The document discusses concept learning algorithms. It introduces the problem of concept learning as inducing a function to classify examples into categories based on their attributes. The Candidate Elimination Algorithm (CEA) is presented as a method for finding all hypotheses consistent with training examples without enumerating them. CEA works by maintaining the most specific (S) and most general (G) consistent hypotheses. It updates S and G in response to positive and negative examples.
The document discusses the concept of PAC (Probably Approximately Correct) learning. It begins by describing a learning scenario where a hidden hypothesis is chosen by nature, and a learner tries to approximate this hypothesis based on randomly generated training data. It then defines what it means for a learned hypothesis to be "bad" or have high test error, and shows that by choosing a large enough random training set, the probability of learning a bad hypothesis can be bounded. Finally, it provides the formula for calculating the minimum size of the random training set needed to guarantee this probability bound.
"PAC Learning - a discussion on the original paper by Valiant" presentation @...Adrian Florea
This document discusses PAC (probably approximately correct) learning, which was introduced in Valiant's 1984 paper. It defines key concepts in PAC learning like concepts, concept classes, learning algorithms, hypothesis spaces, and error rates. It also proves theorems like the theorem of ε-exhausting the version space, which shows that the number of training examples needed is logarithmic in the size of the hypothesis space. As an example, it shows that learning conjunctions of Boolean literals is PAC learnable, while learning all concepts is not PAC learnable.
A Theory of the Learnable; PAC Learningdhruvgairola
The document presents a theory of PAC (Probably Approximately Correct) learning. It discusses how PAC learning uses probabilities to measure the correctness of a learning algorithm's hypotheses. It is shown that k-decision lists are PAC learnable, having both polynomial sample complexity and efficient learning algorithms. This establishes that k-decision lists are computationally learnable. The theory of PAC learning provides a framework for analyzing machine learning algorithms and their learnability.
Concept learning and candidate elimination algorithmswapnac12
This document discusses concept learning, which involves inferring a Boolean-valued function from training examples of its input and output. It describes a concept learning task where each hypothesis is a vector of six constraints specifying values for six attributes. The most general and most specific hypotheses are provided. It also discusses the FIND-S algorithm for finding a maximally specific hypothesis consistent with positive examples, and its limitations in dealing with noise or multiple consistent hypotheses. Finally, it introduces the candidate-elimination algorithm and version spaces as an improvement over FIND-S that can represent all consistent hypotheses.
1) The document discusses concept learning, which involves inferring a Boolean function from training examples. It focuses on a concept learning task where hypotheses are represented as vectors of constraints on attribute values.
2) It describes the FIND-S algorithm, which finds the most specific hypothesis consistent with positive examples by generalizing constraints. However, FIND-S has limitations like ignoring negative examples.
3) The Candidate-Elimination algorithm represents the version space of all hypotheses consistent with examples to address FIND-S limitations. It outputs the version space rather than a single hypothesis.
This document defines and provides examples of quantifiers - universal and existential quantification. Universal quantification uses "for all" and is represented by ∀, while existential quantification uses "there exists" and is represented by ∃. A counterexample can disprove a universal statement by showing a case that makes the proposition false. De Morgan's laws state that the negation of a universal statement is an existential statement, and vice versa. Examples are provided to illustrate these concepts.
The document provides an overview of Bayesian decision theory and naive Bayesian classification. It discusses how Bayesian decision theory predates other machine learning techniques and forms the basis of classifiers like naive Bayes. It then explains the Bayes theorem and how it is used for probability inference and decision making. An example of predicting whether a customer will buy a computer is used to illustrate Bayesian reasoning. Finally, the document describes how the naive Bayes classifier works by making a strong independence assumption between features to simplify computations. It gives the mathematical formulas and works through an example to classify a customer profile.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
The document discusses the concept of PAC (Probably Approximately Correct) learning. It begins by describing a learning scenario where a hidden hypothesis is chosen by nature, and a learner tries to approximate this hypothesis based on randomly generated training data. It then defines what it means for a learned hypothesis to be "bad" or have high test error, and shows that by choosing a large enough random training set, the probability of learning a bad hypothesis can be bounded. Finally, it provides the formula for calculating the minimum size of the random training set needed to guarantee this probability bound.
"PAC Learning - a discussion on the original paper by Valiant" presentation @...Adrian Florea
This document discusses PAC (probably approximately correct) learning, which was introduced in Valiant's 1984 paper. It defines key concepts in PAC learning like concepts, concept classes, learning algorithms, hypothesis spaces, and error rates. It also proves theorems like the theorem of ε-exhausting the version space, which shows that the number of training examples needed is logarithmic in the size of the hypothesis space. As an example, it shows that learning conjunctions of Boolean literals is PAC learnable, while learning all concepts is not PAC learnable.
A Theory of the Learnable; PAC Learningdhruvgairola
The document presents a theory of PAC (Probably Approximately Correct) learning. It discusses how PAC learning uses probabilities to measure the correctness of a learning algorithm's hypotheses. It is shown that k-decision lists are PAC learnable, having both polynomial sample complexity and efficient learning algorithms. This establishes that k-decision lists are computationally learnable. The theory of PAC learning provides a framework for analyzing machine learning algorithms and their learnability.
Concept learning and candidate elimination algorithmswapnac12
This document discusses concept learning, which involves inferring a Boolean-valued function from training examples of its input and output. It describes a concept learning task where each hypothesis is a vector of six constraints specifying values for six attributes. The most general and most specific hypotheses are provided. It also discusses the FIND-S algorithm for finding a maximally specific hypothesis consistent with positive examples, and its limitations in dealing with noise or multiple consistent hypotheses. Finally, it introduces the candidate-elimination algorithm and version spaces as an improvement over FIND-S that can represent all consistent hypotheses.
1) The document discusses concept learning, which involves inferring a Boolean function from training examples. It focuses on a concept learning task where hypotheses are represented as vectors of constraints on attribute values.
2) It describes the FIND-S algorithm, which finds the most specific hypothesis consistent with positive examples by generalizing constraints. However, FIND-S has limitations like ignoring negative examples.
3) The Candidate-Elimination algorithm represents the version space of all hypotheses consistent with examples to address FIND-S limitations. It outputs the version space rather than a single hypothesis.
This document defines and provides examples of quantifiers - universal and existential quantification. Universal quantification uses "for all" and is represented by ∀, while existential quantification uses "there exists" and is represented by ∃. A counterexample can disprove a universal statement by showing a case that makes the proposition false. De Morgan's laws state that the negation of a universal statement is an existential statement, and vice versa. Examples are provided to illustrate these concepts.
