[Audio] We will discuss probabilistic reasoning techniques for managing uncertainty including Bayesian inference naïve Bayes models Bayesian networks exact inference in Bayesian networks approximate inference in Bayesian networks causal networks and the importance of probability theory in uncertainty handling. We will also examine sources of uncertainty in the real world and the role of probabilistic reasoning in addressing these uncertainties. By the end of this presentation you will have a better understanding of how to apply probabilistic reasoning in your own work in artificial intelligence and machine learning. Therefore let us proceed!. 

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CS3491 – ARTIFICIAL INTELLIGENCE AND MACHINE LEARNING

[Audio] Probabilistic reasoning is a fundamental concept in this field. In this topic we will discuss various methods for acting under uncertainty including Bayesian inference and naïve Bayes models. We will also examine Bayesian networks and the exact and approximate inference methods used within them. Lastly we will delve into causal networks which are used to represent relationships between variables in a probabilistic context. By mastering these probabilistic reasoning techniques you will be better prepared to handle problems in artificial intelligence and machine learning.. 

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Syllabus 	Acting under uncertainty – Bayesian inference – naïve bayes models. Probabilistic reasoning – Bayesian networks – exact inference in BN – approximate inference in BN – causal networks

[Audio] Uncertainty and ignorance are two different concepts. Uncertainty refers to situations where information is incomplete or unknown. However uncertainty is not a straightforward concept without a description. It can arise due to incompleteness or incorrectness in agents understanding the environment. To represent uncertain knowledge uncertain reasoning or probabilistic reasoning is necessary. In this presentation we will discuss the different types of probabilistic reasoning and their applications.. 

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Uncertainty is a situation which involves imperfect and/or unknown information. However, "uncertainty is an unintelligible expression without a straightforward description. Uncertainty can also arise because of incompleteness, incorrectness in agents understanding the properties of environment. So to represent uncertain knowledge, where we are not sure about the predicates, we need uncertain reasoning or probabilistic reasoning.  A key condition that differentiates ignorance from uncertainty is the absence of knowledge about the factors that influence the issues

[Audio] Agents in complex or uncertain environments cannot find definitive answers. 

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Agents almost never have access to the whole truth about their environment. Uncertainty arises because of both laziness and ignorance. It is inescapable in complex, nondeterministic, or partially observable environments Agents cannot find a categorical answer. Uncertainty can also arise because of incompleteness, incorrectness in agent understanding of properties of environment

[Audio] Probability theory can summarize uncertainty caused by our laziness and ignorance. Probability statements have no evidence semantics so it can help in making better decisions and predictions.. 

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Following are some leading causes of uncertainty to occur in the real world. Information occurred from unreliable sources. Experimental Errors Equipment fault Temperature variation Climate change. Need for probability theory in uncertainty    Probability provides the way of summarizing the uncertainty that comes from our laziness and ignorance. Probability statements do not have quite the same kind of semantics known as evidences.

[Audio] We use probability theory and logic to manage uncertainty in higher education. Some individuals may approach probabilistic reasoning sluggishly or with ignorance but we use probability to handle this uncertainty. Probability is a crucial tool for anyone seeking to comprehend artificial intelligence and machine learning.. 

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Probabilistic reasoning is a way of knowledge representation where we apply the concept of probability to indicate the uncertainty in knowledge. In probabilistic reasoning, we combine probability theory with logic to handle the uncertainty. We use probability in probabilistic reasoning because it provides a way to handle the uncertainty that is the result of someone's laziness and ignorance.

[Audio] Probabilistic reasoning is a way of using uncertain knowledge to solve problems. It's especially useful when there are unpredictable outcomes when specifications or possibilities of predicates becomes too large to handle or when an unknown error occurs during an experiment. There are two main ways to solve problems with uncertain knowledge using probabilistic reasoning: Bayes' rule and Bayesian Statistics. These methods allow us to make predictions and make decisions based on the available data even when we don't have all the information we need. Understanding the basics of probabilistic reasoning is important if you're working with (A-I ). It's a powerful tool that can help you make more informed decisions and solve complex problems.. 

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When there are unpredictable outcomes. When specifications or possibilities of predicates becomes too large to handle. When an unknown error occurs during an experiment. In probabilistic reasoning, there are two ways to solve problems with uncertain knowledge:  Bayes' rule Bayesian Statistics

[Audio] Probability is a measure of the likelihood that an uncertain event will occur. It is defined as the numerical measure of the likelihood that an event will occur. The value of probability always remains between 0 and 1 representing ideal uncertainties.. 

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Probability can be defined as a chance that an uncertain event will occur. It is the numerical measure of the likelihood that an event will occur. The value of probability always remains between 0 and 1 that represent ideal uncertainties. 0 ≤ P (A )≤ 1, where P (A )is the probability of an event A. P(A)= 0, indicates total uncertainty in an event A.  P(A)=1, indicates total certainty in an event A. P(¬A) = probability of a not happening event.  P(¬A) + P(A) = 1.

