Machine Learning: Generative and Discriminative Models

• Gaussians, Naïve Bayes, Mixtures of multinomials • Mixtures of Gaussians, Mixtures of experts, Hidden Markov Models (HMM) ... – by fitting Gaussian class-conditional densities will result in . 2M . parameters for means, M(M+1)/2 ... Markov Random Field (MRF)

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Machine Learning Basics: Supervised Learning Algorithms

Deep Learning Probabilistic Supervised Classification Srihari • If we only have two classes we only need to specify the distribution for one of these classes – The probability of the other class is known – Linear regression has a closed-form solution – But …

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Gaussian Derivatives - University at Buffalo

• Slice the surface with horizontal planes which the locus of points with the quadratic form. INFORMATION THEORY. Relative Entropy. ... great generality, and it is useful when we seek to know whether something other than the assumed case …

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Multiclass Logistic Regression - University at Buffalo

Probabilistic Discriminative Models •Generative vsDiscriminative 1.Fixed basis functions in linear classification 2.Logistic Regression (two-class) 3.Iterative Reweighted Least Squares (IRLS) 4.Multiclass Logistic Regression 5.ProbitRegression 6.Canonical Link Functions 2 Machine Learning Srihari

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Machine Learning Basics: Estimators, Bias and Variance

Machine Learning Basics: Estimators, Bias and Variance Sargur N. Srihari srihari@cedar.buffalo.edu ... • To distinguish estimates of parameters from their true value, a point estimate of a parameter θ is represented by • Let {x(1), x(2),..x(m)} be m independent and

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The Hessian Matrix - University at Buffalo

Diagonal Approximation • In many case inverse of Hessian is needed • If Hessian is approximated by a diagonal matrix (i.e., off-diagonal elements are zero), its inverse is trivially computed • Complexity is O(W) rather than O(W2) for full Hessian 7

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Backpropagation - University at Buffalo

Machine Learning Srihari Matrix Multiplication: Forward Propagation •Each layer is a function of layer that preceded it •First layer is given by z =h(W(1)T x +b(1)) •Second layer is y = σ(W(2)T x +b(2)) •Note that W is a matrix rather than a vector

Iterative Reweighted Least Squares

•Derivative has the form •Setting equal to zero and solving we get Machine Learning Srihari 4 y(x,w)=w j ... 3.Gradient 4.Hessian 5.Newton-Raphsonupdate …

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The Gaussian distribution

The Gaussian distribution has a number of convenient analytic properties, some of which we describe below. Marginalization Often we will have a set of variables x with a joint multivariate Gaussian distribution, but only be interested in reasoning about a subset of these variables. Suppose x has a multivariate Gaussian distribution: p(x j ...

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The Multivariate Gaussian Distribution

The Multivariate Gaussian Distribution Chuong B. Do October 10, 2008 A vector-valued random variable X = X1 ··· Xn T is said to have a multivariate normal (or Gaussian) distribution with mean µ ∈ Rnn

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Chapter 13 The Multivariate Gaussian - People

The multivariate Gaussian distribution is commonly expressed in terms of the parameters µ for now that Σ is also positive definite, but later on we will have occasion to relax that constraint).

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1 Multivariate Normal Distribution - Princeton University

Gaussian Models (9/9/13) Lecturer: Barbara Engelhardt Scribes: Xi He, Jiangwei Pan, Ali Razeen, Animesh Srivastava 1 Multivariate Normal Distribution The multivariate normal distribution (MVN), also known as multivariate gaussian, is a generalization of the one-dimensional normal distribution to higher dimensions.

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More on Multivariate Gaussians - Stanford University

A vector-valued random variable x ∈ Rn is said to have a multivariate normal (or Gaus-sian) distribution with mean µ ∈ Rnn ++ 1 if its probability density function is given by p(x;µ,Σ) = 1 (2π)n/2|Σ|1/2 exp − 1 2 (x−µ)TΣ−1(x−µ) . 2 Gaussian facts Multivariate Gaussians turn out to be extremely handy in practice due to the ...

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Multivariate normal distribution

or multivariate Gaussian distribution, is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One possible definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate

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Multivariate Gaussian Distribution - Mathematics Home

Multivariate Gaussian Distribution The random vector X = (X 1,X 2,...,X p) is said to have a multivariate Gaussian distribution if the joint distribution of X 1,X 2,...,X p has density f X(x 1,x 2,...,x p) = 1 (2π)p/2 det(Σ)1/2 exp − 1 2 (x−µ)tΣ−1(x−µ) (1) follows: x is the column vector x = x 1 x 2... x p , µ is the column vector ...

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An Intuitive Tutorial to Gaussian Processes Regression

An Intuitive Tutorial to Gaussian Processes Regression 3 Gaussian vector x2 = [x1 2, x 2 2,. . ., x n 2] in the same coordinates at Y = 1 shown in Fig.3. Keep in mind that either x 1 or x2 is a uni-variate normal distribution shown in Fig.2. 0.0 0.2 0.4 0.6 0.8 1.0

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Conjugate Bayesian analysis of the Gaussian distribution

The Gaussian or normal distribution is one of the most widely used in statistics. Estimating its parameters using Bayesian inference and conjugate priors is also widely used. The use of conjugate priors allows all the results to be derived in closed form. Unfortunately, different books use different conventions on how to parameterize the various

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