The document provides an overview of Bayesian decision theory and naive Bayesian classification. It discusses how Bayesian decision theory predates other machine learning techniques and forms the basis of classifiers like naive Bayes. It then explains the Bayes theorem and how it is used for probability inference and decision making. An example of predicting whether a customer will buy a computer is used to illustrate Bayesian reasoning. Finally, the document describes how the naive Bayes classifier works by making a strong independence assumption between features to simplify computations. It gives the mathematical formulas and works through an example to classify a customer profile.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
This document provides an overview of Bayesian learning methods. It discusses key concepts like Bayes' theorem, maximum a posteriori hypotheses, maximum likelihood hypotheses, and how Bayesian learning relates to concept learning problems. Bayesian learning allows prior knowledge to be combined with observed data, hypotheses can make probabilistic predictions, and new examples are classified by weighting multiple hypotheses by their probabilities. While computationally intensive, Bayesian methods provide an optimal standard for decision making.
Uncertainty & Probability
Baye's rule
Choosing Hypotheses- Maximum a posteriori
Maximum Likelihood - Baye's concept learning
Maximum Likelihood of real valued function
Bayes optimal Classifier
Joint distributions
Naive Bayes Classifier
This document provides an introduction to probability and statistics concepts over 2 weeks. It covers basic probability topics like sample spaces, events, probability definitions and axioms. Conditional probability and the multiplication rule for conditional probability are explained. Bayes' theorem relating prior, likelihood and posterior probabilities is introduced. Examples on probability calculations for coin tosses, dice rolls and medical testing are provided. Key terms around experimental units, populations, descriptive and inferential statistics are also defined.
Lecture 2 predicates quantifiers and rules of inferenceasimnawaz54
1) Predicates become propositions when variables are quantified by assigning values or using quantifiers. Quantifiers like ∀ and ∃ are used to make statements true or false for all or some values.
2) ∀ (universal quantifier) means "for all" and makes a statement true for all values of a variable. ∃ (existential quantifier) means "there exists" and makes a statement true if it is true for at least one value.
3) Predicates with unbound variables are neither true nor false. Binding variables by assigning values or using quantifiers turns predicates into propositions that can be evaluated as true or false.
This document introduces predicates and quantifiers in predicate logic. It defines predicates as functions that take objects and return propositions. Predicates allow reasoning about whole classes of entities. Quantifiers like "for all" (universal quantifier ∀) and "there exists" (existential quantifier ∃) are used to make general statements about predicates over a universe of discourse. Examples demonstrate how predicates and quantifiers can express properties and relationships for objects. Laws of quantifier equivalence are also presented.
The document discusses inference rules for quantifiers in discrete mathematics. It provides examples of using universal instantiation, universal generalization, existential instantiation, and existential generalization. It also discusses the rules of universal specification and universal generalization in more detail with examples. Finally, it presents proofs involving quantifiers over integers to demonstrate techniques like direct proof, proof by contradiction, and proving statements' contrapositives.
Predicates and quantifiers presentation topicsR.h. Himel
This document discusses predicates and quantifiers in predicate logic. It begins by defining predicate logic as an extension of propositional logic that allows reasoning about whole classes of entities. It then discusses predicates, subjects, and the universal and existential quantifiers. The universal quantifier is defined using the example "All parking spaces at BU are full" while the existential quantifier is defined using the example "There is a parking space at BU that is full." Finally, it discusses some quantifier equivalence laws.
This document discusses predicates and quantifiers in predicate logic. Predicate logic can express statements about objects and their properties, while propositional logic cannot. Predicates assign properties to variables, and quantifiers specify whether a predicate applies to all or some variables in a domain. There are two types of quantifiers: universal quantification with ∀ and existential quantification with ∃. Quantified statements involve predicates, variables ranging over a domain, and quantifiers to specify the scope of the predicate.
This document discusses predicates and quantifiers in predicate logic. It begins by explaining the limitations of propositional logic in expressing statements involving variables and relationships between objects. It then introduces predicates as statements involving variables, and quantifiers like universal ("for all") and existential ("there exists") to express the extent to which a predicate is true. Examples are provided to demonstrate how predicates and quantifiers can be used to represent statements and enable logical reasoning. The document also covers translating statements between natural language and predicate logic, and negating quantified statements.
Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities.
We elaborate on hierarchical credal sets, which are sets of probability mass functions paired with second-order distributions. A new criterion to make decisions based on these models is proposed. This is achieved by sampling from the set of mass functions and considering the Kullback-Leibler divergence from the weighted center of mass of the set. We evaluate this criterion in a simple classification scenario: the results show performance improvements when compared to a credal classifier where the second-order distribution is not taken into account.
This document discusses deductive closure of partially ordered propositional belief bases. It begins with an introduction and example. It then provides background on possibilistic logic and semantics for partially ordered bases. The main sections describe a sound and complete approach to deduction with partially ordered bases using axioms and inference rules. It also compares this approach to encoding a partially ordered base as a symbolic possibilistic base. While encoding adds flexibility, it can introduce unwanted information not implied by the original partial order.
The document defines logical quantifiers such as existence and uniqueness quantifiers. It discusses how quantifiers can be used to restrict domains and bind variables. It provides examples of translating English statements to logical expressions using quantifiers and discusses precedence, logical equivalences, and negating quantifier expressions.
The document discusses propositional logic and covers topics like propositional variables, truth tables, logical equivalence, predicates, and quantifiers. It defines key concepts such as propositions, tautologies, contradictions, predicates, universal and existential quantifiers. Examples are provided to illustrate different types of truth tables, logical equivalences like De Morgan's laws, and uses of quantifiers.
Quantum optical models in noncommutative spacesSanjib Dey
Several quantum optical models, such as, coherent states, cat states and squeezed states are constructed in a noncommutative space arising from the generalised uncertainty relation. We explore some advantages of utilising noncommutative models by comparing the nonclassicality and entanglement properties with that of the usual quantum mechanical systems.
11.[29 35]a unique common fixed point theorem under psi varphi contractive co...Alexander Decker
This document presents a unique common fixed point theorem for two self maps satisfying a generalized contraction condition in partial metric spaces using rational expressions. It begins by introducing basic definitions and lemmas related to partial metric spaces. It then presents the main theorem, which states that if two self maps T and f satisfy certain contractive and completeness conditions, including being weakly compatible, then they have a unique common fixed point. The proof considers two cases - when the sequences constructed from the maps are eventually equal, and when they are not eventually equal but form a Cauchy sequence. It is shown in both cases that the maps must have a unique common fixed point.
The document describes the simplex method for solving linear programming problems. It begins with an example problem of maximizing beer production given constraints on barley and corn supplies. It introduces slack variables to transform inequalities into equalities. The coefficients are written in a tableau and an initial basic feasible solution is chosen. Gaussian elimination is performed to introduce new basic variables while removing others. The process is repeated, moving through the feasible space, until an optimal solution is found without any negative entries in the objective function row. Duality between minimizing costs and maximizing profits is also discussed.
This document discusses quantification in logic. Quantification transforms a propositional function into a proposition by expressing the extent to which a predicate is true. There are two main types of quantification: universal quantification and existential quantification. Universal quantification expresses that a predicate is true for every element, while existential quantification expresses that a predicate is true for at least one element. The document provides examples and pros and cons of each type of quantification and notes that quantification operators like ∀ and ∃ take precedence over logical operators.
