By Nilsson R.

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**Example text**

1/ n), that is, the hyperplane normal is the same as the +1 class mean. 11). Hence, more features improves the optimal prediction performance, as every new feature contributes some extra information about the target variable, even 44 Statistical Data Models though this added information becomes smaller for large n. As n tends to infinity, R(g ∗ ) → 0. Now, consider that w is unknown, so that we have to estimate this parameter from a data set z (1:l) (we assume that Σ = I is known, though). 19) i=1 Note that the ML estimate is simply the mean of the data points ”weighted” by their label, which seems reasonable considering the symmetry of the distribution.

I=1 For a given sample x(1:l) , these θk have the straightforward ML estimates 1 θˆk = l n (j)i =k x(j) : 2x . i=1 Clearly, the number of samples l required for an accurate estimate of θ is on the order of 2k . However, if p(x) can be represented by a Bayesian network such that each node i has at most K < n parents, then each local distributions p(xi | xΠi ) involve no more than 2K parameters. Thus, for such a Bayesian network, no more than n2K 2n are non-zero, simplifying the estimation problem considerably.

Rn , targets Y ∈ {−1, +1}, with each feature distributed as a Gaussian √ f (xi | y) = N (xi | y/ i, 1). All features Xi are independent (identity covariance matrices for both classes), and we set the class probabilities p(y) = 1/2, so that √ f (x, y) = p(y) N (xi | y/ i, 1). i The Bayes classifier g ∗ for this problem can then be expressed as g ∗ (x) = sign(wT x) √ √ with w = (1, 1/ 2, . . , 1/ n), that is, the hyperplane normal is the same as the +1 class mean. 11). Hence, more features improves the optimal prediction performance, as every new feature contributes some extra information about the target variable, even 44 Statistical Data Models though this added information becomes smaller for large n.