Generative Learning Algorithms Quiz
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What is the primary second step when using generative learning algorithms to classify animals?

  • To gather more training data
  • To create a decision boundary
  • To model the features of each class separately (correct)
  • To use Bayes theorem to predict classes

    • Generative learning algorithms try to learn the conditional distribution of features given the class label.

    False

    What is the role of Bayes theorem in generative learning algorithms?

    It is used to derive the posterior distribution.

    In generative learning algorithms, the feature distribution of dogs is modeled as P(X | Y = ______).

    <p>0</p> Signup and view all the answers

    Match the following algorithms with their classification approach:

    <p>Logistic Regression = Discriminative Naïve Bayes = Generative Perceptron = Discriminative Linear Discriminant Analysis = Generative</p> Signup and view all the answers

    What type of algorithms are logistic regression and the perceptron algorithm classified as?

    <p>Discriminative learning algorithms</p> Signup and view all the answers

    Quadratic Discriminant Analysis is a type of generative learning algorithm.

    <p>True</p> Signup and view all the answers

    What types of distributions are typically used to model classes in generative learning algorithms?

    <p>Normal (Gaussian) distributions</p> Signup and view all the answers

    What is the primary concept represented by Bayes theorem in classification?

    <p>It determines the probability that an observation belongs to a class.</p> Signup and view all the answers

    Estimating the density function $f_k(x)$ is typically straightforward and easy.

    <p>False</p> Signup and view all the answers

    What does the symbol $ u_k$ represent in the context of Bayes theorem?

    <p>The prior probability for class k.</p> Signup and view all the answers

    The Bayes classifier classifies an observation to the class for which the posterior probability $p(Y = k | X = x)$ is _____.

    <p>largest</p> Signup and view all the answers

    Match the following classifiers with their characteristics:

    <p>Linear Discriminant Analysis = Assumes linear boundaries between classes Quadratic Discriminant Analysis = Allows for quadratic decision boundaries Naive Bayes = Assumes independence among predictors Bayes Classifier = Classifies based on maximum posterior probability</p> Signup and view all the answers

    Which of the following classifiers does NOT require a linear assumption?

    <p>Quadratic Discriminant Analysis</p> Signup and view all the answers

    The differentiation between class densities can be ignored when calculating posterior probabilities.

    <p>False</p> Signup and view all the answers

    Which three classifiers are mentioned that use estimates of $f_k(x)$ to approximate the Bayes classifier?

    <p>Linear Discriminant Analysis, Quadratic Discriminant Analysis, Naive Bayes.</p> Signup and view all the answers

    When performing discriminant analysis, what is the primary goal?

    <p>To determine which class a new data point belongs to</p> Signup and view all the answers

    Linear discriminant analysis can handle more than two response classes effectively.

    <p>True</p> Signup and view all the answers

    What common assumption is made about the variance in Linear Discriminant Analysis?

    <p>The variance is assumed to be the same for all classes.</p> Signup and view all the answers

    In discriminant analysis, the decision boundary is located at ________ when there are two classes with equal priors.

    <p>𝜋1 + 𝜋2 / 2</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>Discriminant Score = The value used to assign a new point to a class Gaussian Distribution = A normal distribution used to model data in LDA Bayes Classifier = Uses prior probabilities and likelihoods to classify data Common Variance = An assumption in LDA that classes share the same variance</p> Signup and view all the answers

    Which of the following statements about Linear Discriminant Analysis (LDA) is correct?

    <p>LDA assumes normality in the distribution of predictors</p> Signup and view all the answers

    As the number of classes increases, discriminant analysis provides higher dimensionality views of the data.

    <p>False</p> Signup and view all the answers

    To estimate the probability P(Y = k|X = x), LDA relies on the ________ function to classify points.

    <p>discriminant</p> Signup and view all the answers

    What assumption is made about the covariance matrix in Linear Discriminant Analysis (LDA)?

    <p>A common covariance matrix is used across classes.</p> Signup and view all the answers

    In Quadratic Discriminant Analysis (QDA), all classes are assumed to have the same covariance matrix.

    <p>False</p> Signup and view all the answers

    What is the primary difference between LDA and QDA?

    <p>LDA assumes a common covariance matrix, while QDA allows each class to have its own covariance matrix.</p> Signup and view all the answers

    In LDA, the observations are drawn from a multivariate Gaussian distribution with a class-specific mean vector and a common __________ matrix.

    <p>covariance</p> Signup and view all the answers

    In the context of multivariate Gaussian distribution, what is represented by the symbol μ?