[image] Number of desired outcomes
Probability of occurrence =
Total number of outcomes

[Audio] We can define an event as a possible outcome of a variable. For example a coin can have two events: heads or tails. The sample space is the collection of all possible events. For example the sample space of a coin can be heads and tails. Random variables are used to represent events and objects in the real world. For example a person's height can be a random variable. Prior probability is the probability computed before observing new information. It is a way of making a guess or prediction about the likelihood of an event happening. Posterior probability is the probability that is calculated after all evidence or information has taken into account. It is a combination of prior probability and new information. For example if we knew that the coin was fair the prior probability of heads would be 0.5 and the prior probability of tails would also be 0.5. If we flip the coin and get heads the posterior probability of heads would be 0.5 but the posterior probability of tails would now be 0.. 

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Event: Each possible outcome of a variable is called an event. Sample space: The collection of all possible events is called sample space. Random variables: Random variables are used to represent the events and objects in the real world. Prior probability: The prior probability of an event is probability computed before observing new information. Posterior Probability: The probability that is calculated after all evidence or information has taken into account. It is a combination of prior probability and new information.

[Audio] We will discuss the concept of conditional probability in probability theory. Conditional probability is the probability of an event occurring under specific conditions. For example if we want to determine the probability of a person going to the doctor based on whether they have a fever we would use conditional probability. We can calculate the probability of a person going to the doctor given that they have a fever using the formula: P(A⋀B) = P(A and B) / P(B) where P(A⋀B) denotes the probability of A given B P(A) denotes the probability of A and P(B) denotes the probability of B This formula enables us to determine the probability of an event occurring under specific circumstances which is useful in many scenarios.. 

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Conditional probability is a probability of occurring an event when another event has already happened. Let's suppose, we want to calculate the event A when event B has already occurred, "the probability of A under the conditions of B", it can be written as: If the probability of A is given and we need to find the probability of B, then it will be given as
 Where P(A⋀B)= Joint probability of a and B P(B)= Marginal probability of B.

[Audio] We need to understand how to calculate the probability of an event given that another event has a probability. This can be done using the formula: P(A⋀B) = P(A) * P(B|A) / P(B). This formula involves representing A and B as sets using Venn diagrams. If event B happens the sample space is decreased to set B and we can calculate the probability of event A given that event B has already occurred by dividing P(A⋀B) by P(B). This is an essential concept in probabilistic reasoning and is widely used in many areas such as machine learning and artificial intelligence.. 

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If the probability of A is given and we need to find the probability of B, then it will be given as: It can be explained by using the below Venn diagram, where B is occurred event, so sample space will be reduced to set B, and now we can only calculate event A when event B is already occurred by dividing the probability of P (A⋀B ) by P ( B ).

[Audio] Discuss the topic of probabilistic reasoning.. 

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In a class, there are 70% of the students who like English and 40% of the students who likes English and mathematics, and then what is the percent of students those who like English also like mathematics? Solution: Let, A is an event that a student likes Mathematics, B is an event that a student likes English.

 Hence, 57% are the students who like English also like Mathematics.

[Audio] Bayes' theorem establishes a relationship between the conditional probability and marginal probabilities of two random events. Through Bayesian inference we can calculate the probability of an event with uncertain knowledge. With this knowledge we can update our probability prediction of an event by observing new information from the real world.. 

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Bayes' theorem is also known as Bayes' rule, Bayes' law, or Bayesian reasoning, which determines the probability of an event with uncertain knowledge. In probability theory, it relates the conditional probability and marginal probabilities of two random events. Bayes' theorem was named after the British mathematician Thomas Bayes.  The Bayesian inference is an application of Bayes' theorem, which is fundamental to Bayesian statistics. It is a way to calculate the value of P (B|A ) with the knowledge of P (A|B ). Bayes' theorem allows updating the probability prediction of an event by observing new information of the real world.

[Audio] Bayes' theorem is a tool that calculates the probability of an event occurring based on prior knowledge and observed evidence. In the context of cancer diagnosis Bayes' theorem can be used to calculate the probability of a person having cancer given their age and other relevant information. Using Bayes' theorem doctors can make more accurate diagnoses and provide more effective treatment options. To calculate the probability of cancer using Bayes' theorem we need to know the prior probability of having cancer which is often estimated based on age and other risk factors. We also need to know the conditional probability of having cancer given age and other relevant information. Once we have these probabilities we can use Bayes' theorem to calculate the posterior probability of having cancer which represents the updated probability of having cancer based on the observed evidence. This posterior probability can be used to inform treatment decisions and guide patient care. In summary Bayes' theorem is a powerful tool that can be used to accurately determine the probability of cancer diagnosis. By using this theorem we can provide more effective treatment options and improve patient outcomes.. 

