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For events B such that P(B) > 0, the conditional probability of A given B is P(A|B) = P(A ∩ B)/P(B). It can be shown that A and B are independent if and only if P(A|B) = P(A). From the facts that P(A ∩ B) = P(A|B)P(B) = P(B|A)P(A) and P(B) = P(B|A)P(A) + P(B|Ac )P(Ac ), where Ac indicates the complement of A within Ω , we obtain Bayes rule, P(A|B) = P(B|A)P(A) . 4) For three events A, B, and C, the events A and B are said to be conditionally independent given C if P(A ∩ B |C) = P(A |C)P(B |C) .

16) Eh (g(X)) = (1/n) ∑ni=1 w(xi ) using draws x1 , , . . , xn from h. 13), by concentrating our sampling in that part of the support of f where it is most needed. , Markov chain Monte Carlo (MCMC) methods. 9), for the case of discrete states) is in fact the distribution f of interest to us. MCMC methods can therefore be substantially more complicated than methods like those just described. However, designed appropriately, they allow us to generate samples from quite complex distributions and are used frequently in network modeling.

1 Background on Graphs 19 called a root, which is distinguished by being the only vertex from which there is a directed path to every other vertex in the graph. Such a graph is called a rooted tree. 2. A vertex preceding another vertex on a path from the root is called an ancestor, while a vertex following another vertex is called a descendant. Immediate ancestors are called parents, and immediate descendants, children. A vertex without any children is called a leaf. Given a rooted tree of this sort, it is not uncommon to represent it diagrammatically without any indication of its directedness, as this is to be understood from the definition of the root.

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A 2E4-time algorithm for MAX-CUT by Kulikov A. S.

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