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Reasoning with Uncertainty Definition: - So far as the laws of mathematics are concerns they are not certain. And so far they are certain they do not refer to reality. Albert Einstein. Introduction: - We must draw useful conclusions from poorly formed and uncertain evidence using unsound inference rule.

Drawing useful conclusion from incomplete and imprecise data with un sound reasoning is not an impossible task, we do it very successfully in almost every aspect of our life. We deliver correct diagnoses and recommend treatment from ambiguous symptoms, we analyze problems with our cars or stereos, we comprehend language statements that are often ambiguous or incomplete, we recognize friends from their voices or their gestures and so on. What is reasoning? When we require any knowledge system to do something it has not been explicitly told how to do it must reason. The system must figure out what it needs to know from what it already knows. For example if we know: Robins are birds.

All birds have wings. Then if we ask: Do robins have wings? Some reasoning (albeit very simple) has to go on answer the question. How can we reason: - To a certain extent this will depend on the knowledge representation chosen. Although a good knowledge representation scheme has to allow easy, natural and plausible reasoning. Listed below are very broad methods of how we may reason.

Formal reasoning: - -- Basic rules of inference with logic knowledge representations. Procedural reasoning: - -- Uses procedures that specify how to perhaps solve (sub) problems. Reasoning by analogy: - -- Humans are good at this, more difficult for AI systems. e.g. If we are asked Can robins fly? The system might reason that robins are like sparrows and it knows sparrows can fly so... Generalization and abstraction: - -- Again humans effective at this. This is basically getting towards learning and understanding methods.

Uncertain Reasoning: - Unfortunately the world is an uncertain place. Any AI system that seeks to model and reasoning in such a world must be able to deal with this. In particular it must be able to deal with: Incompleteness -- compensate for lack of knowledge. Inconsistencies -- resolve ambiguities and contradictions.

Change -- it must be able to update its world knowledge base over time. Clearly in order to deal with this some decision that a made are more likely to be true (or false) than others and we must introduce methods that can cope with this uncertainty. There are three basic methods that can do this: Symbolic methods. Statistical methods. Fuzzy logic methods. Reasoning Problem demonstration: - "^3 Car battery problem If The engine does not turn over, and The lights do not come on Then The problem is battery or cables.

"^3 Further problem with battery or cable if the problem is battery or cables then the engine does not turn over, and the lights do not come on. Our expert system offers an example of abduct ive reasoning. Formally, abduction states that from p "^3 q and Q is it possible to infer P. Abduction is an unsound rule of inference, meaning that the conclusion is not necessarily true for every interpretation in which the premises are true. In knowledge based systems, we often attach certainty factor to rule to measure our confidence in its conclusion. For example the rule P "^3 Q (. 9) expresses the belief!

SS if you believe p to be true, then you believe Q will happen 90% of the time!" Statistical approach of Uncertainty: - Using probability theory, we can determine often from a priori argument, the chances of events occurring, we also describe who combinations of events influence each other. There are a number of situations when probabilistic analysis is appropriate. First when the world is genuinely random as in playing games. With well shuffled cards or spinning a fair roulette wheel.

In cards for example the next card to be dealt is function of type of deck (poker) and cards already seen. In knowledge based problem solving we often find over selves reasoning with limited knowledge and incomplete information. A number of research groups have used forms of probabilistic reasoning to guide their work. Bayesian reasoning: - Bayesian reasoning is based in formal probability theory and is used extensively in several current areas of research. Including Pattern recognition and classification.

Bayesian theory supports the calculation of more complex probabilities from previously known results in simple probability calculation, we are able to conclude, for example how cards might be distributed o number of players. Suppose that I am one person out of a four person card game where all the cards are equally distributed. If I do not have the queen of spades, then I can conclude that each player has the have the queen of spades with probability 1/3. Similarly I can conclude that ach player has the ace of heart with a probability 1/3. And that any one player has both cards with probability 1/3 1/3 or 1/9. Assuming that event of getting the two cards are independent.

