Minimum Error-Rate Discriminant

Transcription

1 Discriminants Minimum Error-Rate Discriminant In the case of zero-one loss function, the Bayes Discriminant can be further simplified: g i (x) =P (ω i x). (29) J. Corso (SUNY at Buffalo) Bayesian Decision Theory 22 / 59

2 Discriminants Uniqueness Of Discriminants Is the choice of discriminant functions unique? J. Corso (SUNY at Buffalo) Bayesian Decision Theory 23 / 59

3 Discriminants Uniqueness Of Discriminants Is the choice of discriminant functions unique? No! Multiply by some positive constant. Shift them by some additive constant. J. Corso (SUNY at Buffalo) Bayesian Decision Theory 23 / 59

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5 Discriminants Visualizing Discriminants Decision Regions The effect of any decision rule is to divide the feature space into decision regions. Denote a decision region R i for ω i. One not necessarily connected region is created for each category and assignments is according to: If g i (x) >g j (x) j i, then x is in R i. (33) Decision boundaries separate the regions; they are ties among the discriminant functions. J. Corso (SUNY at Buffalo) Bayesian Decision Theory 24 / 59

6 Discriminants Visualizing Discriminants Decision Regions 0.3 p(x ω 1 )P(ω 1 ) p(x ω 2 )P(ω 2 ) R 2 5 R 1 decision boundary 0 R J. Corso (SUNY at Buffalo) Bayesian Decision Theory 25 / 59

7 Discriminants Two-Category Discriminants Dichotomizers In the two-category case, one considers single discriminant What is a suitable decision rule? g(x) =g 1 (x) g 2 (x). (34) J. Corso (SUNY at Buffalo) Bayesian Decision Theory 26 / 59

8 Discriminants Two-Category Discriminants Dichotomizers In the two-category case, one considers single discriminant g(x) =g 1 (x) g 2 (x). (34) The following simple rule is then used: Decide ω 1 if g(x) > 0; otherwise decide ω 2. (35) J. Corso (SUNY at Buffalo) Bayesian Decision Theory 26 / 59

9 Discriminants Two-Category Discriminants Dichotomizers In the two-category case, one considers single discriminant g(x) =g 1 (x) g 2 (x). (34) The following simple rule is then used: Decide ω 1 if g(x) > 0; otherwise decide ω 2. (35) Various manipulations of the discriminant: g(x) =P (ω 1 x) P (ω 2 x) (36) g(x) =ln p(x ω 1) p(x ω 2 ) +lnp (ω 1) P (ω 2 ) (37) J. Corso (SUNY at Buffalo) Bayesian Decision Theory 26 / 59

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15 The Normal Density Entropy Entropy is the uncertainty in the random samples from a distribution. H(p(x)) = p(x)lnp(x)dx (44) The normal density has the maximum entropy for all distributions have a given mean and variance. What is the entropy of the uniform distribution? J. Corso (SUNY at Buffalo) Bayesian Decision Theory 31 / 59

16 The Normal Density Entropy Entropy is the uncertainty in the random samples from a distribution. H(p(x)) = p(x)lnp(x)dx (44) The normal density has the maximum entropy for all distributions have a given mean and variance. What is the entropy of the uniform distribution? The uniform distribution has maximum entropy (on a given interval). J. Corso (SUNY at Buffalo) Bayesian Decision Theory 31 / 59

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19 The Normal Density The Covariance Matrix Σ E[(x µ)(x µ) T ]= (x µ)(x µ) T p(x)dx Symmetric. Positive semi-definite (but DHS only considers positive definite so that the determinant is strictly positive). The diagonal elements σ ii are the variances of the respective coordinate x i. The off-diagonal elements σ ij are the covariances of x i and x j. What does a σ ij =0imply? That coordinates x i and x j are statistically independent. J. Corso (SUNY at Buffalo) Bayesian Decision Theory 33 / 59

20 The Normal Density The Covariance Matrix Σ E[(x µ)(x µ) T ]= (x µ)(x µ) T p(x)dx Symmetric. Positive semi-definite (but DHS only considers positive definite so that the determinant is strictly positive). The diagonal elements σ ii are the variances of the respective coordinate x i. The off-diagonal elements σ ij are the covariances of x i and x j. What does a σ ij =0imply? That coordinates x i and x j are statistically independent. What does Σ reduce to if all off-diagonals are 0? J. Corso (SUNY at Buffalo) Bayesian Decision Theory 33 / 59

21 The Normal Density The Covariance Matrix Σ E[(x µ)(x µ) T ]= (x µ)(x µ) T p(x)dx Symmetric. Positive semi-definite (but DHS only considers positive definite so that the determinant is strictly positive). The diagonal elements σ ii are the variances of the respective coordinate x i. The off-diagonal elements σ ij are the covariances of x i and x j. What does a σ ij =0imply? That coordinates x i and x j are statistically independent. What does Σ reduce to if all off-diagonals are 0? The product of the d univariate densities. J. Corso (SUNY at Buffalo) Bayesian Decision Theory 33 / 59