This document discusses algorithms for predictive modeling, including logistic regression. It presents a medical dataset containing measurements of heart patients and whether they survived. Logistic regression is applied to predict survival using maximum likelihood estimation. Numerical optimization techniques like BFGS and Fisher's algorithm are discussed for maximum likelihood estimation of logistic regression. Iteratively reweighted least squares is also presented as an alternative approach.
1. The document discusses universal quantification and quantifiers. Universal quantification refers to statements that are true for all variables, while quantifiers are words like "some" or "all" that refer to quantities.
2. It explains that a universally quantified statement is of the form "For all x, P(x) is true" and is defined to be true if P(x) is true for every x, and false if P(x) is false for at least one x.
3. When the universe of discourse can be listed as x1, x2, etc., a universal statement is the same as the conjunction P(x1) and P(x2) and etc., because this
Este documento argumenta que cuando los proyectos escolares son compartidos y elegidos colectivamente por estudiantes y maestros, en lugar de impuestos unilateralmente, esto permite que la escuela se desarrolle de una manera democrática e inclusiva que beneficia a todos, en lugar de favorecer a algunos sobre otros. Compartir proyectos de una manera que involucre a todos fomenta una educación para todos basada en los derechos democráticos.
This document provides an overview of Bayesian learning methods. It discusses key concepts like Bayes' theorem, maximum a posteriori hypotheses, maximum likelihood hypotheses, and how Bayesian learning relates to concept learning problems. Bayesian learning allows prior knowledge to be combined with observed data, hypotheses can make probabilistic predictions, and new examples are classified by weighting multiple hypotheses by their probabilities. While computationally intensive, Bayesian methods provide an optimal standard for decision making.
Uncertainty & Probability
Baye's rule
Choosing Hypotheses- Maximum a posteriori
Maximum Likelihood - Baye's concept learning
Maximum Likelihood of real valued function
Bayes optimal Classifier
Joint distributions
Naive Bayes Classifier
This document provides an introduction to probability and statistics concepts over 2 weeks. It covers basic probability topics like sample spaces, events, probability definitions and axioms. Conditional probability and the multiplication rule for conditional probability are explained. Bayes' theorem relating prior, likelihood and posterior probabilities is introduced. Examples on probability calculations for coin tosses, dice rolls and medical testing are provided. Key terms around experimental units, populations, descriptive and inferential statistics are also defined.
Lecture 2 predicates quantifiers and rules of inferenceasimnawaz54
1) Predicates become propositions when variables are quantified by assigning values or using quantifiers. Quantifiers like ∀ and ∃ are used to make statements true or false for all or some values.
2) ∀ (universal quantifier) means "for all" and makes a statement true for all values of a variable. ∃ (existential quantifier) means "there exists" and makes a statement true if it is true for at least one value.
3) Predicates with unbound variables are neither true nor false. Binding variables by assigning values or using quantifiers turns predicates into propositions that can be evaluated as true or false.
This document introduces predicates and quantifiers in predicate logic. It defines predicates as functions that take objects and return propositions. Predicates allow reasoning about whole classes of entities. Quantifiers like "for all" (universal quantifier ∀) and "there exists" (existential quantifier ∃) are used to make general statements about predicates over a universe of discourse. Examples demonstrate how predicates and quantifiers can express properties and relationships for objects. Laws of quantifier equivalence are also presented.
The document discusses inference rules for quantifiers in discrete mathematics. It provides examples of using universal instantiation, universal generalization, existential instantiation, and existential generalization. It also discusses the rules of universal specification and universal generalization in more detail with examples. Finally, it presents proofs involving quantifiers over integers to demonstrate techniques like direct proof, proof by contradiction, and proving statements' contrapositives.
Predicates and quantifiers presentation topicsR.h. Himel
This document discusses predicates and quantifiers in predicate logic. It begins by defining predicate logic as an extension of propositional logic that allows reasoning about whole classes of entities. It then discusses predicates, subjects, and the universal and existential quantifiers. The universal quantifier is defined using the example "All parking spaces at BU are full" while the existential quantifier is defined using the example "There is a parking space at BU that is full." Finally, it discusses some quantifier equivalence laws.
This document discusses predicates and quantifiers in predicate logic. Predicate logic can express statements about objects and their properties, while propositional logic cannot. Predicates assign properties to variables, and quantifiers specify whether a predicate applies to all or some variables in a domain. There are two types of quantifiers: universal quantification with ∀ and existential quantification with ∃. Quantified statements involve predicates, variables ranging over a domain, and quantifiers to specify the scope of the predicate.
This document discusses predicates and quantifiers in predicate logic. It begins by explaining the limitations of propositional logic in expressing statements involving variables and relationships between objects. It then introduces predicates as statements involving variables, and quantifiers like universal ("for all") and existential ("there exists") to express the extent to which a predicate is true. Examples are provided to demonstrate how predicates and quantifiers can be used to represent statements and enable logical reasoning. The document also covers translating statements between natural language and predicate logic, and negating quantified statements.
Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities.
We elaborate on hierarchical credal sets, which are sets of probability mass functions paired with second-order distributions. A new criterion to make decisions based on these models is proposed. This is achieved by sampling from the set of mass functions and considering the Kullback-Leibler divergence from the weighted center of mass of the set. We evaluate this criterion in a simple classification scenario: the results show performance improvements when compared to a credal classifier where the second-order distribution is not taken into account.
This document discusses deductive closure of partially ordered propositional belief bases. It begins with an introduction and example. It then provides background on possibilistic logic and semantics for partially ordered bases. The main sections describe a sound and complete approach to deduction with partially ordered bases using axioms and inference rules. It also compares this approach to encoding a partially ordered base as a symbolic possibilistic base. While encoding adds flexibility, it can introduce unwanted information not implied by the original partial order.
The document defines logical quantifiers such as existence and uniqueness quantifiers. It discusses how quantifiers can be used to restrict domains and bind variables. It provides examples of translating English statements to logical expressions using quantifiers and discusses precedence, logical equivalences, and negating quantifier expressions.
The document discusses propositional logic and covers topics like propositional variables, truth tables, logical equivalence, predicates, and quantifiers. It defines key concepts such as propositions, tautologies, contradictions, predicates, universal and existential quantifiers. Examples are provided to illustrate different types of truth tables, logical equivalences like De Morgan's laws, and uses of quantifiers.
Quantum optical models in noncommutative spacesSanjib Dey
Several quantum optical models, such as, coherent states, cat states and squeezed states are constructed in a noncommutative space arising from the generalised uncertainty relation. We explore some advantages of utilising noncommutative models by comparing the nonclassicality and entanglement properties with that of the usual quantum mechanical systems.
11.[29 35]a unique common fixed point theorem under psi varphi contractive co...Alexander Decker
This document presents a unique common fixed point theorem for two self maps satisfying a generalized contraction condition in partial metric spaces using rational expressions. It begins by introducing basic definitions and lemmas related to partial metric spaces. It then presents the main theorem, which states that if two self maps T and f satisfy certain contractive and completeness conditions, including being weakly compatible, then they have a unique common fixed point. The proof considers two cases - when the sequences constructed from the maps are eventually equal, and when they are not eventually equal but form a Cauchy sequence. It is shown in both cases that the maps must have a unique common fixed point.
