Consider the data set D= {(n,n)}=1, where the ordered pairs are formed by feature vectors 2,

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Consider the data set D= {(n,n)}=1, where the ordered pairs are formed by feature vectors 2, Rd and labels te {-1, +1}. By Bayes Theorem we

have that the posterior probability P(ta) for the class t is given by:

P(tx)=

P(xt)P(t) P(x-1)P(-1)+P(x+1)P(+1)

where the priors P(t) satisfy that P(-1) + P(+1) = 1. If we model the distri- bution of features inside class t by a Gaussian distribution with mean μ, and covariance matrix 2,demonstrate that:

1. The function σ Rd (-1,0,1} defined as:

(x)= sgn log P(x+1)-log P(x-1)+log;

1-P(+1),

P(+1))).

where sgn(x) = 1 if x > 0, -1 if x < 0, and 0 otherwise, is a discriminant

function.

 

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