The Inferential Complexity of Bayesian and Credal Networks

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1 The Inferential Coplexity of Bayesian and Credal Networks Cassio Polpo de Capos,2 Fabio Gagliardi Cozan Universidade de São Paulo - Escola Politécnica 2 Pontifícia Universidade Católica de São Paulo cassio@pucsp.br, fgcozan@usp.br Abstract This paper presents new results on the coplexity of graph-theoretical odels that represent probabilities (Bayesian networks) and that represent interval and set valued probabilities (credal networks). We define a new class of networks with bounded width, and introduce a new decision proble for Bayesian networks, the axiin a posteriori. We present new links between the Bayesian and credal networks, and present new results both for Bayesian networks (ost probable explanation with observations, axiin a posteriori) and for credal networks (bounds on probabilities a posteriori, ost probable explanation with and without observations, axiu a posteriori). Introduction This paper builds a picture of inferential coplexity in graphtheoretical odels of uncertainty that goes significantly beyond existing results. We focus on Bayesian and credal networks the forer is a purely probabilistic odel, while the latter adits interval and set valued probabilities and generalizes several theories of uncertainty. There already is quite a solid understanding about the inferential coplexity of Bayesian networks; we add to this picture a new class of networks that strengthens existing results, and a new type of proble, axiin a posteriori (MAP), that can be of interest in gae-theoretic settings. Our ain contributions are related to credal networks, as little is known at this point concerning their inferential coplexity: we present new results concerning coputation of probability bounds, ost probable explanations (MPE) and axiu a posteriori (MAP). We show that, rather surprisingly, the MPE proble without observations in credal networks of bounded width is polynoial, while the MPE proble with observations and the MAP proble are Σ p 2-coplete for credal networks. We also strengthen the connection between Bayesian and credal networks by showing links between the coputation of probability bounds, the MAP and the MAP probles. Our results suggest that oving fro point to interval/set valued probabilities takes us one step up in ters of coplexity classes. Section 2 presents a few definitions and, ost iportantly, tries to justify our interest in credal networks. Section 3 describes the probles we are interested in. Section 4 contains our new results; in particular, Table offers a suary of our contributions. Section 5 concludes the paper and suggests future research. 2 Bayesian and Credal Networks A Bayesian network (or BN) represents a single joint probability density over a collection of rando variables. We assue throughout that variables are categorical; variables are uppercase and their assignents are lowercase. Definition A Bayesian network is a pair (G, P), where: G = (V G, E G ) is a directed acyclic graph, with V G a collection of vertices associated to rando variables X (a node per variable), and E G a collection of arcs; P is a collection of conditional probability densities p(x i pa(x i )) where pa(x i ) denotes the parents of X i in the graph (pa(x i ) ay be epty), respecting the relations of E G. In a BN every variable is independent of its nondescendants nonparents given its parents (Markov condition). This structure induces a joint probability density by the expression p(x,..., X n ) = i p(x i pa(x i )). Given a BN, E denotes the set of observed variables in the network (the evidence); e denotes the observed value of E. We consider the class of networks with bounded induced-width (or BIW); that is, networks where the subjacent graph (obtained reoving arc s directions) of G has axiu degree and induced-width bounded by log(f(s)). We define f(s) as a polynoial function in the size s of input (this size is evaluated over all inforation needed to specify the proble). We also consider the class of polytrees (or PT); that is, a BIW network where the subjacent graph of G has no cycles. Otherwise, a network is said ultiply-connected. Note that our definition of BIW networks is not, as usually done, based on fixed inducedwidth; rather, we allow the induced-width to vary with the network size. Credal networks generalize Bayesian networks by allowing each variable to be associated with sets of joint probability easures rather than single probability easures [Cano et al., 993; Cozan, 2000a]. Such graph-theoretical odels can be

