Bayesian Classification

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1 CS 650: Computer Vision Bryan S. Morse BYU Computer Science Statistical Basis Training: Class-Conditional Probabilities Suppose that we measure features for a large training set taken from class ω i. Each of these training patterns has a different value x for the features. This can be written as the class-conditional probability: p(x ω i ) In other words, How often do things in class ω i exhibit features x?

2 Statistical Basis Classification When we classify, we measure the feature vector x, then we ask this question: Given that this has features x, what is the probability that it belongs to class ω i?. Mathematically, this is written as P(ω i x) Statistical Basis Why We Care About Conditional Probabilities Training gives us But we want p(x ω i ) P(ω i x) These are not the same! How are they related?

3 Statistical Basis Bayes Theorem (Revisited) Generally: P(A B) = P(B A) P(A) P(B) For our purposes: P(ω i x) = p(x ω i) P(ω i ) p(x) Statistical Basis Definitions P(ω i x) = p(x ω i) P(ω i ) p(x) p(x ω i ) P(ω i ) p(x) P(ω i x) class conditioned probability or likelihood a priori or prior probability evidence (usually ignored) measurement-conditioned or posterior probability

4 The Bayesian Classifier Structure of a Bayesian Classifier Training: Measure p(x ω i ) for each class. Prior Knowledge: Measure or estimate P(ω i ) in the general population. (Can sometimes aggregate the training set if it is a reasonable sampling of the population.) Classification: 1. Measure feature (x) for new pattern. 2. Calculate posterior probabilities P(ω i x) for each class. 3. Choose the one with the larger posterior P(ω i x). The Bayesian Classifier Example Normally distributed class-conditional probabilities: p(x ω i ) = 1 2πσi e 1 2 (x µ i ) 2 /σ 2 i

5 The Bayesian Classifier From Probabilities to Discriminants: 1-D Case Want to maximize P(ω i x) = p(x ω i ) P(ω i ) p(x) same as maximizing p(x ω i ) P(ω i ) which for a normal distribution is 1 2πσi e 1 2 (x µ i )2 /σ 2 i P(ω i ) applying logarithm log 1 2π log σ i 1 2 (x µ i) 2 /σ 2 i + log P(ω i ) dropping constants log P(ω i ) log σ i 1 2 (x µ i) 2 /σ 2 i The Bayesian Classifier Extending to Multiple Features Note that the key term for a 1-D normal distribution is (x µ i ) 2 /σ 2 i the squared distance from the mean in standard deviations Can extend to multiple features by simply normalizing each feature s distance by the respective standard deviation, then just use minimum distance classification (remembering to use the priors as well)

6 The Bayesian Classifier Extending to Multiple Features Some call normalizing each feature by its variance naive Bayes So what s naive about it? It ignores relationships between features Multivariate Normal Distributions The Multivariate Normal Distribution In multiple dimensions, the normal distribution takes on the following form: p(x) = ( 1 2π ) d 1 C 1/2 e 1 2 (x m)t C 1 (x m) = (2π) d/2 C 1/2 e 1 2 (x m)t C 1 (x m) [See examples in Mathematica]

7 Multivariate Normal Distributions Multivariate Normal For multiple classes, each class ω i has its own mean vector m i covariance matrix C i The class-conditional probabilities are p(x ω i ) = (2π) d/2 C i 1/2 e 1 2 (x m i ) T C 1 i (x m i ) Multivariate Normal Distributions From Probabilities to Discriminants Want to maximize P(ω i x) = p(x ω i ) P(ω i ) p(x) so maximize p(x ω i ) P(ω i ) so maximize log p(x ω i ) + log P(ω i ) for normal distribution: d 2 log 2π 1 2 log C i 1 2 (x m i) T C 1 i (x m i ) + log P(ω i ) maximize log P(ω i ) 1 2 log C i 1 2 (x m i) T C 1 i (x m i )

8 Multivariate Normal Distributions Mahalonobis Distance The expression can be thought of as (x m i ) T C 1 (x m i ) x m i 2 C 1 This looks like squared distance, but the inverse covariance matrix C 1 acts like a metric (stretching factor) on the space. This is the Mahalonobis distance. Pattern recognition using multivariate normal distributions is simply a minimum (Mahalonobis) distance classifier. Case 1: Identity Matrix Case 1: Identity Matrix Suppose that the covariance matrix for all classes is the identity matrix I: C i = I or C i = σ 2 I Discriminant becomes g i (x) = 1 2 (x m i) T (x m i ) + log P(ω i ) Assuming all classes ω i are a priori equally likely, g i (x) = 1 2 (x m i) T (x m i ) Ignoring the constant 1 2, we can use g i (x) = (x m i ) T (x m i )

9 Case 1: Identity Matrix Example: Equal Priors Case 1: Identity Matrix Examples: Different Priors

10 Case 2: Same Covariance Matrix Case 2: Same Covariance Matrix If each class has the same covariance matrix, g i (x) = 1 2 (x m i) T C(x m i ) + log P(ω i ) Loci of constant probability are hyperellipes oriented with the eigenvectors of C: eigenvectors directions of ellipse axes eigenvalues variance (squared axis length) in axis directions The decision boundaries are still hyperplanes, though they may no longer be normal to the lines between the respective class means. Case 2: Same Covariance Matrix Examples

11 Case 3: Different Covariances for Each Class Case 3: Different Covariances for Each Class Suppose that each class has its own arbitrary covariance matrix (the most general case): C i C j Loci of constant probability for each class are hyperellipes oriented with the eigenvectors of C i for that class. Decision boundaries are quadratic, specifically, hyperellipses or hyperhyperboloids. [See examples in Mathematica] Case 3: Different Covariances for Each Class Examples: 2-D

12 Case 3: Different Covariances for Each Class Examples: 3-D Case 3: Different Covariances for Each Class Example: Multiple Classes

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