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If cancer corresponds to one's age then by using Bayes' theorem, we can determine the probability of cancer more accurately with the  Bayes' theorem can be derived using product rule and conditional probability of event A with known event B:As from product rule we can write: P(A ⋀ B)= P(A|B)P(B)or Similarly, the probability of event B with known event A: P(A ⋀ B)= P(B|A)P(A) Equating right hand side of both the equations, we will get: help of age.

[Audio] Posterior probability is the probability of hypothesis A given evidence B It is calculated as P(A|B) = P(B|A) \* P(A) / P(B). This means that we calculate the probability of hypothesis A given evidence B by considering both the probability of evidence B given hypothesis A and the prior probability of hypothesis A Likelihood is the probability of evidence B given that hypothesis A is true and is calculated as P(B|A). Understanding these concepts is essential for developing effective probabilistic models in artificial intelligence and machine learning.. 

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P(A|B) is known as posterior, which we need to calculate, and it will be read as Probability of hypothesis A when we have occurred an evidence B. P(B|A) is called the likelihood, in which we consider that hypothesis is true, then we calculate the probability of evidence. P(A) is called the prior probability, probability of hypothesis before considering the evidence P(B) is called marginal probability, pure probability of an evidence.

[Audio] Bayes' rule is a fundamental concept in probabilistic reasoning. It enables us to compute the probability of an event given some evidence particularly useful when we have good probability of the evidence. In the context of perception Bayes' rule enables us to determine the cause of some unknown event by using the evidence of the effect such as computing the probability that the cause of a tree falling in a forest was a lightning strike. Bayes' rule is a powerful tool in artificial intelligence and machine learning with many applications.. 

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Bayes' rule allows us to compute the single term P(B|A) in terms of P(A|B), P(B), and P(A).  This is very useful in cases where we have a good probability of these three terms and want to determine the fourth one. Suppose we want to perceive the effect of some unknown cause, and want to compute that cause, then the Bayes' rule becomes:

[image] P(effectlcause) P(cause)
P(cause I effect) =
P (effect)

[Audio] We will discuss example-1 which looks at the probability of a patient having meningitis with a stiff neck. Specifically we will examine the probability of a patient having meningitis given that they have a stiff neck. The known probability of a patient having meningitis is 1/30
000 which means that P(a|b) = 0.8 where P(a) is the probability of having meningitis and P(b) is the probability of having a stiff neck. Therefore we can conclude that 1 patient out of 750 patients has meningitis with a stiff neck which gives us a probability of 2%. This example is a great way to understand how probability can be used in artificial intelligence. By understanding the probability of certain events happening we can make better decisions and predictions.. 

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Example-1:What is the probability that a patient has diseases meningitis with a stiff neck?

Given Data: The Known probability that a patient has meningitis disease is 1/30,000. P(a|b) = 0.8 P(b) = 1/30000 P(a)= .02 Hence, we can assume that 1 patient out of 750 patients has meningitis disease with a stiff neck. probability that a patient has a stiff neck is 2%

[Audio] Consider a standard deck of playing cards where there is a king card in 4 out of 52 cards. Now imagine that a single card is drawn from the deck and it is revealed to be a face card. The probability that the face card is the king card is 4/52 which can be calculated as the ratio of the number of king cards to the total number of cards in the deck. To calculate the posterior probability of the event occurring we use Bayes' theorem which states that P(A|B) = P(B|A) * P(A) / P(B) In this case A represents the event of the face card being the king card and B represents the drawn face card. The probability of the event A occurring is P(A) = 4/52 which is the probability of drawing a king card. The probability of the event B occurring given that A occurs is P(B|A) = 1 which means that if the drawn face card is the king card it is certain that the event B will occur. The probability of the event B occurring is P(B) = P(B|A) * P(A) = 4/52 which is the probability of drawing a face card. Finally we can calculate the posterior probability of the event A occurring given the observed event B as P(A|B) = P(B|A) * P(A) / P(B) = 4/52 * 4/52 / 4/52 = 1 which means that given the observed event B it is certain that the event A occurred. This example demonstrates how to calculate the posterior probability of an event occurring given prior knowledge using Bayes' theorem.. 

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From a standard deck of playing cards, a single card is drawn. The probability that the card is king is 4/52, then calculate posterior probability P(King|Face), which means the drawn face card is a king card. Solution:

[Audio] 1. Remove the welcome sentences: We will discuss the solution to a problem involving card probabilities. 2. Rewrite the text: We will look at the probability of the card being a King the probability of the card being a face card and the probability of a face card given that it is a King. These probabilities will be calculated using the formula P(A or B) = P(A) plus P(B) P(A and B). We will start by looking at the probability of the card being a King. The probability of a card being a King is 1/13. Next we will look at the probability of the card being a face card. The probability of a card being a face card is 3/13. Finally we will look at the probability of a face card given that it is a King. The probability of a face card given that it is a King is 1. To put all of these values in equation (i) we can use the formula P(A or B) = P(A) plus P(B) P(A and B). This will give us: P(king or face) = 1/13 plus 3/13 0 = 16/13. So the probability of the card being a King or a face card is 16/13.. 