2 viewed as Bayesian networks with relaxed nuerical paraeters; they can be used to study robustness of probabilistic odels, to investigate the behavior of groups of experts, or to represent incoplete or vague knowledge about probabilities. Sets of probability easures are sufficiently powerful to represent belief functions, possibility easures, qualitative probabilities, and probabilistic logic stateents, thus offering a general language that can convey any odels of interest in artificial intelligence [Walley, 996]. A set of probability distributions is called a credal set [Levi, 980]. A credal set for X is denoted by K(X). A conditional credal set is a set of conditional distributions, obtained applying Bayes rule to each distribution in a credal set of joint distributions [Walley, 99]. Given a credal set K(X), the lower probability and the upper probability of event A are defined respectively as P (A) = in p(x) K(X) P (A) and P (A) = ax p(x) K(X) P (A). We assue that, given a credal set, finding a lower/upper probability is a polynoial operation. Definition A credal network is a pair (G, K), where: G = (V G, E G ) is a directed acyclic graph, with V G a collection of vertices associated to rando variables X (a node per variable), and E G a collection of arcs; K is a collection of conditional credal sets K(X i pa(x i )), respecting the relations iplied by E G. We assue a Markov condition in credal networks: every variable is independent of its nondescendants nonparents given its parents. In this paper we adopt the concept of strong independence; that is, the joint credal set represented by a credal network is a set where each vertex factorizes as a Bayesian network [Cozan, 2000b]. Thus we can really view a credal network as a set of Bayesian networks, all with identical graphs. Credal networks can also be classified as polytree, bounded induced-width or ultiply-connected. 3 Reasoning In this section we present decision versions of the inferences we focus in this paper. Definition BN-Pr: given a Bayesian network (G, P), evidence E = e with E X, a query variable Q X \ E and its category q, and a rational nuber r, is P (q e) > r? Note the following distinction between MPE probles with or without evidence; the reason for this will be indicated later: Definition BN-MPE: given a Bayesian network (G, P), evidence E = e with E X and a rational nuber r, is there an instantiation x for X \ E such that P (x, e) > r? Definition BN-MPEe: given a Bayesian network (G, P), evidence E = e with E X and a rational nuber r, is there an instantiation x for X \ E such that P (x e) > r? Definition BN-MAP: given a Bayesian network (G, P), evidence E = e with E X, a set Q X \ E and a rational nuber r, is there an instantiation q for Q such that P (q e) > r? We introduce the axiin a posteriori proble, which ay be of interest in applications involving gae-theoretic behavior with axiizers and iniizers [Kakade et al., 200]: Definition BN-MAP: given a Bayesian network (G, P), soe evidence E = e with E X, the sets A X \ E and B X \ E, with A B = and a rational nuber r, is there an instantiation a for the A variables such that in b P (a, b e) > r? The corresponding probles in CN are now defined. Definition CN-Pr: given a credal network (G, K), evidence E = e with E X, a query variable Q X \ E and its category q and a rational nuber r, is P (q e) > r? In this definition we use upper queries, but lower queries ay be of interest too. We can show that both queries lead to identical coplexity results: Lea Evaluating arginal lower probabilities in CN is as hard as evaluating arginal upper probabilities. Proof Suppose we have a CN-Pr with arginal query Q = q. The calculation of P (q e) can be done by inserting a binary child Q to Q, where P (q Q) = if Q q and 0 otherwise. Now P (q e) = ax Q q P (Q e) = P (q e). We now define axiin versions of MPE and MAP in credal networks (we could alternatively define axiax versions for these probles, with possibly different coplexities). Definition CN-MAP: given a credal network (G, K), soe evidence E = e with E X, a set Q X \ E and a rational nuber r, is there an instantiation q for the Q variables such that P (q e) > r? CN-MPEe is obtained when CN-MAP has Q = X \ E. CN-MPE is CN-MPEe without evidence, that is, Q = X. We use abbreviations to refer to these probles (e.g. PT-CN-Pr is the belief updating proble in a polytree credal network). 4 Coplexity results Table suarizes relevant coplexity results. We start by explaining the origin of the results in this table (nubering atches the nubers in the table):. [Dechter, 996] describes algoriths with exponential tie coplexity on the induced-width of an eliination order; [Eyal, 200] shows how to obtain a constantfactor approxiation to optial order in polynoial tie for networks with bounded degree and bounded induced-width by log(f(s)). Note that the result here is sightly different fro the one in [Dechter, 996]; we refer to the actual induced width of the graph, not the induced width of an ordering. 2. [Roth, 996] shows coplexity of functional version, [Litan et al., 200] takes the decision version. 3. [Shiony, 994] shows (by reduction fro the vertex cover proble) that BN-MPE is NP-Coplete, while Theore 2 shows that BN-MPEe is PP-Coplete. We assue that the reader is failiar with notions of coplexity theory; for an introduction see [Papadiitriou, 994]. Polynoial tie eans polynoial tie in the size of input. A reduction eans a polynoial tie reduction; when a proble is solvable by another, there is a reduction fro the forer to the latter.