Rule: - P (A B) = P (A) + (b)! V P (A+B) In case of independent probability. P (A B) = P (A) (b) Definition: - Prior probability: - The prior probability, often called the unconditional probability, of an even is the probability assigned to an event in the absence of knowledge supporting its occurrence or absence, that is the probability of the event prior to any evidence. The prior probability of an event is symbolized as. P (Event). Posterior probability: - The Posterior!

SS after the fact! SS probability often called the conditional probability, of an event is the probability of an event given some evidence. Conditional probability is symbolized as P (A / b) = P (A B) /P (B). The impotent thing about bayes theorem is that the numbers on the right hand side of equation are easily available. At least when compared to the left hand side of equation.

P (A / B) is the probability that A is true given evidence B P (A) is the probability that A is true over all P (B / A) is the probability of observing evidence B when A is true. There are two major requirements for the use of bayes theorem First all the probabilities on the relationships of the evidence with the various hypotheses must be known, as well as the probabilistic relationship among the pieces of evidence. Second and some thing more difficult to establish is that all relationship between evidence and hypotheses, must be independent In general and especially in areas such as medicine these assumptions of independence cannot be justified. Bayesian reasoning requires complete and up to date probabilities, including joint probabilities, if its conclusions are to be correct. In many domains such extensive data collection and verification are not possible.

Bayesian Networks and Decision Networks Belief networks (also known as Bayesian networks, Bayes networks and causal probabilistic networks), provide a method to represent relationships between propositions or variables, even if the relationships involve uncertainty, unpredictability or imprecision. They may be learned automatically from data files, created by an expert, or developed by a combination of the two. They capture knowledge in a modular form that can be transported from one situation to another; it is a form people can understand, and which allows a clear visualization of the relationships involved. By adding decision variables (things that can be controlled), and utility variables (things we want to optimize) to the relationships of a belief network, a decision network (also known as an influence diagram) is formed. This can be used to find optimal decisions, control systems, or plans. Bayesian belief networks relax many constraints of the full Bayesian model and show how the data of a domain can partition and focus reasoning.

Several observation first the modularity of the problem domain often allows the relaxation of many of the dependence / independence constraints required for bayes in most reasoning situations. it is not necessary to build a large joint probability table in which the probabilities for all possible combinations of events and evidence are listed rather we selects the local phenomena that we know will interact and obtain probability. Assuming all other events are either conditionally independent or their correlations are so small that they may be ignored. The links between the nodes of belief network represent the conditioned probabilities for causal influence. Expert reasoning is that the presence or evidence of data in a domain can partition and focus the search for explanations. Fuzzy logic What is fuzzy logic: - Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth- truth-values between "completely true" and "completely false". As its name suggests, it is the logic underlying modes of reasoning which are approximate rather than exact.

The importance of fuzzy logic derives from the fact that most modes of human reasoning and especially common sense reasoning are approximate in nature. Boolean vs. Fuzzy 300 years B.C., the Greek philosopher, Aristotle came up with binary logic (0, 1), which is now the principle foundation of Mathematics. It came down to one law: A or not -A, either this or not this. For example, a typical rose is either red or not red. It cannot be red and not red. Every statement or sentence is true or false or has the truth-value 1 or 0.

This is Aristotle's law of bivalence and was philosophically correct for over two thousand years. Two centuries before Aristotle, Buddha, had the belief which contradicted the black-and-white world of worlds, which went beyond the bivalent cocoon and see the world as it is, filled with contradictions, with things and not things. He stated that a rose, could be to a certain degree completely red, but at the same time could also be at a certain degree not red. Meaning that it can be red and not red at the same time. Conventional (Boolean) logic states that a glass can be full or not full of water.

However, suppose one were to fill the glass only halfway. Then the glass can be half-full and half-not-full. Clearly, this disproves Aristotle's law of bivalence. This concept of certain degree or multi valence is the fundamental concept, which propelled Zander Loft i of University Berkeley in the 1960's to introduce fuzzy logic. The essential characteristics of fuzzy logic founded by him are as follows.

In fuzzy logic, exact reasoning is viewed as a limiting case of approximate reasoning. In fuzzy logic everything is a matter of degree. Any logical system can be fuzz ified In fuzzy logic, knowledge is interpreted as a collection of elastic or, equivalently, fuzzy constraint on a collection of variables Inference is viewed as a process of propagation of elastic constraints.