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P(king): probability that the card is King= 4/52= 1/13 P(face): probability that a card is a face card= 3/13 P(Face|King ): probability of face card when we assume it is a king = 1 Putting all values in equation (i ) we will get:

[image] P(king I face) =
= 1/3, it is a probability that a face card is a king card.

[Audio] Bayes' theorem is a powerful tool that can be used to calculate the next step of the robot when the already executed step is given. It is particularly useful in weather forecasting where it can help predict the likelihood of certain weather patterns occurring. Additionally Bayes' theorem can be used to solve the Monty Hall problem a classic problem in probability theory. It is also a valuable tool in the field of Artificial Intelligence that can help us make more informed decisions and improve the performance of our machines.. 

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Application of Bayes' theorem in Artificial intelligence:

It is used to calculate the next step of the robot when the already executed step is given. Bayes' theorem is helpful in weather forecasting. It can solve the Monty Hall problem.

[Audio] Bayesian Networks are a type of probabilistic graphical model that represent a set of variables and their conditional dependencies using a directed acyclic graph. They are also known as belief networks decision networks or Bayesian models. One of the key features of Bayesian networks is that they are probabilistic meaning that they are built from a probability distribution and use probability theory for prediction and anomaly detection. Bayesian networks can be used for tasks such as classification prediction and decision making.. 

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"A Bayesian network is a probabilistic graphical model which represents a set of variables and theirconditional dependencies using a directed acyclic graph." It is also called a Bayes network, belief network, decision network, or Bayesian model. Bayesian networks are probabilistic, because these networks are built from a probability distribution, and also use probability theory for prediction and anomaly detection.

[Audio] Bayesian Networks are a fundamental concept in probabilistic reasoning. They are used for building models from data and expert opinions and consist of two parts: a Directed Acyclic Graph and a Table of conditional probabilities. Bayesian Networks use nodes to represent variables and edges to represent the relationships between them. The Directed Acyclic Graph shows the causal relationships between these variables and the Table of conditional probabilities stores the probability of each variable given its parents allowing to calculate the probability of any variable given its parents useful for making predictions and solving decision problems. Bayesian Networks are a powerful tool for building models in various fields including healthcare finance and engineering and can be used to model complex systems and make decisions based on uncertain knowledge.. 

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Bayesian Network can be used for building models from data and experts opinions, and it consists oftwo parts: Directed Acyclic Graph  Table of conditional probabilities. The generalized form of Bayesian network that represents and solve decision problems under uncertain knowledge is known as an Influence diagram.

[Audio] Today we will discuss Bayesian Networks. Bayesian networks are useful for probabilistic reasoning and can be applied in various fields.. 

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A Bayesian network graph is made up of nodes and Arcs (directed links), where:  each node corresponds to the random variables, and a variable can be continuous or discrete.  Arc or directed arrows represent the causal relationship or conditional probabilities between random variables. These directed links or arrows connect the pair of nodes in the graph.

[Audio] We will discuss probabilistic reasoning and its importance in understanding the relationships between random variables represented by nodes in a network graph. We can see from the diagram that there are links between nodes A B C and D and each node represents a random variable. The arrows in the diagram indicate the direction of influence between the nodes. If there is a directed link between nodes A and B then A is considered the parent of B meaning A has an influence on B On the other hand if there is no directed link between A and C then A and C are independent of each other. Understanding these relationships between random variables is crucial in probabilistic reasoning. By analyzing the probabilities of different outcomes we can make predictions and make informed decisions. In summary probabilistic reasoning is an important tool in understanding the relationships between random variables in a network graph. By analyzing the probabilities of different outcomes we can make predictions and make informed decisions.. 

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These links represent that one node directly influence the other node, and if there is no directed link that means that nodes are independent with each other In the above diagram, A, B, C, and D are random variables represented by the nodes of the network graph. If we are considering node B, which is connected with node A by a directed arrow, then node A is called the parent of Node B. Node C is independent of node A.

[Audio] The joint probability distribution in Bayesian networks describes the probability of all possible outcomes of a set of random variables. This distribution is used to determine the probability of each node given its parent nodes. To calculate the joint probability of each node we multiply the condition probability distribution of its parent nodes and the likelihood function of the node. The likelihood function describes the probability of observing the data given the state of the random variable. With the joint probability distribution we can make predictions and perform causality reasoning in Bayesian networks.. 

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Bayesian network has mainly two components:

Causal Component Actual numbers Each node in the Bayesian network has condition probability distribution P(Xi |Parent(Xi) ), whichdetermines the effect of the parent on that node. Bayesian network is based on Joint probability distribution and conditional probability. So let's first understand the joint probability distribution:

[Audio] When we have variables x1 x2 x3
..... xn the probabilities of different combinations of x1 x2 x3.. xn are known as Joint probability distribution. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities. 

