Approximations of Bayes classifiers for statistical learning by Magnus Ekdahl.

By Magnus Ekdahl.

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2 [5] An Occam-algorithm with constant parameters c and 0 α < 1 is an algorithm that given 1 1. a sample (xl , cB (xl ))nl=1 . 2. s. 3. cB (ξ) = ς. produces 1. a cˆ ξ|x(n) that needs at most nc2 nα bits to be represented and 2. cˆ ξ|x(n) that is such that for all xl ∈ x(n) we have cˆ xl |x(n) = cl 3. Runs in time polynomial in n. 1 [5] Given independent observations of (ξ, cB (ξ)), where cB (ξ) needs n2 bits to be represented, an Occam-algorithm with parameters c 1 and 0 α < 1 produces a cˆ ξ|x(n) such that P P cˆ ξ|x(n) = cB (ξ) 1−δ ε (25) using sample size O 1 δ ln ε + nc2 ε 1 1−α February 13, 2006 (13:19) .

1, equation (53)) Pξi (xi |Πi ) arg max PV (v|G) = arg max vi ∈Vi v∈V Pξi (xi |Πi ). = arg max PR (r) vi ∈Vi i∈V i∈V \{u} Then using the distributive law (equation (52)) ⎡ ⎡ = arg max ⎣PR (r) arg max ⎣ vu ∈Vu v\{vu } ⎤⎤ Pξi (xi |Πi )⎦⎦ . 2 (equation (53) and (54)) can be used to construct a recursive algorithm for finding the x with maximal probability recursively. e. v’s ξj such that ξi ∈ Πj . 1 (i, l) Algorithms for SC Optimizing the unsupervised classification SC In section 3 we established how to calculate SC for a specific classification c(n) .

1 and the induction assumption 1 2 log 2 x (1 − Pξj+1 |ξ(j) (y|x(j) ))Pξ(j) (x1 , . . ,xj = j(1 − Pξ (y)) 2 log 2 (j + 1)(1 − Pξ (y)) . 2 log 2 3. By 1, 2 and the induction axiom equation (37) holds. 3 1 − Pξ (y) Eξ(n) − log Pξ(n) (ξ (n) ) Eξ(n) − log Pˆξ(n) (ξ (n) ) n · 2 log 2 n · 2 log 2 . 10 and Eξ(n) − log Pˆξ(n) (ξ (n) ) . 5) for a Bayesian Network. As seen in the end of section 2 the SC will be minimized with Jeffreys’ prior. Jeffreys’ prior for a Bayesian Network is calculated in [50]. To present the result some notation (mostly from [41]) are introduced.

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