3 Proble Polytree Bounded induced-width Multiply-connected BN-Pr Polynoial () Polynoial () PP-Coplete (2) BN-MPE Polynoial () Polynoial () NP-Coplete (3) BN-MPEe Polynoial () Polynoial () PP-Coplete (3) BN-MAP NP-Coplete (4) NP-Coplete (4) NP PP -Coplete (5) BN-MAP Σ P 2-Coplete (6) Σ P 2-Coplete (6) NP PP -Hard (7) CN-Pr NP-Coplete (8) NP-Coplete (8) NP PP -Coplete (9) CN-MPE Polynoial (0) Polynoial (0) NP-Coplete () CN-MPEe CN-MAP Σ P 2-Coplete (2) Σ P 2-Coplete (2) Σ P 2-Coplete (2) Σ P 2-Coplete (2) Σ P 2-Hard and PP-Hard (3) NP PP -Hard (7) Table : Coplexity results; nubers in parenthesis indicate the ite that discusses the result. 4. [Park, 2002] reduces MAXSAT proble to BN-MAP in polytrees, a result that can be extended to BIW networks as both probles belong to NP (given the polynoial nature of BN-Pr in BIW networks). 5. [Park and Darwiche, 2004] by reduction fro E-MAJSAT. 6. Theore The coplexity of BN-MAP iplies it. 8. Theore [Cozan et al., 2004] by a reduction fro E-MAJSAT. 0. Theore 5.. Theore Theore BN-MPE and PT-CN-MPEe ensure it. In the context of Bayesian networks, the difference between MPE and MPEe ay see acadeic, because any ost probable explanation can be found with BN-MPE. However the sae is not true for credal networks, where one cannot find a ost probable explanation with evidence by siply running a version of MPE and note that CN-MPE and CN-MPEe do display non-trivial differences. In fact, these differences were our otivation for differentiating MPE fro MPEe. The following theore clarifies the difference between these probles for Bayesian networks. Theore 2 BN-MPEe is PP-Coplete. Proof Pertinence is obtained fro the fact that, after aking a PP query to find P (e) (this query is ade once), we can decide whether a given instantiation x has P (x e) > r in linear tie, by ultiplying the probabilities. To show hardness, we reduce the decision proble #3SAT( 2 n/2 ) to it, which is PP-Coplete [Bailey et al., 200] and can be stated as: Given a set of boolean variables X = {X,..., X n } and a 3CNF forula φ(x) with clauses {C,..., C }, is φ(x) satisfied by at least 2 n/2 of the instantiations of X? We construct a BN with binary nodes X,..., X n (x i and x i are the categories) and C,..., C (c i and c i ), where X i has no parents and unifor prior and C i has three parents (the variables contained in the clause) with probabilities respecting the truth table for the clause. Furtherore, we insert a duy binary node Y appearing non-negated in every clause (there will be 2 n instantiations with {Y = y} satisfying φ; this ensures that the forula is satisfiable). Now we solve the BN-MPEe proble with queries X,..., X n, Y and evidence C i = c i for all i (indicating that all C i are true). Then P (X, Y c,..., c ) is equal to = P (c,..., c X,..., X n, Y ) P (X,..., X n, Y ) P (c,..., c ) = = 2 n+ P (c,..., c X,..., X n, Y ) X,...,X n,y [ P (c,..., c X,..., X n, Y ) 2 n+ ] #sats if X,..., X n, Y satisfies φ and 0 otherwise, where #sats is the total nuber of satisfying instantiations. So, ax P (X, Y c,..., c ) iplies that forula φ(x) is satisfied by at least 2 n/2 of all X instantiations. 