The document describes the simplex method for solving linear programming problems. It begins with an example problem of maximizing beer production given constraints on barley and corn supplies. It introduces slack variables to transform inequalities into equalities. The coefficients are written in a tableau and an initial basic feasible solution is chosen. Gaussian elimination is performed to introduce new basic variables while removing others. The process is repeated, moving through the feasible space, until an optimal solution is found without any negative entries in the objective function row. Duality between minimizing costs and maximizing profits is also discussed.
This document discusses quantification in logic. Quantification transforms a propositional function into a proposition by expressing the extent to which a predicate is true. There are two main types of quantification: universal quantification and existential quantification. Universal quantification expresses that a predicate is true for every element, while existential quantification expresses that a predicate is true for at least one element. The document provides examples and pros and cons of each type of quantification and notes that quantification operators like ∀ and ∃ take precedence over logical operators.
This document discusses algorithms for predictive modeling, including logistic regression. It presents a medical dataset containing measurements of heart patients and whether they survived. Logistic regression is applied to predict survival using maximum likelihood estimation. Numerical optimization techniques like BFGS and Fisher's algorithm are discussed for maximum likelihood estimation of logistic regression. Iteratively reweighted least squares is also presented as an alternative approach.
1. The document discusses universal quantification and quantifiers. Universal quantification refers to statements that are true for all variables, while quantifiers are words like "some" or "all" that refer to quantities.
2. It explains that a universally quantified statement is of the form "For all x, P(x) is true" and is defined to be true if P(x) is true for every x, and false if P(x) is false for at least one x.
3. When the universe of discourse can be listed as x1, x2, etc., a universal statement is the same as the conjunction P(x1) and P(x2) and etc., because this
Este documento argumenta que cuando los proyectos escolares son compartidos y elegidos colectivamente por estudiantes y maestros, en lugar de impuestos unilateralmente, esto permite que la escuela se desarrolle de una manera democrática e inclusiva que beneficia a todos, en lugar de favorecer a algunos sobre otros. Compartir proyectos de una manera que involucre a todos fomenta una educación para todos basada en los derechos democráticos.
Ms. Afrika Abney has received praise from many for her community work and dedication to promoting arts education. She is described as a tireless promoter, passionate about children's literacy, and committed to her students. Testimonials highlight her skill in managing communications, organizing communities, and bringing her artistic talents and energy to diverse groups. Ms. Abney is seen as a role model, inspiration, and positive influence on youth.
Dual Credit Survey: Accelerating Educational Readiness, Progress, and CompletionHobsons
The document contains a survey about dual credit courses, which allow high school students to earn college credit. The survey asks respondents about their school district, current dual credit offerings, partnerships, benefits, and barriers to dual credit courses. It also asks about the respondents' views on issues like whether dual credit demonstrates college readiness and whether K-12 education should evolve into a K-14 system.
The document summarizes Robert Gibbons' experience as a co-op at Southco, including:
1) Creating displays and prototypes for Olympus Keymed products and ensuring their safe transport to trade shows.
2) Recognizing a need for additional tools in the tool room and procuring a toolbox to conveniently store them.
3) Suggesting areas for potential future work with other Southco departments to gain broader experience.
4) Assisting engineers with drawings, models, and prototypes and working with machinists on parts and displays.
5) Offering advice to future co-ops to absorb information, manage time effectively, and take notes.
6) Impression of Southco as the best
Este documento describe la metodología de diseño combinacional. Un sistema combinacional tiene salidas que dependen únicamente de las combinaciones de entrada. La metodología incluye especificar el sistema, determinar las entradas y salidas, construir la tabla de verdad, minimizar, crear un diagrama esquemático e implementar. Se provee un ejemplo de diseño de un sistema de alarma para una granja.
This is the first in our five-part series orienting investors to modern commercial real estate investing: the key terminology and principles, and things to be aware of when getting started. This series will also cover the contemporary landscape of real estate crowdfunding. www.equitymultiple.com
Iso 22000 2005 food safety management system certification practice guideHenry Nelson
ISO 22000 food safety management systems - Requirements for any organization in the food chain. The standard provides for international harmonization in the field of food safety standards, providing a tool for implementing the HACCP system (Hazard Analysis and Critical Control Points) throughout the food supply chain.
The State Library of Kansas created the Kansas Library Collaborative in December 2005 and subscribed to the OverDrive platform for downloadable content. In 2010, OverDrive proposed a 700% fee increase and removing ownership of purchased titles from Kansas libraries. As a result, the State Library began looking for new platforms and negotiating with publishers to transfer purchased content. They transitioned from OverDrive to 3M Cloud Library and OneClickdigital in 2011 while promoting free ebook collections. The State Library advocated for library users by launching a Facebook page in 2012 to raise awareness of publishers' restrictions on ebook lending.
Este documento discute los árboles de decisión, incluyendo cómo inducir árboles a partir de conjuntos de datos clasificados, implementar árboles mediante criterios de partición y parada, y abordar problemas como el sobreajuste. Explica conceptos como la ganancia de información para seleccionar atributos y dividir los datos, así como diferentes enfoques para determinar la profundidad óptima de un árbol.
sharechart Technical Analysis from Student Shailesh Shrestha Presentation on...Sharechart Shrestha
Chilime Hydropower Company Limited was incorporated in 1995 with the objective of generating hydroelectricity through optimal utilization of resources in Nepal. It owns and operates a 22.1 MW power plant in Rasuwa district that generates around 150 GWh annually and sells the electricity to Nepal Electricity Authority. Chilime has also established three subsidiary companies for hydropower development. The company's vision is to be the largest public hydropower company in Nepal and its mission includes harnessing hydropower potential, ensuring sustainable returns to shareholders, creating career opportunities for employees, increasing public participation, and improving local communities.
This document provides an agenda for a hands-on workshop on using the GoodRelations ontology, RDFa, and Yahoo SearchMonkey to publish structured data on e-commerce websites. The workshop covers an overview of the semantic web and GoodRelations ontology, using RDFa to embed semantic annotations in web pages, hands-on exercises for annotating a sample web shop with GoodRelations, and techniques for publishing and querying semantic web data. Attendees will learn how to represent e-commerce data using GoodRelations and RDFa, publish their structured data on the web, and write SPARQL queries to search over semantic web datasets.
The photo shows five men posing for a picture. The man sitting appears to be the oldest and most respected as the others have let him sit in the center. The man on the far right smiles more relaxedly than the others. The two standing men have similar serious facial expressions and stand in similar poses with hands behind their backs, suggesting they are trying to fit in. The suits imply the men are businessmen.
The document discusses concept learning and the candidate elimination algorithm. It defines concept learning as inducing a function that maps examples into categories. It then describes concept learning as a search problem to find the most specific hypothesis consistent with training examples. The candidate elimination algorithm maintains the most specific and general hypotheses consistent with examples to represent the version space of possible concepts. It updates these lists by generalizing specific hypotheses or specializing general hypotheses based on new positive and negative examples.