Drawing useful conclusion from incomplete and imprecise data with un sound reasoning is not an impossible task, we do it very successfully in almost every aspect of our life. We deliver correct diagnoses and recommend treatment from ambiguous symptoms, we analyze problems with our cars or stereos, we comprehend language statements that are often ambiguous or incomplete, we recognize friends from their voices or their gestures and so on. What is reasoning? When we require any knowledge system to do something it has not been explicitly told how to do it must reason. The system must figure out what it needs to know from what it already knows. For example if we know: Robins are birds.

All birds have wings. Then if we ask: Do robins have wings? Some reasoning (albeit very simple) has to go on answer the question. How can we reason: - To a certain extent this will depend on the knowledge representation chosen. Although a good knowledge representation scheme has to allow easy, natural and plausible reasoning. Listed below are very broad methods of how we may reason.

Formal reasoning: - -- Basic rules of inference with logic knowledge representations. Procedural reasoning: - -- Uses procedures that specify how to perhaps solve (sub) problems. Reasoning by analogy: - -- Humans are good at this, more difficult for AI systems. e.g. If we are asked Can robins fly? The system might reason that robins are like sparrows and it knows sparrows can fly so... Generalization and abstraction: - -- Again humans effective at this. This is basically getting towards learning and understanding methods.

Uncertain Reasoning: - Unfortunately the world is an uncertain place. Any AI system that seeks to model and reasoning in such a world must be able to deal with this. In particular it must be able to deal with: Incompleteness -- compensate for lack of knowledge. Inconsistencies -- resolve ambiguities and contradictions.

Change -- it must be able to update its world knowledge base over time. Clearly in order to deal with this some decision that a made are more likely to be true (or false) than others and we must introduce methods that can cope with this uncertainty. There are three basic methods that can do this: Symbolic methods. Statistical methods. Fuzzy logic methods. Reasoning Problem demonstration: - "^3 Car battery problem If The engine does not turn over, and The lights do not come on Then The problem is battery or cables.

"^3 Further problem with battery or cable if the problem is battery or cables then the engine does not turn over, and the lights do not come on. Our expert system offers an example of abduct ive reasoning. Formally, abduction states that from p "^3 q and Q is it possible to infer P. Abduction is an unsound rule of inference, meaning that the conclusion is not necessarily true for every interpretation in which the premises are true. In knowledge based systems, we often attach certainty factor to rule to measure our confidence in its conclusion. For example the rule P "^3 Q (. 9) expresses the belief!

SS if you believe p to be true, then you believe Q will happen 90% of the time!" Statistical approach of Uncertainty: - Using probability theory, we can determine often from a priori argument, the chances of events occurring, we also describe who combinations of events influence each other. There are a number of situations when probabilistic analysis is appropriate. First when the world is genuinely random as in playing games. With well shuffled cards or spinning a fair roulette wheel.

In cards for example the next card to be dealt is function of type of deck (poker) and cards already seen. In knowledge based problem solving we often find over selves reasoning with limited knowledge and incomplete information. A number of research groups have used forms of probabilistic reasoning to guide their work. Bayesian reasoning: - Bayesian reasoning is based in formal probability theory and is used extensively in several current areas of research. Including Pattern recognition and classification.

Bayesian theory supports the calculation of more complex probabilities from previously known results in simple probability calculation, we are able to conclude, for example how cards might be distributed o number of players. Suppose that I am one person out of a four person card game where all the cards are equally distributed. If I do not have the queen of spades, then I can conclude that each player has the have the queen of spades with probability 1/3. Similarly I can conclude that ach player has the ace of heart with a probability 1/3. And that any one player has both cards with probability 1/3 1/3 or 1/9. Assuming that event of getting the two cards are independent.

Rule: - P (A B) = P (A) + (b)! V P (A+B) In case of independent probability. P (A B) = P (A) (b) Definition: - Prior probability: - The prior probability, often called the unconditional probability, of an even is the probability assigned to an event in the absence of knowledge supporting its occurrence or absence, that is the probability of the event prior to any evidence. The prior probability of an event is symbolized as. P (Event). Posterior probability: - The Posterior!