2 n +2 n/2 Note that if ax P (X, Y c,..., c ) = α and α >, 2 n +2 n/2 then P (X, Y c,..., c ) = α for all satisfying instantiations, which iplies that there are less than 2 n/2 instantiations of X satisfying φ(x). The hardness of PT-CN-Pr was stated by [da Rocha and Cozan, 2002], but the proof there was flawed, and the central arguent used zero probabilities and vertex-based description in an essential way. The following proof corrects these difficulties. Theore 3 PT-CN-Pr and BIW-CN-Pr are NP- Coplete. Proof Pertinence of BIW-CN-Pr (which ensures pertinence of PT-CN-Pr) is iediate, as choosing a vertex of each credal set we have a BIW-BN-Pr proble to solve, which is polynoial. To show hardness we reduce the MAX-3-SAT proble to PT-CN-Pr. It can be forulated as follows: Given a set of boolean variables {X,..., X n }, a 3CNF forula with clauses {C,..., C } and an integer 0 k, is there an assignent for the variables that satisfies at least k clauses? Initially we reove all clauses that have both x i and x i and decreent k for each eliination (because those clauses are already satisfied). For each variable X i we construct two nodes, naely X i and S i. The forer is binary, has no parents and represents the state of X i ; the probabilities P (X i = x i ) and P (X i = x i ) are in [ε, ε] (0 < ε < + is a sall constant). The latter ay assue + categories

4 S 0 X X2 X n X n S S 2 S n S n Figure : Polytree used in the network of Theore 3. (fro 0 to ), has S i and X i as parents and is defined by P (S i = c S i = c, x i ) = 0 if x i C c, or otherwise P (S i = c S i = c, x i ) = 0 if x i C c, or otherwise P (S i = c S i c, X i ) = 0 for X i {x i, x i }, for c 0. When S i = 0, we have: P (S i = 0 S i = c, X i ) = P (S i = c S i = c, X i ) P (S i = 0 S i = 0, X i ) = for X i {x i, x i }. The rules above guarantee coherency in probabilities; note that we include a duy node S 0 with P (S 0 = c) = + for all c. Now, consider P (S n = c) for c 0. Note that P (S n = c) = P (S 0 = c) i P (S i = c S i = c) because, for c 0, every tie {S i = c} and {S i c} appear together we are led to zero. Let A c,i be defined as follows: A c,i = P (S i = c S i = c, X i ) P (X i ). X i {x i,x i} We get P (S n = c) = + i {,...,n} A c,i. Note that A c,i ay assue three values: A c,i = if X i does not influence C c ; A c,i = ε if X i satisfies C c and A c,i = ε if X i does not satisfy C c. We ay conclude that if P (S n = c) α = ( ε) 2 ε, then soe X i satisfied C c. Furtherore, we know that if C c was not satisfied, then P (S n = c) = β = ( ε) 3. Note that α < β. To find out how any clauses were not satisfied, we have to su over all categories of S n, obtaining P (S n = 0) = c {,...,} P (S n = c) and thus P (S n = 0) iniizes this su. We define r h = [( + )( P (S n = 0)) hβ]/( h). and then calculate r 0, r,... until r h α or h =. We know that P (S n = 0) is the iniu su of all P (S n = c), for c 0. This su is coposed by two types of ters: those which are equal to β and those which are less than or equal to α. So what we are verifying with r h is whether there are ore than h ters of the su that are equal to β or not. The last thing should be noted is that ε < + ensures that if just one ter of the su equals to β, then the su is greater than α, that is, if the su is coposed by ters equal to ε 3 and just one equal to β, it ust su greater than α, because one clause was not satisfied. ε < + ensures that everywhere it is necessary (for any h). Thus the inference P (S n = 0) solves MAX-3-SAT proble. If r h > α for all 0 h, then no clause was satisfied. Otherwise h counts how any clauses were not satisfied. Because of the reduction fro MAX-3-SAT, we can state that there is no polynoial tie approxiation schee for PT-CN-Pr (or BIW-CN-Pr) unless P=NP. Reark The proof still holds if we substitute all ε by zero; the proof becoes sipler but depends