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If we have variables x1, x2, x3,....., xn, then the probabilities of a different combination of x1, x2, x3.. xn, are known as Joint probability distribution. P[x1, x2, x3,....., xn], it can be written as the following way in terms of the joint probability distribution.     = P[x1| x2, x3,....., xn]P[x2, x3,....., xn] = P[x1| x2, x3,....., xn]P[x2|x3,....., xn]....P[xn-1|xn]P[xn] In general for each variable Xi, we can write the equation as: P(Xi|Xi-1,........., X1) = P(Xi |Parents(Xi))

[Audio] Discuss Bayesian networks and their real-world applications.. 

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Example:  Harry installed a new burglar alarm at his home to detect burglary. The alarm reliably responds at detecting a burglary but also responds for minor earthquakes.  Harry has two neighbour David and Sophia, who have taken a responsibility to inform Harry at work when they hear the alarm David always calls Harry when he hears the alarm, but sometimes he got confused with the phone ringing and calls at that time too.  On the other hand, Sophia likes to listen to high music, so sometimes she misses to hear the alarm. Here we would like to compute the probability of Burglary Alarm.

[Audio] The probability that the alarm has sounded is 20.4%.. 

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Calculate the probability that alarm has sounded, but there is neither a burglary, nor an earthquake occurred, and David and Sophia both called the Harry.
 Solution : List of all events occurring in this network:
 o Burglary (B) o Earthquake(E) o Alarm(A) o David Calls(D) o Sophia calls(S)

[Audio] Probabilistic reasoning is a technique used in (A-I ) and ML to make predictions and decisions based on the probability of different outcomes. In the context of (A-I ) and ML probabilistic reasoning can be used to predict the likelihood of different events occurring such as a burglary or an earthquake. The image shows probabilities of different outcomes for various events. In this case the probability of burglary is 0.998 while the probability of earthquake is 0.75. Probabilistic reasoning can also be used to make decisions based on these probabilities such as installing an alarm to prevent burglary. Overall probabilistic reasoning is an important tool in (A-I ) and ML for making predictions and decisions based on the likelihood of different outcomes.. 

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[image] 0.002
Burglary
0.998
B
Alarm
E
Earthquake
B
F
s
Sophia
calls
E
F
0.91
0.05
David Calls
0.09
0.95
0.94
0.95
0.69
0.999
0.75
0.02
0.001
0.06
0.69
0.999
0.25
0.98

[Audio] Discuss observed probabilities for the Burglary and earthquake components. Examine values of P(B= True) = 0.002 P(B= False)= 0.998 P(E= True) = 0.001 and P(E= False)= 0.999. These probabilities can be used to make predictions and decisions about the potential occurrence of these events.. 

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Let's take the observed probability for the Burglary and earthquake component: P(B= True) = 0.002, which is the probability of burglary. P(B= False)= 0.998, which is the probability of no burglary. P(E= True)= 0.001, which is the probability of a minor earthquake P(E= False)= 0.999, Which is the probability that an earthquake not occurred.

[Audio] 1. Discuss the conditional probability of Alarm A in the context of burglar and earthquake. 2. 3. Discuss. 

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Conditional probability table for Alarm A:

The Conditional probability of Alarm A depends on Burglar and earthquake:

[image] True)
True
True
False
False
False)
True
False
True
False
0.94
0.95
0.31
0.001
0.06
0.04
0.69
0.999

[Audio] We will discuss the mathematical concepts behind conditional probability tables. These concepts are widely used in the telecommunications industry. In the context of David calls the random variables are the different events that can occur during a call such as the caller's location the type of call and the duration of the call. The conditional probability table for David calls shows the probabilities of different events occurring given the state of a call. For example the table can show the probability of a call lasting a certain duration given the caller's location. Understanding conditional probability tables is crucial for building accurate and reliable David call models. With conditional probability tables we can make predictions about the likelihood of different events occurring during a call based on the state of the call.. 

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Conditional probability table for David Calls:

[image] True
False
True)
0.91
0.05
False)
0.09
0.95

[Audio] The conditional probability table for Sophia Calls shows the probability of Sophia making a specific call based on the presence or absence of certain attributes and conditions. The table includes mutually exclusive attributes such as B and E as well as the probability of Sophia making a call based on the presence or absence of certain conditions such as S and D The table is calculated by multiplying the probabilities of each individual attribute and condition which helps to provide a more accurate representation of Sophia's behavior and decision-making process.. 

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Conditional probability table for Sophia Calls:

[image] True
False
True)
0.75
0.02
False)
0.25
0.98

P(S, D, A, ¬B, ¬E)  	= P (S|A) *P (D|A)*P (A|¬B ^ ¬E) *P (¬B) *P(¬E). 	= 0.75* 0.91* 0.001* 0.998*0.999 	= 0.00068045.

