Approaches to Classification Probabilistic Inference stage – use training data to learn a model for p(C k |x) Decision stage – use posterior probabilities p(C k |x) to make optimal class assignments Discriminant fn. Solve both problems together and simply learn a fn. that maps x directly to a class Probabilistic Generative First solve inference problem of determining class conditional densities p(x|C k ) for each class C k and then determine posterior probabilities Discriminative
Probabilistic Models Probabilistic view of classification! Models with linear decision boundaries arise from simple assumptions about the distribution of data Two approaches: Generative Models (2 steps) Discriminative Models (1 step) In both cases use decision theory to assign a new x to a class
Probabilistic Generative Models 2 step process Model class conditional densities p(x|Ck) & class priors p(Ck) Use them to compute class posterior probabilities p(Ck|x) according to Bayes’ theorem 2 class case Posterior prob. for class C1 = p(C1|x) = σ (a), the logistic sigmoid fn. a = & Sigmoid fn is S-shaped and is also called the squashing fn because it maps the whole real line into a finite interval (maps real a ε (-∞, +∞) to finite (0,1) interval) Plays an important role in many classification algorithms
Probabilistic Generative Models Symmetry property Inverse of the logistic sigmoid fn is given by & is called the logit or log odds function because it represents the log of the ratio of the probabilities for the 2 classes
Probabilistic Generative Models Posterior probabilities have been written in an equivalent form using σ! p(C1|x) = σ (a) What’s the significance of doing so? We shall see this shortly when a(x) takes a simple functional form If a(x) is a linear fn. of x, then then posterior prob. is governed by a generalized linear model Generalized Linear Model?
Probabilistic Generative Models Generalized Linear Model In linear regression models, the model prediction y(x,w)was given by a linear function of the parameters w In the simplest case, the model is also linear in the input variables x so that y is a real no. In classification, we wish to predict discrete class labels, or more generally posterior probabilities in (0,1) To achieve this, we consider a generalization of this model in which we transform the linear fn of w using a non-linear fn f(.) so that In ML, f(.) is know as an activation fn., whereas its inverse is called a link fn. in statistical literature
Probabilistic Generative Models Generalized Linear Model Decision surface corresponds to so th Decision surfaces are linear fns of x, even if the fn f(.) is non-linear Generalized linear models are no longer linear in the parameters due to non-linear fn f(.) More complex in terms of analytical and computational properties! Still simpler that the more general non-linear models
Probabilistic Generative Models K>2 classes: Softmax Function
Probabilistic Generative Models K>2 classes: Softmax Function
Probabilistic Generative Models Softmax Function
Probabilistic Generative Models Softmax Function
Probabilistic Generative Models Softmax Function The soft maximum approximates the hard maximum and is a convex function just like the hard maximum. But the soft maximum is smooth. It has no sudden changes in direction and can be differentiated as many times as you like. These properties make it easy for convex optimization algorithms to work with the soft maximum. In fact, the function may have been invented for optimization;
Probabilistic Generative Models Softmax Function Accuracy of the soft maximum approximation depends on scale Multiplying x and y by a large constant brings the soft maximum closer to the hard maximum For example, g(1, 2) = 2.31, but g(10, 20) = 20.00004 “hardness” of the soft maximum can be controlled by generalizing the soft maximum to depend on a parameter k. g(x, y; k) = log( exp(kx) + exp(ky) ) / k Soft maximum can be made as close to the hard maximum as desired by making k large enough For every value of k the soft maximum is differentiable, but the size of the derivatives increase as k increases. In the limit the derivative becomes infinite as the soft maximum converges to the hard maximum
Probabilistic Generative Models Forms of class-conditional densities Continuous Inputs (x follows Gaussian distribution) Discrete Inputs (for example )
Probabilistic Generative Models Continuous Inputs (x follows Gaussian distribution) All classes share the same covariance matrix All classes do not share the same covariance matrix
Probabilistic Discriminative Models Logistic Regression 2-class Multi-class Parameters using Maximum Likelihood Iterative Reweighted Least Squares (IRLS) Probit Regression
Probabilistic Discriminative Models Logistic Regression Logistic regression is a form of regression analysis in which the outcome variable is binary or dichotomous Consider a binary response variable Variable with two outcomes One outcome represented by a 1 and the other represented by a 0 Examples: Does the person have a disease? Yes or No Who is the person voting for? McCain or Obama Outcome of a baseball game? Win or loss
Probabilistic Discriminative Models Logistic Regression Example Data Set Response Variable –> Admission to Grad School (Admit) 0 if admitted, 1 if not admitted Predictor Variables GRE Score (gre) Continuous University Prestige (topnotch) 1 if prestigious, 0 otherwise Grade Point Average (gpa) Continuous
Probabilistic Discriminative Models First 10 Observations of the Data Set ADMITGRETOPNOTCHGPA 138003.61 066013.67 080014 064003.19 152002.93 076003 056002.98 140003.08 054003.39 170013.92
Logistic Regression Consider the linear probability model Issue: π (X i ) can take on values less than 0 or greater than 1 Issue: Predicted probability for some subjects fall outside of the [0,1] range.