The Find-S algorithm finds the most specific hypothesis that is consistent with positive training examples by starting with the most specific hypothesis and gradually generalizing it only as far as needed to be consistent with each new positive example seen. The final hypothesis output by Find-S will be the most specific hypothesis within the hypothesis space that is consistent with all positive examples, and also consistent with negative examples if the target concept is representable. Consistency means the hypothesis agrees with all training examples - it outputs the correct label for each example.
This document summarizes lecture notes on learning theory from CS229. It discusses the bias-variance tradeoff when fitting models to data. Simply models have high bias but low variance, while complex models can have high variance but low bias. The optimal model balances these factors. It then provides preliminaries on learning theory, defining key concepts like training error, generalization error, and hypothesis classes. For a finite hypothesis class, it shows that training error is a reliable estimate of generalization error, and this implies an upper bound on the generalization error of the model selected by empirical risk minimization.
Bayesian Learning- part of machine learningkensaleste
This module provides an overview of Bayesian learning methods. It introduces Bayesian reasoning and Bayes' theorem as a probabilistic approach to inference. Key concepts covered include maximum likelihood hypotheses, naive Bayes classifiers, Bayesian belief networks, and the Expectation-Maximization (EM) algorithm. The EM algorithm is described as a method for estimating parameters of probability distributions when some variables are hidden or unobserved.
This document summarizes lecture notes from Andrew Ng on learning theory. It discusses the bias-variance tradeoff in machine learning models and introduces key concepts like generalization error, training error, and hypothesis classes. The document proves that if the hypothesis class H is finite, then with high probability the training error of all hypotheses in H will be close to their true generalization errors, provided the training set is sufficiently large. This uniform convergence guarantee allows relating the performance of the empirical risk minimization algorithm to the best possible hypothesis in H.
The document discusses concept learning through inductive logic. It introduces the concept learning task of predicting when a person will enjoy a sport based on attributes of the day. It describes representing hypotheses as conjunctions of attribute values and the version space approach of tracking the most specific and most general consistent hypotheses. The document explains the candidate elimination algorithm, which uses positive and negative examples to generalize the specific boundary and specialize the general boundary, respectively, until the version space is fully resolved.
The document discusses concept learning and the general-to-specific ordering of hypotheses. It describes how concept learning can be framed as a search problem through a hypothesis space to find the hypothesis that best fits training examples. The Find-S algorithm performs a specific-to-general search to find the most specific hypothesis, while the Candidate-Elimination algorithm computes the version space by iteratively updating the sets of most specific and most general hypotheses consistent with the data. The Candidate-Elimination algorithm provides a framework for concept learning but may not be robust to noisy data or situations where the target concept is not expressible in the hypothesis space. Inductive bias, such as the assumption that the target concept exists in the hypothesis space
-BayesianLearning in machine Learning 12Kumari Naveen
1. Bayesian learning methods are relevant to machine learning for two reasons: they provide practical classification algorithms like naive Bayes, and provide a useful perspective for understanding many learning algorithms.
2. Bayesian learning allows combining observed data with prior knowledge to determine the probability of hypotheses. It provides optimal decision making and can accommodate probabilistic predictions.
3. While Bayesian methods may require estimating probabilities and have high computational costs, they provide a standard for measuring other practical methods.
Statistical machine learning aims to develop algorithms that can detect meaningful patterns in large, complex datasets. It focuses on tasks like classification, clustering, and prediction. Support vector machines (SVMs) are a common approach that learns by finding a hyperplane that maximizes the margin between examples of separate classes. SVMs map data into a high-dimensional feature space to allow for linear separation. The kernel trick allows efficient learning without explicitly computing the mapping, by defining a kernel function measuring similarity. SVMs balance expressiveness, statistical soundness, and computational feasibility.
This document summarizes a lecture on computational learning theory and machine learning. It discusses the difference between training error and generalization error, and how having a small training error does not necessarily mean good generalization. It introduces the Probably Approximately Correct (PAC) learning framework for relating training examples, hypothesis complexity, accuracy, and probability of successful learning. Key concepts discussed include the version space, sample complexity, VC dimension, and uniform convergence. The goal of computational learning theory is to understand what general laws constrain inductive learning and relate various factors like training examples, hypothesis complexity, and accuracy.
The document summarizes algorithms for learning first-order logic rules from examples, including:
1) A sequential covering algorithm that learns one rule at a time to cover examples, removing covered examples and repeating until all examples are covered or rules have low performance.
2) The learn-one-rule sub-algorithm uses a decision tree-like approach to greedily select the attribute that best splits examples according to a performance metric.
3) Variations include allowing low probability classes and using a seed example approach instead of removing covered examples between rules.
1. The document discusses machine learning and provides an overview of key concepts like inductive reasoning, learning from examples, and the constituents of machine learning problems.
2. It explains that machine learning problems involve an example set, background concepts, background axioms, and potential errors in data. Common machine learning tasks are categorization and prediction.
3. The document also outlines the constituents of machine learning methods, including representation schemes, search methods, and approaches for selecting hypotheses when multiple solutions are produced.
1. The document discusses machine learning and provides an overview of key concepts like inductive reasoning, learning from examples, and the constituents of machine learning problems.
2. It explains that machine learning problems involve an example set, background concepts, background axioms, and potential errors in data. Common machine learning tasks are categorization and prediction.
3. The document also outlines the constituents of machine learning methods, including representation schemes, search methods, and approaches for selecting hypotheses when multiple solutions are produced.
This document discusses machine learning concepts of concept learning and decision-tree learning. It describes concept learning as inferring a boolean function from training examples and using algorithms like Candidate Elimination to search the hypothesis space. Decision tree learning is explained as representing classification functions as trees with nodes testing attributes, allowing disjunctive concepts. The ID3 algorithm is presented as a greedy top-down search that selects the best attribute at each node using information gain, potentially overfitting data without pruning or a validation set.
Bayesian learning uses prior knowledge and observed training data to determine the probability of hypotheses. Each training example can incrementally increase or decrease the estimated probability of a hypothesis. Prior knowledge is provided by assigning initial probabilities to hypotheses and probability distributions over possible observations for each hypothesis. New instances can be classified by combining the predictions of multiple hypotheses, weighted by their probabilities. Even when computationally intractable, Bayesian methods provide an optimal standard for decision making.
This document provides an overview of supervised learning and linear regression. It introduces supervised learning problems using an example of predicting house prices based on living area. Linear regression is discussed as an initial approach to model this relationship. The cost function is defined as the mean squared error between predictions and targets. Gradient descent and stochastic gradient descent are presented as algorithms to minimize this cost function and learn the parameters of the linear regression model.
1. Bayesian learning provides a probabilistic approach to inference based on probability distributions of quantities of interest together with observed data.
2. The maximum a posteriori (MAP) hypothesis is the most probable hypothesis given observed training data. Consistent learning algorithms that make no errors on training data will always output a MAP hypothesis under certain assumptions.