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26 The Normal Density Simple Case: Statistically Independent Features with Same Variance What do the decision boundaries look like if we assume Σ i = σ 2 I? They are hyperplanes ω 1 ω p(x ω i ) ω 1 ω ω 1 P(ω 2 )=.5 ω R R 1 R 2 P(ω 1 )=.5 P(ω 2 )=.5 x -2 P(ω 1 )= R 1 R 2 4 P(ω 2 )= P(ω 1 )= R 1 2 Let s see why... J. Corso (SUNY at Buffalo) Bayesian Decision Theory 37 / 59

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33 The Normal Density General Case: Arbitrary Σ i J. Corso (SUNY at Buffalo) Bayesian Decision Theory 44 / 59

34 The Normal Density General Case for Multiple Categories R 4 R 3 R 2 R 4 R 1 nquite regions A Complicated for four normal Decisiondistribut Surface! J. Corso (SUNY at Buffalo) Bayesian Decision Theory 45 / 59

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36 Receiver Operating Characteristics Signal Detection Theory The detector uses x to decide if the external signal is present. Discriminability characterizes how difficult it will be to decide if the external signal is present without knowing x. d = µ 2 µ 1 σ (63) Even if we do not know µ 1, µ 2, σ, orx,wecanfindd by using a receiver operating characteristic or ROC curve, as long as we know the state of nature for some experiments J. Corso (SUNY at Buffalo) Bayesian Decision Theory 47 / 59

37 Receiver Operating Characteristics Receiver Operating Characteristics Definitions A Hit is the probability that the internal signal is above x given that the external signal is present P (x >x x ω 2 ) (64) J. Corso (SUNY at Buffalo) Bayesian Decision Theory 48 / 59

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39 Receiver Operating Characteristics Receiver Operating Characteristics Definitions A Hit is the probability that the internal signal is above x given that the external signal is present P (x >x x ω 2 ) (64) A Correct Rejection is the probability that the internal signal is below x given that the external signal is not present. P (x <x x ω 1 ) (65) A False Alarm is the probability that the internal signal is above x despite there being no external signal present. P (x >x x ω 1 ) (66) J. Corso (SUNY at Buffalo) Bayesian Decision Theory 48 / 59

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42 Missing or Bad Features Missing Features Suppose we have built a classifier on multiple features, for example the lightness and width. What do we do if one of the features is not measurable for a particular case? For example the lightness can be measured but the width cannot because of occlusion. J. Corso (SUNY at Buffalo) Bayesian Decision Theory 50 / 59

43 Missing or Bad Features Missing Features Suppose we have built a classifier on multiple features, for example the lightness and width. What do we do if one of the features is not measurable for a particular case? For example the lightness can be measured but the width cannot because of occlusion. Marginalize! Let x be our full feature feature and x g be the subset that are measurable (or good) and let x b be the subset that are missing (or bad/noisy). We seek an estimate of the posterior given just the good features x g. J. Corso (SUNY at Buffalo) Bayesian Decision Theory 50 / 59

44 Missing or Bad Features Missing Features P (ω i x g )= p(ω i, x g ) p(x g ) (68) = = p(ωi, x g, x b )dx b p(x g ) p(ωi x)p(x)dx b p(x g ) (69) (70) = gi (x)p(x)dx b p(x)dxb (71) We will cover the Expectation-Maximization algorithm later. This is normally quite expensive to evaluate unless the densities are special (like Gaussians). J. Corso (SUNY at Buffalo) Bayesian Decision Theory 51 / 59

45 Bayesian Belief Networks Statistical Independence Two variables x i and x j are independent if p(x i,x j )=p(x i )p(x j ) (72) x 3 x 1 x 2 FIGURE Athree-dimensionaldistributionwhichobeysp(x 1, x 3 ) = p(x 1 )p(x 3 ); thus here x 1 and x 3 are statistically independent but the other feature pairs are not. From: Richard O. Duda, Peter E. Hart, and David G. Stork, Pattern Classification. Copyright c 2001 by John Wiley & Sons, Inc. J. Corso (SUNY at Buffalo) Bayesian Decision Theory 52 / 59

46 Bayesian Belief Networks Simple Example of Conditional Independence From Russell and Norvig Consider a simple example consisting of four variables: the weather, the presence of a cavity, the presence of a toothache, and the presence of other mouth-related variables such as dry mouth. The weather is clearly independent of the other three variables. And the toothache and catch are conditionally independent given the cavity (one as no effect on the other given the information about the cavity). Weather Cavity Toothache Catch J. Corso (SUNY at Buffalo) Bayesian Decision Theory 53 / 59

47 Bayesian Belief Networks Naïve Bayes Rule If we assume that all of our individual features x i,i=1,...,d are conditionally independent given the class, then we have p(ω k x) d i=1 p(x i ω k ) (73) Circumvents issues of dimensionality. Performs with surprising accuracy even in cases violating the underlying independence assumption. J. Corso (SUNY at Buffalo) Bayesian Decision Theory 54 / 59

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