Logistic Regression Consider the logistic regression model GLM with binomial random component and identity link g( μ ) = logit( μ ) Range of values for π (X i ) is 0 to 1
Logistic Regression Consider the logistic regression model And the linear probability model Then the graph of the predicted probabilities for different grade point averages:
What is Logistic Regression? In a nutshell: A statistical method used to model dichotomous or binary outcomes (but not limited to) using predictor variables. Used when the research method is focused on whether or not an event occurred, rather than when it occurred (time course information is not used).
What is Logistic Regression? What is the “Logistic” component? Instead of modeling the outcome, Y, directly, the method models the log odds(Y) using the logistic function.
Maximum Likelihood Flipped a fair coin 10 times: T, H, H, T, T, H, H, T, H, H What is the Pr(Heads) given the data? 1/100? 1/5? 1/2? 6/10?
T, H, H, T, T, H, H, T, H, H What is the Pr(Heads) given the data? Most reasonable data-based estimate would be 6/10. In fact, is the ML estimator of p. Maximum Likelihood
Discrete distribution, finite parameter space How biased an unfair coin is?unfair coin Call the probability of tossing a HEAD p. Determine p. Toss the coin 80 times Outcome is 49 HEADS and 31 TAILS, Suppose the coin was taken from a box containing three coins: one which gives HEADS with probability p = 1/3, one which gives HEADS with probability p = 1/2 and another which gives HEADS with probability p = 2/3. NO labels on these coins Using maximum likelihood estimation the coin that has the largest likelihood can be found, given the data that were observed. By using the probability mass function of the binomial distribution with sample size equal to 80, number successes equal to 49 but different values of p (the "probability of success"), the likelihood function (defined below) takes one of three values:probability mass functionbinomial distribution Maximum Likelihood: Example
The likelihood is maximized when p = 2/3, and so this the maximum likelihood estimate for p. Maximum Likelihood: Example Discrete distribution, finite parameter space
Discrete distribution, continuous parameter space Now suppose that there was only one coin but its p could have been any value 0 ≤ p ≤ 1. The likelihood function to be maximized is: and the maximization is over all possible values 0 ≤ p ≤ 1. differentiatingdifferentiating with respect to p (solutions p = 0, p = 1, and p = 49/80) The solution which maximizes the likelihood is clearly p = 49/80 Thus the maximum likelihood estimator for p is 49/80. Maximum Likelihood: Example
Continuous distribution, continuous parameter space Do it for Gaussian Distribution yourself! Two parameters, μ & σ Maximum Likelihood: Example Its expectation value is equal to the parameter μ of the given distribution,expectation value which means that the maximum-likelihood estimator μ is unbiased. This means that the estimator is biased. However, is consistent. In this case it could be solved individually for each parameter. In general, it may not be the case.
The method of maximum likelihood estimation chooses values for parameter estimates (regression coefficients) which make the observed data “maximally likely.” Standard errors are obtained as a by-product of the maximization process Maximum Likelihood
The Logistic Regression Model intercept is the log(odds) of the outcome. model coefficients
Maximum Likelihood We want to choose β ’s that maximizes the probability of observing the data we have: Assumption: independent y’s
Obvious possibility is to use traditional linear regression model But this has problems Distribution of dependent variable hardly normal Predicted probabilities cannot be less than 0, greater than 1 Linear probability model
Least-squares no longer best way of estimating parameters of logistic regression model Instead use maximum likelihood estimation Finds values of parameters that have greatest probability, given data Estimation of logistic regression model
Data on 24 space shuttle launches prior to Challenger Dependent variable, whether shuttle flight experienced thermal distress incident Independent variables Date – whether shuttle changes or age has effect Temperature – whether joint temperature on booster has effect Space shuttle data
Dependent variable Any thermal distress on launch Independent variable Date (days since 1/1/60) SPSS procedure Regression, Binary logistic First model—date as single independent variable
Predicted probability of thermal distress using date
Odds is the ratio of probability of success to probability of failure Like odds on horse races Even odds, odds = 1, implies probability equals 0.5 Odds = 2 means 2 to 1 in favor of success, implies probability of 0.667 Odds = 0.5 means 1 to 2 in favor (or 2 to 1 against) success, implies probability of 0.333 What does “odds” mean?
Logistic regression can be extended to use multiple independent variables exactly like linear regression Multiple logistic regression
Dependent variable Any thermal distress on launch Independent variables Date (days since 1/1/60) Joint temperature, degrees F Adding joint temperature to the logistic regression model
Probabilistic Discriminative Models Find posterior class probabilities directly Use functional form of generalized linear model and determine its parameters directly by using maximum likelihood principle Iterative Reweighted Least Squares (IRLS) Maximize a likelihood fn defined through the conditional distribution p(Ck|x), which represents a form of discriminative training Advantage of discriminative approach – fewer no. of adaptive parameters to be determined (linear in M) Improved predictive performance particularly when the class-conditional density assumptions give a poor approximation of the true distributions
Probabilistic Discriminative Models Classification methods work directly with the original input vector All such algorithms are still applicable if we first make a fixed NL transformation of the inputs using a vector of basis fns ϕ(x) Decision boundaries linear in the feature space ϕ In linear models of regression, one of the basis fn is typically set to a constant say, so that the corresponding parameter plays the role of bias. A fixed basis fn transformation ϕ(x) will be used in
Probabilistic Discriminative Models Original Input Space (x1,x2) Feature Space ( φ 1, φ 2) Although we use linear classification models Linear-separability in feature space does not imply linear-separability in input space