    <p>The mean vector of X</p> Signup and view all the answers

    Match the following concepts with their descriptions:

    <p>LDA = Assumes a common covariance matrix across classes QDA = Assumes each class has its own covariance matrix Multivariate Gaussian = Distribution represented by a mean vector and covariance matrix Bayes Decision Boundary = Decision boundary derived from Bayes' theorem</p> Signup and view all the answers

    The ellipses representing probability density in a multivariate Gaussian distribution are the same for all classes in LDA.

    <p>True</p> Signup and view all the answers

    What does π represent in the context of LDA?

    <p>The prior probabilities of each class.</p> Signup and view all the answers

    What does the covariance matrix $\Sigma_k$ represent in the observation from the kth class?

    <p>The relationship between the features in the class</p> Signup and view all the answers

    LDA and QDA are effective when the covariance matrices of the classes are identical.

    <p>True</p> Signup and view all the answers

    What is the main assumption of the naive Bayes classifier regarding the features?

    <p>Features are independent</p> Signup and view all the answers

    The fraction of negative examples classified as positive is known as the ____.

    <p>false positive rate</p> Signup and view all the answers

    What is the consequence of using a higher threshold when classifying with a Bayesian approach?

    <p>Increased false negative rate</p> Signup and view all the answers

    Naive Bayes always produces poor classification results due to its strong independence assumptions.

    <p>False</p> Signup and view all the answers

    In the credit data example, what was the training error rate achieved by LDA?

    <p>2.75%</p> Signup and view all the answers

    What is the aim of reducing the threshold in a classification model?

    <p>Decrease the false negative rate</p> Signup and view all the answers

    The Equal Error Rate (EER) is the point at which false positive and false negative rates are identical.

    <p>True</p> Signup and view all the answers

    What does AUC stand for in the context of ROC curves?

    <p>Area Under the Curve</p> Signup and view all the answers

    Logistic regression is popular for classification when K = ______.

    <p>2</p> Signup and view all the answers

    Match the following terms with their corresponding definitions:

    <p>Logistic Regression = Uses conditional likelihood LDA = Uses full likelihood Naive Bayes = Useful when p is very large EER = Point of identical false positive and false negative rates</p> Signup and view all the answers

    Which statement correctly describes LDA and Logistic Regression?

    <p>LDA uses generative learning while Logistic Regression uses discriminative learning</p> Signup and view all the answers

    Both LDA and Logistic Regression will produce drastically different results in most scenarios.

    <p>False</p> Signup and view all the answers

    What is the main advantage of using LDA when n is small?

    <p>It is useful for classification due to well-separated classes and reasonable Gaussian assumptions.</p> Signup and view all the answers

    Study Notes

    Introduction to Machine Learning - AI 305

    • The lecture covers generative learning algorithms
    • These algorithms model p(y|x; θ) - the conditional distribution of y given x
    • Logistic regression is an example
    • A classification problem that distinguishes between elephants (y=1) and dogs (y=0) based on features is discussed.
    • Algorithms (like logistic regression or perceptron) find a decision boundary (a straight line) to separate elephants and dogs.

    Agenda

    • Linear Discriminant Analysis
    • Quadratic Discriminant Analysis
    • Naïve Bayes

    Generative Learning Algorithms

    • Algorithms that learn p(x|y) and p(y) directly are called generative
    • These algorithms model the distribution of x for each class separately: p(x|y=0) and p(x|y=1)
    • If y = 0, p(x|y = 0) models distributions of dog features
    • If y = 1, p(x|y = 1) models distributions of elephant features

    Bayes Theorem for Classification

    • Thomas Bayes developed a subfield of statistical and probabilistic modelling
    • Bayes theorem: $p(Y=k|X=x) = \frac{p(X=x|Y=k)p(Y=k)}{p(X=x)} $
    • Rewritten for discriminant analysis: $p(Y=k|X=x) = \frac{f_k(x) \pi_k}{\sum_{l=1}^K f_l(x)\pi_l}$
    • $f_k(x)$ is the density of x in class k
    • $\pi_k$ is the prior marginal probability for class k
    • $p(Y = k|X = x)$ is the posterior probability that x belongs to the kth class

    Bayes Theorem for Classification - Continued

    • Estimating $\pi_k$ is straightforward using training data
    • Estimating $f_k(x)$ requires assumptions
    • Simplifying assumptions are needed to estimate $f_k(x)$

    Classify to the Highest Density

    • Classifying a new point is based on which density is higher
    • If priors are different, they are considered when comparing $p(x|y)p(y)$
    • Decision boundaries shift differently according to prior probabilities

    Why Discriminant Analysis?

