the present. However, the process of induction is not as straightforward as it might seem. It involves complex cognitive processes and computations that are the subject of ongoing research in the field of cognitive science.



In this chapter, we will explore the computational models of induction, discussing how these models attempt to replicate the human ability to induce general principles from specific instances. We will delve into the mathematical foundations of these models, using the popular Markdown format and MathJax library to present equations in a clear and comprehensible manner.



We will also discuss the challenges and limitations of computational induction, highlighting the areas where current models fall short and the directions for future research. This will provide a comprehensive understanding of the state of the art in computational cognitive science as it pertains to induction.



By the end of this chapter, you will have a solid understanding of the role of induction in cognitive science, the computational models that attempt to explain it, and the challenges that lie ahead in this fascinating area of research. Whether you are a student, a researcher, or simply someone interested in the workings of the human mind, this chapter will provide you with valuable insights into the computational aspects of induction.



### Section: 3.1 Goodman's grue problem



Goodman's grue problem, named after philosopher Nelson Goodman, is a paradox that challenges our understanding of the process of induction. It is a thought experiment that questions the assumptions we make when we generalize from specific instances, and it has significant implications for the computational models of induction that we will discuss in this chapter.



#### The Grue Paradox



The grue paradox is based on the concept of 'grue' objects, which Goodman defined as objects that are observed to be green before a certain time t, and blue thereafter. The paradox arises when we try to predict the color of a grue object after time t based on our observations before time t.



Suppose we have observed a large number of emeralds, all of which have been green. Based on these observations, we might induce that all emeralds are green. However, according to Goodman's definition, it is equally valid to induce that all emeralds are grue, since all emeralds observed before time t have been green.



The problem arises when we try to predict the color of an emerald that we will observe after time t. If we have induced that all emeralds are green, we will predict that the emerald will be green. But if we have induced that all emeralds are grue, we will predict that the emerald will be blue. This leads to a contradiction, as the same set of observations leads to two incompatible predictions.



#### Implications for Computational Models of Induction



Goodman's grue problem poses a significant challenge for computational models of induction