Problem_Set_2_Solutions.pdf - ECEN 649 Pattern Recognition...

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ECEN 649 Pattern Recognition – Spring 2020 Problem Set 2 Due on: Feb 26 1. Suppose that X is a discrete feature vector over a feature space D . Define the discrete versions of the weighted class-conditional densities and posterior probabilities, and derive the discrete versions of equations (2.3), (2.4), (2.8), (2.9), (2.11), (2.30), (2.34), and (2.36) in Chapter 2 of the book. Hint: Note that all events { X = x } and { Y = y } have positive probability (no conditioning on events of probability zero occcurs) and integration in the discrete case reduces to summation.
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[ ψ ] = E [ [ ψ | X = x k ]] = X x k D [ ψ | X = x k ] P ( X = x k ) . (6) * = X x k D ( I η ( x k ) 1 - η ( x k ) η ( x k ) + I η ( x k ) > 1 - η ( x k ) (1 - η ( x k ))) P ( X = x k ) = E [min { η ( X ) , 1 - η ( X ) } ] . (7) * = P ( Y = 0) 0 [ ψ * ] + P ( Y = 1) 1 [ ψ * ] = X x k ; P ( Y =1) P ( X = x k | Y =1) >P ( Y =0) P ( X = x k | Y =0) P ( Y = 0) P ( X = x k | Y = 0) + X x k ; P ( Y =1) P ( X = x k | Y =1) P ( Y =0) P ( X = x k | Y =0) } P ( Y = 1) P ( X = x k | Y = 1) . (8) E [ η ( X )] = X x k D P ( Y = 1 | X = x k ) P ( X = x k ) = P ( Y = 1) . (9) 2. Concerning Examples 2.1 and 2.2. (a) Repeat Example 2.1 if (i) only H is observable and (ii) if no observations are available. Solution:
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