    • In well-separated classes, logistic regression parameter estimates are unstable
    • Linear Discriminant Analysis avoids instability
    • LDA is more stable than logistic regression when $n$ is small and predictors $X$ approximately normal in each class
    • Also useful with more than two response classes to provide low-dimensional views of data.

    Linear Discriminant Analysis when p = 1

    • To estimate $f_k(x)$ when p = 1 (one predictor)
    • Assumption: $f_k(x)$ is normal/Gaussian
    • $f_k(x) = \frac{1}{\sqrt{2\pi\sigma_k}} exp(-\frac{1}{2\sigma_k^2}(x - \mu_k)^2)$
    • $\mu_k$: mean in class k
    • $\sigma_k$: variance in class k (for simplicity, assumed equal)

    Discriminant Functions

    • To classify a new value of X, find the class with the highest discriminant score.
    • $ δ_k(x) = \frac{(x - \mu_k)^2}{2\sigma^2} + log(\pi_k)$
    •  $δ_k(x)$ is a linear function of x
    • If there are two classes and prior probabilities are equal, the decision boundary is $x = \frac{\mu_1 + \mu_2}{2}$

    Example with µ1 = −1.5, µ2 = 1.5, π1 = π2 = 0.5, and σ^2 = 1

    • Show examples of different densities in different scenarios

    Estimating the parameters

    • $\pi_k = \frac{n_k}{n}$, where $n_k$ = observations in class k
    • $\mu_k = \frac{1}{n_k} \sum_{i:y_i =k} x_i$
    • $\sigma^2 = \frac{1}{n - K} \sum_{k=1}^K \sum_{i:y_i = k} (x_i - \mu_k)^2$

    LDA - Continued

    • Assumes observations within each class follow a normal distribution with a common variance and class-specific mean

    Linear Discriminant Analysis when p > 1

    • Extends LDA to multiple predictors
    • Assumes each predictor follows a multivariate Gaussian distribution
    • Multivariate Gaussian has class-specific mean vectors and a common covariance matrix

    Linear Discriminant Analysis when p > 1 - Continued

    • Formally, multivariate Gaussian density: $f(x) = \frac{1}{(2\pi)^{p/2} |\Sigma|^{1/2}} exp(-\frac{1}{2}(x - \mu)^T\Sigma^{-1}(x - \mu))$
    • Discriminant function $δ_k(x) = \frac{1}{2} x^T \Sigma^{-1} \mu_k - \frac{1}{2} \mu_k^T \Sigma^{-1} \mu_k + log(\pi_k)$

    Example

    • Show examples of applying LDA to real data

    Quadratic Discriminant Analysis

    • When class covariance matrices are different, QDA is used
    • QDA’s discriminant function is quadratic in $x$.

    LDA and QDA in two scenarios

    • Show examples of applying LDA and QDA in scenarios with different correlations or variables

    Naïve Bayes

    • Features are assumed independent in each class in Naive Bayes
    • $f_k(x) = \prod_{j=1}^p f_{kj}(x_j)$

    Naïve Bayes - Continued

    • $f_{kj}(x_j)$ is probability distribution of feature j in class k
    • Useful when p is large, or when LDA breaks down

    Gaussian Naïve Bayes

    • $δ_k(x) = log(\pi_k) + \sum_{j=1}^p log(f_{kj}(x_j))$
    • If x is qualitative, use probability mass function of feature values instead of normal distribution

    LDA on Credit Data

    • Example of applying LDA to credit data
    • Issues with training error vs test error are discussed

    Types of Errors

    • False positive rate and false negative rate are defined
    • Error rates can be changed by changing the threshold

    Varying the threshold

    • The effects of changing threshold on error rates are discussed
    • Equal Error Rate (EER) is identified as point where False Positive and False Negative rates are equal.

    ROC Curve

    • ROC plot displays true positive rate vs false positive rate
    • AUC (Area Under Curve) is calculated to summarize performance; Higher is better.

    Logistic Regression versus LDA

    • Both can be shown as log function, but parameters estimated differently
    • Logistic regression uses conditional likelihood (discriminative) , while LDA uses full likelihood (generative)

    Summary

    • Summary of when to use each classification method (Logistic Regression, LDA, QDA, Naive Bayes) based on data characteristics

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    AI 305 - Generative Learning Algorithms PDF

    Description

    Test your knowledge on generative learning algorithms and their application in animal classification. This quiz covers key concepts, including Bayes theorem, feature distribution, and various classification algorithms. Perfect for students studying machine learning and artificial intelligence.

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