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We will discuss probabilistic reasoning techniques for managing uncertainty
 including Bayesian inference
 naïve Bayes models
 Bayesian networks
 exact inference in Bayesian networks
 approximate inference in Bayesian networks
 causal networks
 and the importance of probability theory in uncertainty handling. We will also examine sources of uncertainty in the real world and the role of probabilistic reasoning in addressing these uncertainties. By the end of this presentation
 you will have a better understanding of how to apply probabilistic reasoning in your own work in artificial intelligence and machine learning. Therefore
 let us proceed!

avatar

Probabilistic reasoning is a fundamental concept in this field. In this topic
 we will discuss various methods for acting under uncertainty
 including Bayesian inference and naïve Bayes models. We will also examine Bayesian networks and the exact and approximate inference methods used within them. Lastly
 we will delve into causal networks
 which are used to represent relationships between variables in a probabilistic context. By mastering these probabilistic reasoning techniques
 you will be better prepared to handle problems in artificial intelligence and machine learning.

Uncertainty and ignorance are two different concepts. Uncertainty refers to situations where information is incomplete or unknown. However
 uncertainty is not a straightforward concept without a description. It can arise due to incompleteness or incorrectness in agents understanding the environment. To represent uncertain knowledge
 uncertain reasoning or probabilistic reasoning is necessary. In this presentation
 we will discuss the different types of probabilistic reasoning and their applications.

Agents in complex or uncertain environments cannot find definitive answers

Probability theory can summarize uncertainty caused by our laziness and ignorance. Probability statements have no evidence semantics
 so it can help in making better decisions and predictions.

We use probability theory and logic to manage uncertainty in higher education. Some individuals may approach probabilistic reasoning sluggishly or with ignorance
 but we use probability to handle this uncertainty. Probability is a crucial tool for anyone seeking to comprehend artificial intelligence and machine learning.

Probabilistic reasoning is a way of using uncertain knowledge to solve problems. It's especially useful when there are unpredictable outcomes
 when specifications or possibilities of predicates becomes too large to handle
 or when an unknown error occurs during an experiment.                                  There are two main ways to solve problems with uncertain knowledge using probabilistic reasoning: Bayes' rule and Bayesian Statistics. These methods allow us to make predictions and make decisions based on the available data
 even when we don't have all the information we need.                                 Understanding the basics of probabilistic reasoning is important if you're working with (A-I ). It's a powerful tool that can help you make more informed decisions and solve complex problems.

Probability is a measure of the likelihood that an uncertain event will occur. It is defined as the numerical measure of the likelihood that an event will occur. The value of probability always remains between 0 and 1
 representing ideal uncertainties.

We can define an event as a possible outcome of a variable. For example
 a coin can have two events: heads or tails. The sample space is the collection of all possible events. For example
 the sample space of a coin can be heads and tails. Random variables are used to represent events and objects in the real world. For example
 a person's height can be a random variable. Prior probability is the probability computed before observing new information. It is a way of making a guess or prediction about the likelihood of an event happening. Posterior probability is the probability that is calculated after all evidence or information has taken into account. It is a combination of prior probability and new information. For example
 if we knew that the coin was fair
 the prior probability of heads would be 0.5
 and the prior probability of tails would also be 0.5. If we flip the coin and get heads
 the posterior probability of heads would be 0.5
 but the posterior probability of tails would now be 0.

We will discuss the concept of conditional probability in probability theory. Conditional probability is the probability of an event occurring under specific conditions. For example
 if we want to determine the probability of a person going to the doctor based on whether they have a fever
 we would use conditional probability. We can calculate the probability of a person going to the doctor given that they have a fever using the formula: P(A⋀B) = P(A and B) / P(B)
 where P(A⋀B) denotes the probability of A given B
 P(A) denotes the probability of A
 and P(B) denotes the probability of B This formula enables us to determine the probability of an event occurring under specific circumstances
 which is useful in many scenarios.

We need to understand how to calculate the probability of an event given that another event has a probability. This can be done using the formula: P(A⋀B) = P(A) * P(B|A) / P(B). This formula involves representing A and B as sets using Venn diagrams. If event B happens
 the sample space is decreased to set B
 and we can calculate the probability of event A given that event B has already occurred by dividing P(A⋀B) by P(B). This is an essential concept in probabilistic reasoning and is widely used in many areas
 such as machine learning and artificial intelligence.

Discuss the topic of probabilistic reasoning.

Bayes' theorem establishes a relationship between the conditional probability and marginal probabilities of two random events. Through Bayesian inference
 we can calculate the probability of an event with uncertain knowledge. With this knowledge
 we can update our probability prediction of an event by observing new information from the real world.

