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109Appendix BAppendix to Section 4B.1 ProofsProof of Lemma 6: We first note thatP(α 0 ) . = e −nD(α 0/ρ||p) = e −n ρ D(α 0||ρp)(B.1)where as previously stated D(α 0 ||p) = ∑ f α 0,f log α 0,fp fis the information divergence (alsocalled the Kullback-Leibler distance) between α 0 and p. We use = . to denote exponential equality,i.e., we write f(n) = . e −nd logf(n)to denote lim = d and . .≤, ≥ are similarly defined. Inn→∞ nestablishing P(α 1 |α 0 ), we focus on a specific category f, and look to calculate(P |S ∩C f | = n ρ α 1,f | |C f | = n ρ α 0,f), (B.2)i.e., to calculate the probability that pruning introduces n ρ α 1,f elements, from C f to S, given thatthere are n ρ α 0,f elements of C f . Towards this we note that there is a total of|C f | = n ρ α 0,f (B.3)possible elements in C f which may be categorized, each with probability ɛ f , to belong to C 1 bythe categorization algorithm. The fraction of such elements that are asked to be categorized tobelong to C 1 , is defined by α to bex f := |S ∩C f||C f |=nρ α 1,f|C f |an event which happens with probability(P(x f ) = P |S ∩C f | = n ρ α 1,f | |C f | = n )ρ α 0,f= α 1,fα 0,f,(B.4).= e −|C f|I f (x f ) , (B.5)where in the above,I f (x f ) = x f log( x fɛ f)+(1−x f )log( 1−x f1−ɛ f) is the rate function of the binomialdistribution with parameter ɛ f (cf. [CT06]). Now given thatρ∏(P(α 1 |α 0 ) = P |S ∩C f | = n ρ α 1,f | |C f | = n )ρ α 0,f(B.6)f=1