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Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 06); 2006 p 708 713 13 Buriol L, Frahling G, Leonardi S, Marchetti-Spaccamela A, Sohler C Counting triangles in data streams Proceedings of the 25th ACM Symposium on Principles of Database Systems (PODS 06); 2006 p 253 262 14 Chakrabarti A, Khot S, Sun X Near-optimal lower bounds on the multi-party communication complexity of set disjointness Proceedings of the IEEE Conference on Computational Complexity; 2003 p 107 117 15 Charikar M, Chaudhuri S, Motwani R, Narasayya V Towards estimation error guarantees for distinct values Proceedings of the 19th ACM Symposium on Principles of Database Systems (PODS 00); 2000 p 268 279 16 Charikar M, Chen K, Farach-Colton M Finding frequent items in data streams Proceedings of the 29th International Colloquium on Automata, Languages and Programming (ICALP 02); 2002 p 693 703 17 Charikar M, O Callaghan L, Panigrahy R.

It can be shown that the Kullback Leibler information divergence is nonnegative and that it is zero only if p1 p2 (cf the proof of Lemma A141 in Section A14 in the appendix) Therefore it is plausible that this measure can be used to assess the approximation of a given multidimensional distribution by the distribution that is represented by a given graph: the smaller the value of this measure, the better the approximation Note that this measure does not treat the two distributions p1 and p2 equally, because it uses the values of the distribution p1 as weights in the sum Therefore this measure is not symmetric in general, that is, in general it is IKLdiv (p1 , p2 ) = IKLdiv (p2 , p1 ) Consequently, one has to decide which of p1 and p2 should be the actual distribution and which the approximation.

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We may argue as follows: the quotient of the probabilities captures the difference of the distributions wrt single events E The influence of such a local difference depends on the probability of the occurrence of the corresponding event E Therefore it should be weighted with the (actual) probability of this event and hence p1 should be the actual distribution As an illustration of an approach based on decomposition tests, Figure 75 shows, for the simple three-dimensional probability distribution depicted in Figure 311 on page 75, all eight candidate graphs together with the Kullback Leibler information divergence of the original distribution and its approximation by each of the graphs (upper numbers) This figure is the probabilistic counterpart of Figure 71 Clearly, graph 5 represents an exact decomposition of the distribution, since the Kullback Leibler information divergence is zero for this graph.

Better streaming algorithms for clustering problems Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC 03); 2003 18 Chaudhuri S, Motwani R, Narasayya V Random sampling for histogram construction: How much is enough Proceedings of the ACM SIGMOD International Conference on Management of Data; 1998 p 436 447..

Note that, as in the relational case, the complete graph 8 always receives or shares the best assessment, since it consists of only one maximal clique and thus represents no real decomposition Up to now we have assumed that the multidimensional probability distribution, for which a decomposition is desired, is given However, in applications this is usually not the case Instead we are given a database of sample cases Clearly, the direct approach to handle this situation, namely to estimate the joint probability distribution from this database, so that we can proceed as indicated above, is, in general, infeasible The reason is that the available data are rarely sufficient to make the estimation of the joint probabilities reliable, simply because the number of dimensions is usually large and thus the number of probabilities that have to be estimated is enormous (cf.

For example, if one has a fear of dentists, a trip to the dentist s of ce will be imbued with chronic negative integral affect; however, if this particular trip happens to go very well, the episodic integral affect may end up being quite positive The affective dynamics of social behavior are very likely to involve both of these kinds of integral affect, as well as the incidental affective background of moods that are brought into an interaction from previous unrelated events Whereas a rich set of theory and data has emerged to study the incidental side of the picture, the role of integral affect in social cognition is only beginning to be explored (eg, Moreno & Bodenhausen, 2001; Perrott & Bodenhausen, 2002).

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