3. Bayesian learning can be used to characterize the behavior of learning algorithms like decision tree induction even when the algorithms do not explicitly manipulate probabilities.
This document provides an introduction to machine learning concepts including:
- Machine learning involves learning parameters of probabilistic models from data.
- Maximum likelihood and maximum a posteriori estimation are common techniques for learning parameters.
- Inductive learning involves constructing a hypothesis from examples to generalize the target function to new examples. Cross-validation is used to evaluate hypotheses on held-out data and avoid overfitting.
Bayesian Learning by Dr.C.R.Dhivyaa Kongu Engineering CollegeDhivyaa C.R
This document provides an overview of Bayesian learning methods. It discusses key concepts like Bayes' theorem, maximum a posteriori hypotheses, and maximum likelihood hypotheses. Bayes' theorem allows calculating the posterior probability of a hypothesis given observed data and prior probabilities. The maximum a posteriori hypothesis is the one with the highest posterior probability. Maximum likelihood hypotheses maximize the likelihood of the data. Bayesian learning faces challenges from requiring many initial probabilities and high computational costs but provides a useful perspective on machine learning algorithms.
This document provides an overview of Bayesian learning. It discusses key concepts like Bayes theorem, maximum likelihood hypotheses, minimum description length principle, Bayes optimal classifiers, and Gibbs algorithm. Bayes theorem allows calculating the posterior probability of a hypothesis given observed data and prior probabilities. The maximum likelihood hypothesis is the one that maximizes the likelihood of the data. The minimum description length principle selects the hypothesis that minimizes the total description length of the hypothesis and data. A Bayes optimal classifier combines predictions of multiple hypotheses weighted by their probabilities to classify new instances. The Gibbs algorithm makes predictions by randomly selecting hypotheses based on their posterior probabilities.
Este documento describe reglas de asociación a múltiples niveles y cómo modelar una jerarquía de ítems utilizando una taxonomía. Explica que las reglas de asociación generalizadas permiten encontrar asociaciones entre conjuntos de ítems a diferentes niveles de generalidad. También introduce la noción de reglas R-interesantes para definir cuáles reglas generalizadas son significativas teniendo en cuenta el soporte esperado basado en el soporte de sus ancestros.
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The document discusses instance-based learning methods. It introduces k-nearest neighbors classification and locally weighted regression. For k-nearest neighbors, it explains how to determine the number of neighbors k through validation and describes how to handle both discrete and real-valued classification problems. Locally weighted regression predicts values based on a weighted average of nearby points, where the weights depend on each point's distance from the query instance.
El documento habla sobre el análisis de sentimientos (sentiment analysis), explicando que consiste en identificar y extraer información subjetiva de textos no estructurados para determinar la actitud de un escritor. Explica que existen diferentes enfoques como clasificar la polaridad de un texto o identificar los aspectos mencionados y sus sentimientos asociados. También discute técnicas como n-gramas, detección de características, y modelos generativos vs discriminativos para realizar clasificación de sentimientos.
The document discusses genetic algorithms and genetic programming. It explains that genetic algorithms perform a parallel search of the hypothesis space to optimize a fitness function, mimicking biological evolution. New hypotheses are generated through mutation and crossover of existing hypotheses. Genetic programming similarly evolves computer programs represented as trees through genetic operators. An example shows a genetic programming approach for stacking blocks to spell a word.
Eduardo Poggi presented on explanation-based learning (EBL). EBL is an analytical learning approach that uses background knowledge to analyze training examples and form generalizations. It involves 3 steps: explaining an example using domain knowledge, analyzing the explanation to identify relevant features, and refining the hypothesis. EBL can learn from a single example by using prior knowledge to reduce the hypothesis space. It differs from inductive learning which does not use background knowledge. Potential issues with EBL include developing overly complex theories from numerous rules and imperfect domain theories.
Este documento discute diferentes tipos de órdenes de lenguajes lógicos y formas de inferencia como la deducción, inducción y abducción. Explica que la deducción preserva la verdad al sacar conclusiones de premisas ya conocidas, mientras que la inducción y abducción son conjeturales y no preservan necesariamente la verdad al generar nuevas hipótesis a partir de observaciones. También analiza los límites y usos de cada forma de inferencia.
1) El documento discute los temas de ensambles, bagging, boosting y random forest. 2) Explica que los ensambles combinan múltiples modelos para mejorar el rendimiento y reducir el sesgo y la varianza. 3) Random forest aplica bagging a los árboles de decisión agregando aleatoriedad en la selección de variables para cada nodo con el fin de decorrelacionar los árboles.
El documento describe diferentes conceptos relacionados con el aprendizaje automático y la inteligencia artificial, incluyendo definiciones de aprendizaje automático, sistemas de aprendizaje, métodos para aprender, y discusiones sobre problemas intratables, heurísticas, inferencia, y la diferencia entre aprendizaje automático y estadística. También presenta ejemplos de cómo resolver problemas mediante la búsqueda en un grafo y discute el uso de soluciones aproximadas.
El documento discute la evolución de los sistemas de información y la gestión de datos. Explica cómo los usuarios internos y externos interactúan con los sistemas de una organización y cómo la externalización de servicios e infraestructura ha cambiado la forma en que las organizaciones comparten y consumen datos. También analiza las oportunidades y desafíos que plantea la gestión y el análisis de grandes volúmenes de datos provenientes de múltiples fuentes.
El documento presenta una introducción al aprendizaje bayesiano, comenzando con una aproximación probabilística al aprendizaje automático. Luego revisa conceptos básicos de probabilidad como variables aleatorias y probabilidad condicional, y explica el teorema de Bayes, el cual proporciona un método para calcular la probabilidad posterior de una hipótesis dados los datos. Finalmente, introduce conceptos como la hipótesis de máxima probabilidad a posteriori y la hipótesis de máxima verosimilitud.
This document provides an overview of clustering techniques, including supervised vs. unsupervised learning, clustering concepts, non-hierarchical clustering like k-means, and hierarchical clustering like hierarchical agglomerative clustering. It discusses clustering applications, algorithms like k-means and hierarchical agglomerative clustering, and evaluation metrics like cluster silhouettes. Key clustering goals are to partition unlabeled data into clusters such that examples within a cluster are similar and different between clusters.
Este documento presenta una introducción a las redes neuronales artificiales. Explica que las redes neuronales se inspiran en el funcionamiento del cerebro y están compuestas de unidades simples (neuronas) que se conectan entre sí. También describe el proceso de aprendizaje supervisado mediante el cual las redes neuronales pueden aprender de ejemplos etiquetados y realizar tareas como clasificación y predicción. Finalmente, menciona algunas aplicaciones como reconocimiento de patrones, procesamiento del lenguaje natural y análisis de datos.
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2. Concept Learning
Definitions
Search Space and General-Specific Ordering
Concept learning as search
FIND-S
The Candidate Elimination Algorithm
Inductive Bias
3. First definition
The problem is to learn a function mapping examples into two
classes: positive and negative.