Bayes' theorem is a tool that calculates the probability of an event occurring based on prior knowledge and observed evidence. In the context of cancer diagnosis
 Bayes' theorem can be used to calculate the probability of a person having cancer given their age and other relevant information. Using Bayes' theorem
 doctors can make more accurate diagnoses and provide more effective treatment options.
                                    
                                    To calculate the probability of cancer using Bayes' theorem
 we need to know the prior probability of having cancer
 which is often estimated based on age and other risk factors. We also need to know the conditional probability of having cancer given age and other relevant information. Once we have these probabilities
 we can use Bayes' theorem to calculate the posterior probability of having cancer
 which represents the updated probability of having cancer based on the observed evidence. This posterior probability can be used to inform treatment decisions and guide patient care.
                                    
                                    In summary
 Bayes' theorem is a powerful tool that can be used to accurately determine the probability of cancer diagnosis. By using this theorem
 we can provide more effective treatment options and improve patient outcomes.

Posterior probability is the probability of hypothesis A given evidence B It is calculated as P(A|B) = P(B|A) \* P(A) / P(B). This means that we calculate the probability of hypothesis A given evidence B by considering both the probability of evidence B given hypothesis A and the prior probability of hypothesis A Likelihood is the probability of evidence B given that hypothesis A is true and is calculated as P(B|A). Understanding these concepts is essential for developing effective probabilistic models in artificial intelligence and machine learning.

Bayes' rule is a fundamental concept in probabilistic reasoning. It enables us to compute the probability of an event given some evidence
 particularly useful when we have good probability of the evidence. In the context of perception
 Bayes' rule enables us to determine the cause of some unknown event by using the evidence of the effect
 such as computing the probability that the cause of a tree falling in a forest was a lightning strike. Bayes' rule is a powerful tool in artificial intelligence and machine learning
 with many applications.

We will discuss example-1
 which looks at the probability of a patient having meningitis with a stiff neck. Specifically
 we will examine the probability of a patient having meningitis
 given that they have a stiff neck. The known probability of a patient having meningitis is 1/30
000
 which means that P(a|b) = 0.8
 where P(a) is the probability of having meningitis and P(b) is the probability of having a stiff neck. Therefore
 we can conclude that 1 patient out of 750 patients has meningitis with a stiff neck
 which gives us a probability of 2%. This example is a great way to understand how probability can be used in artificial intelligence. By understanding the probability of certain events happening
 we can make better decisions and predictions.

Consider a standard deck of playing cards
 where there is a king card in 4 out of 52 cards.
                                
Now
 imagine that a single card is drawn from the deck and it is revealed to be a face card.
                                
The probability that the face card is the king card is 4/52
 which can be calculated as the ratio of the number of king cards to the total number of cards in the deck.
                                
To calculate the posterior probability of the event occurring
 we use Bayes' theorem
 which states that P(A|B) = P(B|A) * P(A) / P(B)
                                
In this case
 A represents the event of the face card being the king card
 and B represents the drawn face card.
                                
The probability of the event A occurring is P(A) = 4/52
 which is the probability of drawing a king card.
                                
The probability of the event B occurring given that A occurs is P(B|A) = 1
 which means that if the drawn face card is the king card
 it is certain that the event B will occur.
                                
The probability of the event B occurring is P(B) = P(B|A) * P(A) = 4/52
 which is the probability of drawing a face card.
                                
Finally
 we can calculate the posterior probability of the event A occurring given the observed event B as P(A|B) = P(B|A) * P(A) / P(B) = 4/52 * 4/52 / 4/52 = 1
 which means that given the observed event B
 it is certain that the event A occurred.
                                
This example demonstrates how to calculate the posterior probability of an event occurring given prior knowledge using Bayes' theorem.

1. Remove the welcome sentences: We will discuss the solution to a problem involving card probabilities.
                                    2. Rewrite the text: We will look at the probability of the card being a King
 the probability of the card being a face card
 and the probability of a face card given that it is a King. These probabilities will be calculated using the formula P(A or B) = P(A)  plus  P(B)  P(A and B). We will start by looking at the probability of the card being a King. The probability of a card being a King is 1/13. Next
 we will look at the probability of the card being a face card. The probability of a card being a face card is 3/13. Finally
 we will look at the probability of a face card given that it is a King. The probability of a face card given that it is a King is 1. To put all of these values in equation (i)
 we can use the formula P(A or B) = P(A)  plus  P(B)  P(A and B). This will give us: P(king or face) = 1/13  plus  3/13  0 = 16/13. So the probability of the card being a King or a face card is 16/13.

Bayes' theorem is a powerful tool that can be used to calculate the next step of the robot when the already executed step is given. It is particularly useful in weather forecasting
 where it can help predict the likelihood of certain weather patterns occurring. Additionally
 Bayes' theorem can be used to solve the Monty Hall problem
 a classic problem in probability theory. It is also a valuable tool in the field of Artificial Intelligence that can help us make more informed decisions and improve the performance of our machines.

