Biometrics: A Pattern Recognition System Yes/No Pattern classification Biometrics CSE 190 Lecture 3 Authentication False accept rate (FAR): Proportion of imposters accepted False reject rate (FRR): Proportion of genuine users rejected Failure to enroll rate (FTE): portion of population that cannot be enrolled Failure to acquire rate (FTA): portion of population that cannot be verified Enrollment Error Rates Error vs Threshold False Match (False Accept): Mistaking biometric measurements from two different persons to be from the same person; False Non-match (False reject): Mistaking two biometric measurements from the same person to be from two different persons FAR: False accept rate FRR: False reject rate (c) Jain 2004 (c) Jain 2004 ROC Curve Announcements Readings Project assignment on web page Accuracy requirements of a biometric system are application dependent (c) Jain 2004 1
An Example 8 Pattern Classification Sorting incoming Duck on a conveyor according to vs. diseased Duck 9 10 11 Bayesian Decision Theory Continuous Features (Sections 2.1-2.2) 2
Introduction 13 14 The sea bass/salmon example State of nature, prior State of nature is a random variable The catch of salmon and sea bass is equiprobable (The healthy of ducks probably isn t equiprobable). P(ω 1 ), P(ω 2 ) Prior probabilities P(ω 1 ) = P(ω 2 ) (uniform priors) Decision rule with only the prior information Decide ω 1 if P(ω 1 ) > P(ω 2 ) otherwise decide ω 2 Use of the class conditional information P(x ω 1 ) and P(x ω 2 ) describe the difference in lightness between populations of healthy and diseased ducks P(ω 1 ) + P( ω 2 ) = 1 (exclusivity and exhaustivity) 15 16 Posterior, likelihood, evidence P(ω j x) = (P(x ω j ) * P (ω j )) / P(x) (BAYES RULE) In words, this can be said as: Posterior = (Likelihood * Prior) / Evidence Where in case of two categories 17 18 Intuitive decision rule given the posterior probabilities: Given x: if P(ω 1 x) > P(ω 2 x) True state of nature = ω 1 if P(ω 1 x) < P(ω 2 x) True state of nature = ω 2 Why do this?: Whenever we observe a particular x, the probability of error is : P(error x) = P(ω 1 x) if we decide ω 2 P(error x) = P(ω 2 x) if we decide ω 1 3
19 Bayesian Decision Theory Continuous Features 20 Since decision rule is optimal for each feature value X, there is not better rule for all x. Generalization of the preceding ideas Use of more than one feature Use more than two states of nature Allowing actions and not only decide on the state of nature Introduce a loss of function (more general than the probability of error) Allowing actions other than classification primarily allows the possibility of rejection Refusing to make a decision in close or bad cases! Letting loss function state how costly each action taken is Bayesian Decision Theory Continuous Features 21 What is the Expected Loss for action α i? 22 Let X be a vector of features. For any given x the expected loss is Let {ω 1, ω 2,, ω c } be the set of c states of nature (or classes ) Let {α 1, α 2,, α a } be the set of possible actions Let λ(α i ω j ) be the loss for action α i when the state of nature is ω j R(α i x) is called the Conditional Risk (or Expected Loss) Overall risk R = Sum of all R(α i x) for i = 1,,a Conditional risk Minimizing R Minimizing R(α i x) for i = 1,, a 23 Given a measured feature vector x, which action should we take? Select the action α i for which R(α i x) is minimum R is minimum and R in this case is called the Bayes risk = best performance that can be achieved! 24 for i = 1,,a 4
Two-Category Classification α 1 : deciding ω 1 α 2 : deciding ω 2 λ ij = λ(α i ω j ) loss incurred for deciding ω i when the true state of nature is ω j 25 Our rule is the following: if R(α 1 x) < R(α 2 x) λ 11 P(ω 1 x) + λ 12 P(ω 2 x) < λ 21 P(ω 1 x) + λ 22 P(ω 2 x) action α 1 : decide ω 1 is taken This results in the equivalent rule : decide ω 1 if: 26 Conditional risk: R(α 1 x) = λ 11 P(ω 1 x) + λ 12 P(ω 2 x) R(α 2 x) = λ 21 P(ω 1 x) + λ 22 P(ω 2 x) (λ 21 - λ 11 ) P(x ω 1 ) P(ω 1 ) > (λ 12 - λ 22 ) P(x ω 2 ) P(ω 2 ) and decide ω 2 otherwise 27 x (λ 21 - λ 11 ) x (λ 12 - λ 22 ) 5