We are given a database of examples already classified as positive or
negative.
Concept learning: the process of inducing a function mapping input
examples into a Boolean output.
4. Notation
Set of instances X
Target concept c : X {+,-}
Training examples E = {(x , c(x))}
Data set D X
Set of possible hypotheses H
h H / h : X {+,-}
Goal: Find h / h(x)=c(x)
5. Representation of Examples
Features:
• color {red, brown, gray}
• size {small, large}
• shape {round,elongated}
• land {humid,dry}
• air humidity {low,high}
• texture {smooth, rough}
6. The Input and Output Space
X
Only a small subset is contained
in our database.
Y = {+,-}
X : The space of all possible examples (input space).
Y: The space of classes (output space).
An example in X is a feature vector X.
For instance: X = (red,small,elongated,humid,low,rough)
X is the cross product of all feature values.
7. The Training Examples
D: The set of training examples.
D is a set of pairs { (x,c(x)) }, where c is the target concept
Example of D:
((red,small,round,humid,low,smooth), +)
((red,small,elongated,humid,low,smooth),+)
((gray,large,elongated,humid,low,rough), -)
((red,small,elongated,humid,high,rough), +)
Instances from the input space
Instances from
the output space
8. Hypothesis Representation
Consider the following hypotheses:
(*,*,*,*,*,*): all mushrooms are poisonous
(0,0,0,0,0,0): no mushroom is poisonous
Special symbols:
* Any value is acceptable
0 no value is acceptable
Any hypothesis h is a function from X to Y
h: X Y
We will explore the space of conjunctions.
9. Hypothesis Space
The space of all hypotheses is represented by H
Let h be a hypothesis in H.
Let X be an example of a mushroom.
if h(X) = + then X is poisonous,
otherwise X is not-poisonous
Our goal is to find the hypothesis, h*, that is very “close”
to target concept c.
A hypothesis is said to “cover” those examples it classifies
as positive.
X
h
10. Assumption 1
We will explore the space of all conjunctions.
We assume the target concept falls within this space.
Target concept c
H
11. Assumption 2
A hypothesis close to target concept c obtained after
seeing many training examples will result in high
accuracy on the set of unobserved examples.
Training set D
Hypothesis h* is good
Complement set D’
Hypothesis h* is good
12. Concept Learning as Search
There is a general to specific ordering inherent to any
hypothesis space.
Consider these two hypotheses:
h1 = (red,*,*,humid,*,*)
h2 = (red,*,*,*,*,*)
We say h2 is more general than h1 because h2 classifies
more instances than h1 and h1 is covered by h2.
13. General-Specific
For example, consider the following hypotheses:
h1
h2 h3
h1 is more general than h2 and h3.
h2 and h3 are neither more specific nor more general
than each other.
14. Let hj and hk be two hypotheses mapping examples into {+,-}.
We say hj is more general than hk iff
For all examples X, hk(X) = + hj(X) = +
We represent this fact as hj >= hk
The >= relation imposes a partial ordering over the
hypothesis space H (reflexive, antisymmetric, and transitive).
Definition
15. Lattice
Any input space X defines then a lattice of hypotheses ordered
according to the general-specific relation:
h1
h3 h4
h2
h5 h6
h7 h8
16. Working Example: Mushrooms
Class of Tasks: Predicting poisonous mushrooms
Performance: Accuracy of Classification
Experience: Database describing mushrooms with their class
Knowledge to learn:
Function mapping mushrooms to {+,-}
where -:not-poisonous and +:poisonous
Representation of target knowledge:
conjunction of attribute values.
Learning mechanism:
Find-S
17. Finding a Maximally-Specific Hypothesis
Algorithm to search the space of conjunctions:
Start with the most specific hypothesis
Generalize the hypothesis when it fails to cover a positive
example
Algorithm:
1. Initialize h to the most specific hypothesis
2. For each positive training example X
For each value a in h
If example X and h agree on a, do nothing
else generalize a by the next more general constraint
3. Output hypothesis h
18. Example
Let’s run the learning algorithm above with the
following examples:
((red,small,round,humid,low,smooth), +)
((red,small,elongated,humid,low,smooth),+)
((gray,large,elongated,humid,low,rough), -)
((red,small,elongated,humid,high,rough), +)
We start with the most specific hypothesis:
h = (0,0,0,0,0,0)
The first example comes and since the example is positive and h
fails to cover it, we simply generalize h to cover exactly this
example: h = (red,small,round,humid,low,smooth)
19. Example
Hypothesis h basically says that the first example is the only
positive example, all other examples are negative.
Then comes examples 2:
((red,small,elongated,humid,low,smooth), poisonous)
This example is positive. All attributes match hypothesis h
except for attribute shape: it has the value elongated, not
round.
We generalize this attribute using symbol * yielding:
h: (red,small,*,humid,low,smooth)
The third example is negative and so we just ignore it.
Why is it we don’t need to be concerned with negative
examples?
20. Example
Upon observing the 4th example, hypothesis h is
generalized to the following:
h = (red,small,*,humid,*,*)
h is interpreted as any mushroom that is red, small and
found on humid land should be classified as poisonous.
21. Analyzing the Algorithm
• The algorithm is
guaranteed to find the
hypothesis that is most
specific and consistent with
the set of training
examples.
• It takes advantage of the
general-specific ordering to
move on the corresponding
lattice searching for the
next most specific
hypothesis.
h1
h3 h4
h2
h5 h6
h7 h8
24. Points to Consider
There are many hypotheses consistent with the training data D.
Why should we prefer the most specific hypothesis?
What would happen if the examples are not consistent?
What would happen if they have errors, noise?
What if there is a hypothesis space H where one can find more that one
maximally specific hypothesis h?
The search over the lattice must then be different to allow for this
possibility.
25. Summary FIND-S
The input space is the space of all examples; the output space is the
space of all classes.
A hypothesis maps examples into classes.
We want a hypothesis close to target concept c.
The input space establishes a partial ordering over the hypothesis
space.
One can exploit this ordering to move along the corresponding
lattice.
26. Working Example: Mushrooms
Class of Tasks: Predicting poisonous mushrooms
Performance: Accuracy of Classification
Experience: Database describing mushrooms with their class
Knowledge to learn:
Function mapping mushrooms to {+,-}
where -:not-poisonous and +:poisonous
Representation of target knowledge:
conjunction of attribute values.
Learning mechanism:
candidate-elimination
27. Candidate Elimination
The algorithm that finds the maximally specific hypothesis
is limited in that it only finds one of many hypotheses
consistent with the training data.
The Candidate Elimination Algorithm (CEA) finds ALL
hypotheses consistent with the training data.
CEA does that without explicitly enumerating all
consistent hypotheses.
28. Consistency vs Coverage
h1
h2
h1 covers a different set of examples than h2
h2 is consistent with training set D
h1 is not consistent with training set D
Positive examples
Negative examples
Training set D
-
-
-
-
+
+
++
+
+
+
29. Version Space VS
Hypothesis space H
Version space:
Subset of hypothesis from H consistent with training set D.
