By Alex Smola, S.V.N. Vishwanathan

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For instance, we could use string edit distances to compare two documents or information theory based measures. However, the problem with nearest neighbor classification is that the estimates can be very noisy whenever the data itself is very noisy. For instance, if a spam email is erroneously labeled as nonspam then all emails which are similar to this email will share the same fate. 18 for an example. In this case it is beneficial to pool together a number of neighbors, say the k-nearest neighbors of x and use a majority vote to decide the class membership of x.

E. φZm to that of a normally distributed random variable W with zero mean and unit variance. Expanding the exponential to second order yields: 1 exp(iwx) = 1 + iwx − w2 x2 + o(|w|2 ) 2 1 and hence φX (ω) = 1 + iwEX [x] − w2 VarX [x] + o(|w|2 ) 2 Since the mean of Zm vanishes by centering (Xi − µ) and the variance per variable is m−1 we may write the characteristic function of Zm via φZm (ω) = 1 2 1− w + o(m−1 |w|2 ) 2m m As before, taking limits m → ∞ yields the exponential function. We have that limm→∞ φZm (ω) = exp(− 12 ω 2 ) which is the characteristic function of the normal distribution with zero mean and variance 1.

Denote by mham and mspam the number of ham and spam e-mails in X. In this case we can estimate mspam mham p(ham) ≈ and p(spam) ≈ . m m The key problem, however, is that we do not know p(x|y) or p(x). We may dispose of the requirement of knowing p(x) by settling for a likelihood ratio L(x) := p(x|spam)p(spam) p(spam|x) = . 17) Whenever L(x) exceeds a given threshold c we decide that x is spam and consequently reject the e-mail. If c is large then our algorithm is conservative and classifies an email as spam only if p(spam|x) p(ham|x).