By Alex Smola, S.V.N. Vishwanathan

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1. One of the key ingredients was the ability to use information about word counts for different document classes to estimate the probability p(wj |y), where wj denoted the number of occurrences of word j in document x, given that it was labeled y. In the following we discuss an extremely simple and crude method for estimating probabilities. 22) m→∞ m −1 {xi = x} for all x ∈ X. 23) i=1 Let us discuss a concrete case. We assume that we have 12 documents and would like to estimate the probability of occurrence of the word ’dog’ from it.

Right: 7-nearest neighbour classifier. Note that the regression estimate is much more smooth. come extremely costly, in particular whenever the number of observations is large or whenever the observations xi live in a very high dimensional space. Random projections are a technique that can alleviate the high computational cost of Nearest Neighbor classifiers. A celebrated lemma by Johnson and Lindenstrauss [DG03] asserts that a set of m points in high dimensional Euclidean space can be projected into a O(log m/ 2 ) dimensional Euclidean space such that the distance between any two points changes only by a factor of (1 ± ).

8 (Sums of random variables and convolutions) Denote by X, Y ∈ R two independent random variables. Moreover, denote by Z := X + Y the sum of both random variables. Then the distribution over Z satisfies p(z) = p(x) ◦ p(y). Moreover, the characteristic function yields: φZ (ω) = φX (ω)φY (ω). 10) Proof Z is given by Z = X + Y . Hence, for a given Z = z we have the freedom to choose X = x freely provided that Y = z − x. In terms of distributions this means that the joint distribution p(z, x) is given by p(z, x) = p(Y = z − x)p(x) and hence p(z) = p(Y = z − x)dp(x) = [p(x) ◦ p(y)](z).

### Introduction to Machine Learning by Alex Smola, S.V.N. Vishwanathan

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