30. List-Then-Eliminate Algorithm
Algorithm:
1. Version Space VS: All hypotheses in H
2. For each training example X
Remove every hypothesis h in H inconsistent
with X: h(x) = c(x)
3. Output the version space VS
Comments: This is unfeasible. The size of H is unmanageable.
31. Previous Exercise: Mushrooms
Let’s remember our exercise in which we tried to classify
mushrooms as poisonous (+) or not-poisonous (-).
Training set D:
((red,small,round,humid,low,smooth), +)
((red,small,elongated,humid,low,smooth), +)
((gray,large,elongated,humid,low,rough), -)
((red,small,elongated,humid,high,rough), +)
32. Consistent Hypotheses
Our first algorithm found only one out of six
consistent hypotheses:
(red,small,*,humid,*,*)
(*,small,*,humid,*,*)(red,*,*,humid,*,*) (red,small,*,*,*,*)
(red,*,*,*,*,*) (*,small,*,*,*,*)G:
S:
S: Most specific
G: Most general
34. Candidate-Elimination Algorithm
• Initialize G to the set of maximally general hypotheses in H
• Initialize S to the set of maximally specific hypotheses in H
• For each training example X do
• If X is positive: generalize S if necessary
• If X is negative: specialize G if necessary
• Output {G,S}
35. Candidate-Elimination Algorithm
Initialize G to the set of maximally general hypotheses in H
Initialize S to the set of maximally specific hypotheses in H
For each training example d, do
If d+
Remove from G any hypothesis inconsistent with d
For each hypothesis s in S that is not consistent with d
Remove s from S
Add to S all minimal generalizations h of s such that h is consistent with
d and some member of G is more general than h
Remove from S any hipothesis that is more general than another
hypothesis in S
If d-
Remove from S any hypothesis inconsistent with d
For each hypothesis g in G that is not consistent with d
Remove g from G
Add to G all minimal specializations h of g such that h is consistent with
d and some member of S is more general than h
Remove from G any hipothesis that is less general than another
hypothesis in G
36. Positive Examples
a) If X is positive:
Remove from G any hypothesis inconsistent with X
For each hypothesis h in S not consistent with X
Remove h from S
Add all minimal generalizations of h consistent with X
such that some member of G is more general than h
Remove from S any hypothesis more general than
any other hypothesis in S
G:
S:
h
inconsistent
add minimal generalizations
37. Negative Examples
b) If X is negative:
Remove from S any hypothesis inconsistent with X
For each hypothesis h in G not consistent with X
Remove g from G
Add all minimal generalizations of h consistent with X
such that some member of S is more specific than h
Remove from G any hypothesis less general than any other
hypothesis in G
G:
S: h inconsistent
add minimal specializations
38. An Exercise
Initialize the S and G sets:
S: (0,0,0,0,0,0)
G: (*,*,*,*,*,*)
Let’s look at the first two examples:
((red,small,round,humid,low,smooth), +)
((red,small,elongated,humid,low,smooth), +)
39. An Exercise: two positives
The first two examples are positive:
((red,small,round,humid,low,smooth), +)
((red,small,elongated,humid,low,smooth), +)
S: (0,0,0,0,0,0)
(red,small,round,humid,low,smooth)
(red,small,*,humid,low,smooth)
G: (*,*,*,*,*,*)
generalize
specialize
40. An Exercise: first negative
The third example is a negative example:
((gray,large,elongated,humid,low,rough), -)
S:(red,small,*,humid,low,smooth)
G: (*,*,*,*,*,*)
generalize
specialize
(red,*,*,*,*,*,*) (*,small,*,*,*,*) (*,*,*,*,*,smooth)
Why is (*,*,round,*,*,*) not a valid specialization of G
41. An Exercise: another positive
The fourth example is a positive example:
((red,small,elongated,humid,high,rough), +)
S:(red,small,*,humid,low,smooth)
generalize
specialize
G: (red,*,*,*,*,*,*) (*,small,*,*,*,*) (*,*,*,*,*,smooth)
(red,small,*,humid,*,*)
42. The Learned Version Space VS
G: (red,*,*,*,*,*,*) (*,small,*,*,*,*)
S: (red,small,*,humid,*,*)
(red,*,*,humid,*,*) (red,small,*,*,*,*) (*,small,*,humid,*,*)
43. Points to Consider
Will the algorithm converge to the right hypothesis?
The algorithm is guaranteed to converge to the right hypothesis
provided the following:
No errors exist in the examples
The target concept is included in the hypothesis space H
What happens if there exists errors in the examples?
The right hypothesis would be inconsistent and thus eliminated.
If the S and G sets converge to an empty space we have evidence that
the true concept lies outside space H.
44. Query Learning
Remember the version space VS after seeing our 4 examples
on the mushroom database:
G: (red,*,*,*,*,*,*) (*,small,*,*,*,*)
S: (red,small,*,humid,*,*)
(red,*,*,humid,*,*) (red,small,*,*,*,*) (*,small,*,humid,*,*)
What would be a good question to pose to the algorithm?
What example is best next?
45. Query Learning
Remember there are three settings for learning:
Tasks are generated by a random process outside the learner
The learner can pose queries to a teacher
The learner explores its surroundings autonomously
Here we focus on the second setting; posing queries to an expert.
Version space strategy: Ask about the class of an example that would
prune half of the space.
Example: (red,small,round,dry,low,smooth)
46. Query Learning
In general if we are able to prune the version space by
half on each new query then we can find an optimal
hypothesis in the following
Number of steps: log2 |VS|
Can you explain why?
47. Classifying Examples
What if the version space VS has not collapsed into a
single hypothesis and we are asked to classify a new
instance?
Suppose all hypotheses in set S agree that the instance is
positive.
Then we are sure that all hypotheses in VS agree the instance is
positive. Why?
The same can be said if the instance is negative by all members
of set G. Why?
In general one can vote over all hypotheses in VS if there
is no unanimous agreement.
48. Inductive Bias
Inductive bias is the preference for a hypothesis space H
and a search mechanism over H.
What would happen if we choose an H that contains all
possible hypotheses?
What would the size of H be?
|H| = Size of the power set of the input space X.
Example:
You have n Boolean features. |X| = 2n
And the size of H is 2^2^n
49. Inductive Bias
In this case, the candidate elimination algorithm would simply
classify as positive the training examples it has seen. This is
because H is so large, every possible hypothesis is contained
within it.
A Property of any Inductive Algorithm:
It must have some embedded assumptions about the
nature of H.
Without assumptions learning is impossible.
50. Summary
The candidate elimination algorithm exploits the general-specific
ordering of hypotheses to find all hypotheses consistent with the
training data.
The version space contains all consistent hypotheses and is simply
represented by two lists: S and G.
Candidate elimination algorithm is not robust to noise and assumes
the target concept is included in the hypothesis space.
Any inductive algorithm needs some assumptions about the
hypothesis space, otherwise it would be impossible to perform
predictions.