on events of zero probability, which ay be inconvenient, as pointed out by [Zaffalon, 2003]. Note also that the proof can be rewritten using inequalities instead of vertices, because all credal sets are in binary nodes (pertinence and hardness still hold). It is known that CN-Pr is solvable by BN-MAP, by conducting a CCM transfor in a credal network [Cozan, 2000a]. The following lea presents the reverse connection between inferences in Bayesian and credal networks. Lea 4 BN-MAP is solvable by CN-Pr with joint queries without changing the topology of the network aong the three types defined. Proof Suppose X,..., X n are the MAP variables. Add a binary child X i to each X i with P (X i X i) [0, ] and the constraint X i P (x i X i) =. Now we have ax P (X,..., X n e) = P (x X,...,X n,..., x n e) After evaluating P (x,..., x n e), we just have to look at each X i node and set X i according to which of the P (x i X i) is equal to one (exactly one will be). It should be noted that joint queries are not ore difficult than single arginal queries: we have that CN-Pr with joint queries is still NP PP -Coplete and BIW-CN-Pr with joint queries is still NP-Coplete. Theore 5 PT-CN-MPE and BIW-CN-MPE are solvable in polynoial tie. Proof It is enough to realize that the inner in of the BIW-CN-MPE query ax x in P K(X) i P (x i pa(x i )) factorizes, as the network is locally specified (note that x i s are consistent with the observation e). The optiization becoes ax x i P (x i pa(x i )), which is equivalent to BIW-BN-MPE. Theore 6 CN-MPE is NP-Coplete. Proof Hardness is iediate, because BN-MPE is NP- Coplete and can be trivially transfored to a CN-MPE (we just have to use credal networks coposed by single probability densities). Pertinence is reached because, given an assignent x to the variables, the value of P (x) is given by i P (x i pa(x i )). This holds because each credal set K(x i pa(x i )) is locally specified. Theore 7 PT-CN-MPEe, BIW-CN-MPEe, PT-CN-MAP and BIW-CN-MAP are all Σ P 2-Coplete. Proof Pertinence of the is iediate as BIW-CN-MAP belongs to Σ P 2 (given the MAP variables, we get a BIW-CN-Pr to solve). To see hardness we reduce to PT-CN-MPEe a version of QSAT 2 that is Σ P 2-Coplete: Given a set of variables X,..., X n, an integer 0 < k n and a boolean 3CNF forula φ(x) over these variables, is it true that, for all instantiations to the first k variables, there is an instantiation of the reaining n k that satisfies φ(x)? Initially we construct a network siilar to that of Theore 3. The variables X,..., X k are defined the sae way as there. The variables X k+,..., X n becoe ternary,

5 assuing the categories x i, x i, o i. Their probabilities are: P (X i = o i ) = ε and P (X i = x i ), P (X i = x i ) belong to [0, ε], where 0 < ε < +2 is a sall constant. The probabilities of S i given its parents are the sae as there, except when i > k. In these cases we have P (S i = c S i = c, o i ) = 0, for c 0 (the case when c = 0 reains the sae, that is, equals to ). Furtherore, we add a duy node Q with S n as parent and P (q S n = c) = 0 for c 0 and otherwise. We will solve the PT-CN-MPEe proble ax P (X, S q), where S = {S 0,..., S n } and X = {X,..., X n }. Let X be {X,..., X k } and X + be {X k+,..., X n }; then P ( X, X +, S q ) = P (q S) P (S X) P (X ) P (X + ). P (q) First note that the given q forces S n = 0 to get a nonzero probability. Furtherore, for all instantiations x, x +, s of the variables, there is another instantiation with s = { is i = 