On the Robustness of Most Probable Explanations


 Douglas Houston
 1 years ago
 Views:
Transcription
1 On the Robustness of Most Probble Explntions Hei Chn School of Electricl Engineering nd Computer Science Oregon Stte University Corvllis, OR Adnn Drwiche Computer Science Deprtment University of Cliforni, Los Angeles Los Angeles, CA Abstrct In Byesin networks, Most Probble Explntion (MPE) is complete vrible instntition with the highest probbility given the current evidence. In this pper, we discuss the problem of finding robustness conditions of the MPE under single prmeter chnges. Specificlly, we sk the question: How much chnge in single network prmeter cn we fford to pply while keeping the MPE unchnged? We will describe procedure, which is the first of its kind, tht computes this nswer for ll prmeters in the Byesin network in time O(n exp(w)), where n is the number of network vribles nd w is its treewidth. 1 Introduction A Most Probble Explntion (MPE) in Byesin network is complete vrible instntition which hs the highest probbility given current evidence [1]. Given n MPE solution for some piece of evidence, we concern ourselves in this pper with the following question: Wht is the mount of chnge one cn pply to some network prmeter without chnging this current MPE solution? Our gol is then to deduce robustness conditions for MPE under single prmeter chnges. This problem flls into the relm of sensitivity nlysis. Here, we tret the Byesin network s system which ccepts network prmeters s inputs, nd produces the MPE s n output. Our gol is then to chrcterize conditions under which the output is gurnteed to be the sme (or different) given chnge in some input vlue. This question is very useful in number of ppliction res, including whtif nlysis, in ddition to the This work ws completed while Hei Chn ws t UCLA. Figure 1: An exmple Byesin network where we re interested in the MPE nd its robustness. design nd debugging of Byesin networks. For n exmple, consider Figure 1 which depicts Byesin network for dignosing potentil problems in cr. Suppose now tht we hve the following evidence: the dshbord test nd the lights test cme out positive, while the engine test cme out negtive. When we compute the MPE in this cse, we get scenrio in which ll cr components re working normlly. This seems to be counterintuitive s we expect the most likely scenrio to indicte t lest tht the engine is not working. The methods developed in this pper cn be used to debug this scenrio. In prticulr, we will be ble to identify the mount of chnge in ech network prmeter which is necessry to produce different MPE solution. We will revisit this exmple lter in the pper nd discuss the specific recommendtions computed by our proposed lgorithm. Previous results on sensitivity nlysis hve focused mostly on the robustness of probbility vlues, such s the probbility of evidence, under single or multiple prmeter chnges [2, 3, 4, 5, 6, 7, 8, 9]. Becuse probbility vlues re continuous, while MPE solutions re discrete instntitions, brupt chnges in MPE so
2 lutions my occur when we chnge prmeter vlue. This mkes the sensitivity nlysis of MPE quite different from previous work on the subject. This pper is structured s follows. We first provide the forml definition of Byesin networks nd MPE in Section 2. Then in Section 3, we explore the reltionship between the MPE nd single network prmeter, nd lso look into the cse where we chnge covrying prmeters in Section 4. We deduce tht the reltionship cn be cptured by two constnts tht re independent of the given prmeter. Next in Section 5, we show how we cn compute these constnts for ll network prmeters, llowing us to utomticlly identify robustness conditions for MPE, nd provide complexity nlysis of our proposed pproch. Finlly, we show some concrete exmples in Section 6, nd then extend our nlysis to evidence chnge in Section 7. 