SS after the fact! SS probability often called the conditional probability, of an event is the probability of an event given some evidence. Conditional probability is symbolized as P (A / b) = P (A B) /P (B). The impotent thing about bayes theorem is that the numbers on the right hand side of equation are easily available. At least when compared to the left hand side of equation.

P (A / B) is the probability that A is true given evidence B P (A) is the probability that A is true over all P (B / A) is the probability of observing evidence B when A is true. There are two major requirements for the use of bayes theorem First all the probabilities on the relationships of the evidence with the various hypotheses must be known, as well as the probabilistic relationship among the pieces of evidence. Second and some thing more difficult to establish is that all relationship between evidence and hypotheses, must be independent In general and especially in areas such as medicine these assumptions of independence cannot be justified. Bayesian reasoning requires complete and up to date probabilities, including joint probabilities, if its conclusions are to be correct. In many domains such extensive data collection and verification are not possible.

Bayesian Networks and Decision Networks Belief networks (also known as Bayesian networks, Bayes networks and causal probabilistic networks), provide a method to represent relationships between propositions or variables, even if the relationships involve uncertainty, unpredictability or imprecision. They may be learned automatically from data files, created by an expert, or developed by a combination of the two. They capture knowledge in a modular form that can be transported from one situation to another; it is a form people can understand, and which allows a clear visualization of the relationships involved. By adding decision variables (things that can be controlled), and utility variables (things we want to optimize) to the relationships of a belief network, a decision network (also known as an influence diagram) is formed. This can be used to find optimal decisions, control systems, or plans. Bayesian belief networks relax many constraints of the full Bayesian model and show how the data of a domain can partition and focus reasoning.

Several observation first the modularity of the problem domain often allows the relaxation of many of the dependence / independence constraints required for bayes in most reasoning situations. it is not necessary to build a large joint probability table in which the probabilities for all possible combinations of events and evidence are listed rather we selects the local phenomena that we know will interact and obtain probability. Assuming all other events are either conditionally independent or their correlations are so small that they may be ignored. The links between the nodes of belief network represent the conditioned probabilities for causal influence. Expert reasoning is that the presence or evidence of data in a domain can partition and focus the search for explanations. Fuzzy logic What is fuzzy logic: - Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth- truth-values between "completely true" and "completely false". As its name suggests, it is the logic underlying modes of reasoning which are approximate rather than exact.

The importance of fuzzy logic derives from the fact that most modes of human reasoning and especially common sense reasoning are approximate in nature. Boolean vs. Fuzzy 300 years B.C., the Greek philosopher, Aristotle came up with binary logic (0, 1), which is now the principle foundation of Mathematics. It came down to one law: A or not -A, either this or not this. For example, a typical rose is either red or not red. It cannot be red and not red. Every statement or sentence is true or false or has the truth-value 1 or 0.

This is Aristotle's law of bivalence and was philosophically correct for over two thousand years. Two centuries before Aristotle, Buddha, had the belief which contradicted the black-and-white world of worlds, which went beyond the bivalent cocoon and see the world as it is, filled with contradictions, with things and not things. He stated that a rose, could be to a certain degree completely red, but at the same time could also be at a certain degree not red. Meaning that it can be red and not red at the same time. Conventional (Boolean) logic states that a glass can be full or not full of water.

However, suppose one were to fill the glass only halfway. Then the glass can be half-full and half-not-full. Clearly, this disproves Aristotle's law of bivalence. This concept of certain degree or multi valence is the fundamental concept, which propelled Zander Loft i of University Berkeley in the 1960's to introduce fuzzy logic. The essential characteristics of fuzzy logic founded by him are as follows.

In fuzzy logic, exact reasoning is viewed as a limiting case of approximate reasoning. In fuzzy logic everything is a matter of degree. Any logical system can be fuzz ified In fuzzy logic, knowledge is interpreted as a collection of elastic or, equivalently, fuzzy constraint on a collection of variables Inference is viewed as a process of propagation of elastic constraints.

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