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By Neil Goble

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Similarly, we have cos(A − B) = exp{i(A − B)} = exp(iA) exp(−iB) = (cos A + i sin A)(cos B − i sin B) = cos A cos B + sin A sin B. Finally sin(A − B) = exp{i(A − B)} = exp(iA) exp(−iB) = (cos A + i sin A)(cos B − i sin B) = sin A cos B − cos A sin B. 56 We can most conveniently cast distributions into standard exponential family form by taking the exponential of the logarithm of the distribution. 194) with h(µ) = 1 Γ(a + b) g(a, b) = Γ(a)Γ(b) u(µ) = η(a, b) = (83) ln µ ln(1 − µ) (84) a−1 . 146) we obtain Gam(λ|a, b) = (82) ba exp {(a − 1) ln λ − bλ} .

249), p(x|t) = 1 Nt N n=1 1 k(x, xn )δ(t, tn ). 4 Here Nt is the number of input vectors with label t (+1 or −1) and N = N+1 +N−1 . δ(t, tn ) equals 1 if t = tn and 0 otherwise. Zk is the normalisation constant for the kernel. The minimum misclassification-rate is achieved if, for each new input ˜ , we chose ˜t to maximise p(˜t|˜ vector, x x). With equal class priors, this is equivalent to maximizing p(˜ x|˜t) and thus 1 +1 iff 1 k(˜ x, xi ) k(˜ x, xj ) N+1 N−1 ˜t = i:ti =+1 j :tj =−1 −1 otherwise.

2 (131) Using (130), we can evaluate the integral in (129) to obtain exp {−E(w)} dw = exp {−E(t)} (2π)M/2 |Σ|1/2 . 83), that we only need to deal with the factor exp {−E(t)}. 12 (131) as follows 1 βtT t − mT Σ−1 m 2 1 βtT t − βtT ΦΣΣ−1 ΣΦT tβ = 2 1 T = t βI − βΦΣΦT β t 2 1 T = t βI − βΦ(A + βΦT Φ)−1 ΦT β t 2 1 T −1 −1 t β I + ΦA−1 ΦT t = 2 1 T −1 t C t. s. 85); the two preceding terms are given implicitly, as they form the normalization constant for the posterior Gaussian distribution p(t|X, α, β).