0} that has its probability greater than the forer, that is, P (x, x +, s q) > P (x, x +, s q). This holds because the S i nodes are not credal and the conditional probabilities P (S i = 0 S i = 0) are equal to, for all i. Thus we know that the solution of the MPEe proble will be attained in an instantiation where all S i are set to 0. Besides that, we have that choosing the category o i for all X + variables lead us to greater probabilities than if we choose any other. That is, ax P ( X, X +, S q ) = ax P ( X, o, s q ), X,X +,S X where o denotes {X i = o i for i {k +,..., n}} and s denotes { i S i = 0}. If we choose a category different fro o i whenever possible, the axiu probability would not reach the sae value (in fact it will be zero, because P (X i o i ) [0, ε], and thus it ay assue value zero). P ( X, o, s q ) = + εn k P (X ) c 0 P (S n = c). When finding P (X, o, s q), the nuerator P (X ) will becoe ε k, because any solution that does not ake P (X ) = ε k will not be a iniu for P (X, o, s q) (reeber that ε < +2 ). This holds because just one X variable using the extree point ( ε) instead of ε is enough to ake P (X, o, s q) too large: + εn k ε k ( ε) > R + εn k ε k R where R and R are any possible values for the su c 0 P (S n = c) that appears in the denoinator (note that these sus are restricted in [0, + ] by the probabilities of the network). Suarizing, all S variables in the solution of MPEe are set to zero, all X + variables are set to o, and the instantiation chosen for X akes P (X ) = ε k, which iplies that if x i belongs to the instantiation of X, then P (x i ) = ε and P (x i ) = ε (the opposite case is analogous). This eans that the vertices of the credal sets of the X nodes are copletely fixed by the instantiation chosen. Thus the only credal sets that can float in the denoinator are the X + variables (S i variables are not credal and X are already fixed as indicated). So, processing the MPEe we have P ( X, o, s q ) = + εn ( ), in c 0 P (S n = c) where in c 0 P (S n = c) is evaluated over all possible vertices of the X + credal sets (we know that P (X i = o i ) is set to ε, but the probabilities P (X i = x i ) and P (X i = x i ) ay vary between 0 and ε). The forula φ(x) will be satisfied by X and X + if the su c 0 P (S n = c) is less than δ = + ( ε)2 ε( ε) n k (this is the axiu value that a satisfied φ(x) ay assue). All unsatisfied forulas lead to greater values. In fact the sallest value that a unsatisfied forula iplies in the su is greater than δ 2 = + ( ε)n k+3. Note that δ 2 > δ. If we query the PT-CN-MPEe proble ax P (X, S q) with r = ε n + δ and get a negative answer, then for all instantiations to X, there is an instantiation to X + that satisfy φ(x) (that is, the su is bounded by δ ). If the answer is positive, then there does exist an instantiation to X where no instantiation to X + can ake φ(x) satisfied. Theore 8 PT-BN-MAP and BIW-BN-MAP are Σ P 2- Coplete. Proof Pertinence of BIW-BN-MAP (which ensures pertinence of PT-BN-MAP) is trivial. Given a instantiation for the MAP variables, we need to solve a iniization over the Y variables, which is NP-Coplete (see it as a BIW-CN-Pr using Lea 4). Hardness of PT-BN-MAP (which ensures hardness of BIW-BN-MAP) is reached by a reduction fro another version of QSAT 2 that is Σ P 2-Coplete: Given a set of variables X,..., X n, an integer 0 < k n and a boolean 3DNF forula φ(x) over these variables, is there an instantiation to the first k variables such that, for all instantiations