Consider a two-class, one-dimensional problem where P(W)=P(w) and p(x|w)~ N(,02). Let = 0, 02=1, -, and...

 

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Consider a two-class, one-dimensional problem where P(W₁)=P(w₁) and p(x|w₁)~ N(µ₁,0₁2). Let µ₁= 0, 0₁²2=1, μ₂-μ, and 0²2-0². (a) Derive a general expression for the location of the Bayes optima Idecision boundary as a function of u and 0². (b) With µ= 1 and o²= 2, make two plots one for the class conditional pdfs p(x|w;) and one for the posterior probabilities p(wi|x) with the location of the optimal decision regions. Make surethe plots are correctly labeled (axis, titles, legend, etc) (c) Estimate the Bayes error rate pe (d) Comment on the case where µ= 0, and o2 is much greater than 1. Describe a practical example of a pattern classification problem where such a situation might arise. Consider a two-class, one-dimensional problem where P(W₁)=P(w₁) and p(x|w₁)~ N(µ₁,0₁2). Let µ₁= 0, 0₁²2=1, μ₂-μ, and 0²2-0². (a) Derive a general expression for the location of the Bayes optima Idecision boundary as a function of u and 0². (b) With µ= 1 and o²= 2, make two plots one for the class conditional pdfs p(x|w;) and one for the posterior probabilities p(wi|x) with the location of the optimal decision regions. Make surethe plots are correctly labeled (axis, titles, legend, etc) (c) Estimate the Bayes error rate pe (d) Comment on the case where µ= 0, and o2 is much greater than 1. Describe a practical example of a pattern classification problem where such a situation might arise.

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a The Bayes decision boundary is given by equating the posterior probabilities ... View the full answer