Bayesian Networks are a type of probabilistic graphical model that represent a set of variables and their conditional dependencies using a directed acyclic graph. They are also known as belief networks
 decision networks
 or Bayesian models. One of the key features of Bayesian networks is that they are probabilistic
 meaning that they are built from a probability distribution and use probability theory for prediction and anomaly detection. Bayesian networks can be used for tasks such as classification
 prediction
 and decision making.

Bayesian Networks are a fundamental concept in probabilistic reasoning. They are used for building models from data and expert opinions
 and consist of two parts: a Directed Acyclic Graph and a Table of conditional probabilities.
                                     Bayesian Networks use nodes to represent variables and edges to represent the relationships between them. The Directed Acyclic Graph shows the causal relationships between these variables
 and the Table of conditional probabilities stores the probability of each variable given its parents
 allowing to calculate the probability of any variable given its parents
 useful for making predictions and solving decision problems.
                                     Bayesian Networks are a powerful tool for building models in various fields
 including healthcare
 finance
 and engineering
 and can be used to model complex systems and make decisions based on uncertain knowledge.

Today
 we will discuss Bayesian Networks. Bayesian networks are useful for probabilistic reasoning and can be applied in various fields.

We will discuss probabilistic reasoning and its importance in understanding the relationships between random variables represented by nodes in a network graph.
                                    
                                    We can see from the diagram that there are links between nodes A
 B
 C
 and D
 and each node represents a random variable. The arrows in the diagram indicate the direction of influence between the nodes.
                                    
                                    If there is a directed link between nodes A and B
 then A is considered the parent of B
 meaning A has an influence on B On the other hand
 if there is no directed link between A and C
 then A and C are independent of each other.
                                    
                                    Understanding these relationships between random variables is crucial in probabilistic reasoning. By analyzing the probabilities of different outcomes
 we can make predictions and make informed decisions.
                                    
                                    In summary
 probabilistic reasoning is an important tool in understanding the relationships between random variables in a network graph. By analyzing the probabilities of different outcomes
 we can make predictions and make informed decisions.

The joint probability distribution in Bayesian networks describes the probability of all possible outcomes of a set of random variables. This distribution is used to determine the probability of each node given its parent nodes. To calculate the joint probability of each node
 we multiply the condition probability distribution of its parent nodes and the likelihood function of the node. The likelihood function describes the probability of observing the data given the state of the random variable. With the joint probability distribution
 we can make predictions and perform causality reasoning in Bayesian networks.

When we have variables x1
 x2
 x3
.....
 xn
 the probabilities of different combinations of x1
 x2
 x3.. xn
 are known as Joint probability distribution. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities of different combinations of variables when more than one variable is involved. Joint probability distribution is used to determine the probabilities

Discuss Bayesian networks and their real-world applications.

The probability that the alarm has sounded is 20.4%.

Probabilistic reasoning is a technique used in (A-I ) and ML to make predictions and decisions based on the probability of different outcomes. In the context of (A-I ) and ML
 probabilistic reasoning can be used to predict the likelihood of different events occurring
 such as a burglary or an earthquake. The image shows probabilities of different outcomes for various events. In this case
 the probability of burglary is 0.998
 while the probability of earthquake is 0.75. Probabilistic reasoning can also be used to make decisions based on these probabilities
 such as installing an alarm to prevent burglary. Overall
 probabilistic reasoning is an important tool in (A-I ) and ML for making predictions and decisions based on the likelihood of different outcomes.

Discuss observed probabilities for the Burglary and earthquake components. Examine values of P(B= True) = 0.002
 P(B= False)= 0.998
 P(E= True) = 0.001
 and P(E= False)= 0.999. These probabilities can be used to make predictions and decisions about the potential occurrence of these events.

1. Discuss the conditional probability of Alarm A in the context of burglar and earthquake.
                                    2.
                                    3. Discuss

We will discuss the mathematical concepts behind conditional probability tables. These concepts are widely used in the telecommunications industry. In the context of David calls
 the random variables are the different events that can occur during a call
 such as the caller's location
 the type of call
 and the duration of the call. The conditional probability table for David calls shows the probabilities of different events occurring given the state of a call. For example
 the table can show the probability of a call lasting a certain duration given the caller's location. Understanding conditional probability tables is crucial for building accurate and reliable David call models. With conditional probability tables
 we can make predictions about the likelihood of different events occurring during a call based on the state of the call.

The conditional probability table for Sophia Calls shows the probability of Sophia making a specific call based on the presence or absence of certain attributes and conditions. The table includes mutually exclusive attributes such as B and E
 as well as the probability of Sophia making a call based on the presence or absence of certain conditions
 such as S and D The table is calculated by multiplying the probabilities of each individual attribute and condition
 which helps to provide a more accurate representation of Sophia's behavior and decision-making process.