of the reaining n k, φ(x) is satisfied? We construct again a network following the ideas of Theore 3. It includes a binary node to each X i, without parents and with unifor prior. There are n nodes S i with parents S i and X i. They have + categories and are defined as follows (for c {,..., } and i {,..., n}): if x i clause c P (S i = c S i = c, x i ) = 0 if x i clause c 2 otherwise P (S i = c S i c, x i ) = 0. The conditional probabilities of S i given x i are defined analogously. The probabilities of P (S i = 0 S i, X i ) ensure that they su exactly, as done in Theore 3. Furtherore, P (S 0 ) has unifor prior and we insert an additional binary node Q with S n as parent, having P (q S n = c) = if c 0 and 0 otherwise. So, we have that P (X q) = 2 n P(q X) P(q) = (+)2 c 0 n (+) i A c,i ( 2) n = c 0 is equal to i A c,i

6 where A c,i = X i P (S i = c S i = c, X i ) P (X i ) for i {,..., n}. A c,i assues value zero only when the variable it represents denies the clause c. The nuerator of P (X q) sus how any clauses are satisfied by the instantiation of X variables. Let X = {X,..., X k } and X + = {X k+,..., X n }. We have in X + P (X, X + q) = 0 if, given the instantiation for the X variables, there is an assignent to X + variables that can deny φ(x). This way, questioning if PT-BN-MAP proble with MAP variables X and evidence q (which axiizes in X + P (X q)) has non-zero answer is enough to solve the QSAT 2 proble. Given this X instantiation, all X + will satisfy φ(x). 5 Conclusion We can suarize the contributions of this paper as follows. Concerning Bayesian networks, we have first introduced a ore general definition of bounded induced-width networks (deonstrating that any probles where inducedwidth actually grows with the network reain polynoial), and we have shown the difference between the BN-MPE and the BN-MPEe probles. More iportantly, we have introduced the MAP proble and presented its coplexity. A possible iproveent of our results would be to present a copleteness result for ultiply-connected networks. Our ost significant results pertain to credal networks, with direct iplications to odels that handle interval and set probabilities, belief functions, possibility easures, qualitative probabilities, and failies of probabilistic logic. We have clarified the so far unexplored coplexity of CN-MPE, CN-MPEe, and CN-MAP. The polynoial character of CN-MPE in soe cases is rather surprising. There are several interesting probles still to be explored. For exaple, binary networks (Bayesian and credal) could display lower coplexity than their non-binary counterparts in probles such as PT-BN-MAP, BIW-BN-MAP and BIW-CN-Pr. These probles belong to NP and are clearly related, but are there polynoial tie algoriths to solve the? We conjecture there are not, even as we note that there is a polynoial algorith to solve CN-Pr in binary polytrees [Fagiuoli and Zaffalon, 998]. Acknowledgents We thank Aritanan Gruber for indicating relevant coplexity results, Carlos Guestrin for pointing out a flaw in [da Rocha and Cozan, 2002], Thoas Lukasiewicz for initial thoughts on the coplexity of credal networks, and a reviewer for suggesting a short proof for Theore 5. This research received support fro HP Brazil R&D; the second author was partially supported by CNPq, Brazil. References [Bailey et al., 200] D. D. Bailey, V. Dalau, and P. G. Kolaitis. Phase transitions of PP-coplete satisfiability probles. In Int. 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