2 Most Probble Explntions We will formlly define most probble explntions in this section, but we specify some of our nottionl conventions first. We will denote vribles by uppercse letters (X) nd their vlues by lowercse letters (x). Sets of vribles will be denoted by boldfce uppercse letters (X) nd their instntitions by boldfce lowercse letters (x). For vrible X nd vlue x, we will often write x insted of X = x, nd hence, Pr(x) insted of Pr(X = x). For binry vrible X with vlues true nd flse, we will use x to denote X = true nd x to denote X = flse. Therefore, Pr(X = true) nd Pr(x) represent the sme probbility in this cse. Similrly, Pr(X = flse) nd Pr( x) represent the sme probbility. Finlly, for instntition x of vribles X, we will write x to men the set of ll instntitions x x of vribles X. For exmple, we will write Pr(x) + Pr( x) = 1. A Byesin network is specified by its structure, directed cyclic grph (DAG), nd set of conditionl probbility tbles (CPTs), with one CPT for ech network vrible [1]. In the CPT for vrible X with prents U, we define network prmeter x u for every fmily instntition xu such tht x u = Pr(x u). Given the network prmeters, we cn compute the probbility of complete vrible instntition x s follows: Pr(x) = x u, (1) xu x where is the comptibility reltion between instntitions, i.e., xu x mens tht xu is comptible with x). Now ssume tht we re given evidence e. A most probble explntion (MPE) given e is complete vrible instntition tht is consistent with e nd hs the highest probbility [1]: MPE(e) def = rg mx x e = rg mx x e Pr(x) (2) x u. xu x We note tht the MPE my not be unique instntition s there cn be multiple instntitions with the sme highest probbility. Therefore, we will define MPE(e) s set of instntitions insted of just one instntition. Moreover, we will sometimes use MPE(e, x) to denote the MPE instntitions tht re consistent with e but inconsistent with x. In the following discussion, we will find it necessry to distinguish between the MPE identity nd the MPE probbility. By the MPE identity, we men the set of instntitions hving the highest probbility. By the MPE probbility, we men the probbility ssumed by most likely instntition, which is denoted by: MPE p (e) def = mx x e Pr(x). (3) This distinction is importnt when discussing robustness conditions for MPE since chnge in some network prmeter my chnge the MPE probbility, but not the MPE identity. 3 Reltion Between MPE nd Network Prmeters Assume tht we re given evidence e nd re ble to find its MPE, MPE(e). We now ddress the following question: How much chnge cn we pply to network prmeter x u without chnging the MPE identity of evidence e? To simplify the discussion, we will first ssume tht we cn chnge this prmeter without chnging ny covrying prmeters, such s x u, but we will relx this ssumption lter. Our solution to this problem is bsed on some bsic observtions which we discuss next. In prticulr, we observe tht complete vrible instntitions x which re consistent with e cn be divided into two ctegories: Those tht re consistent with xu. From Eqution 1, the probbility of ech such instntition x is liner function of the prmeter x u. Those tht re inconsistent with xu. From Eqution 1, the probbility of ech such instntition x is constnt which is independent of the prmeter x u. Let us denote the first set of instntitions by Σ e,xu nd the second set by Σ e, (xu). We cn then conclude tht:
3 The set of most likely instntitions in Σ e,xu remins unchnged regrdless of the vlue of prmeter x u, even though the probbility of such instntitions my chnge ccording to the vlue of x u. This is becuse the probbility of ech instntition x Σ e,xu is liner function of the vlue of x u : Pr(x) = r x u, where r is coefficient independent of the vlue of x u. Therefore, the reltive probbilities mong instntitions in Σ e,xu remin unchnged s we chnge the vlue of x u. Note lso tht the most likely instntitions in this set Σ e,xu re just MPE(e, xu) nd their probbility is MPE p (e, xu). Therefore, if we define: r(e, xu) we will then hve: def = MPE p(e, xu) x u, (4) Pr(x) = r(e, xu) x u, for ny x MPE(e, xu). Both the identity nd probbility of the most likely instntitions in Σ e, (xu) re independent of the vlue of prmeter x u. This is becuse the probbility of ech instntition x Σ e, (xu) is independent of the vlue of x u. Note tht the most likely instntition in this set Σ e, (xu) is just MPE(e, (xu)). We will define the probbility of such n instntition s: k(e, xu) def = MPE p (e, (xu)). (5) Given the bove observtions, MPE(e) will either be MPE(e, xu), MPE(e, (xu)), or their union, depending on the vlue of prmeter x u : MPE(e) MPE(e, xu), if r(e, xu) x u > k(e, xu); = MPE(e, (xu)), if r(e, xu) x u < k(e, xu); MPE(e, xu) MPE(e, (xu)), otherwise. Moreover, the MPE probbility cn lwys be expressed s: MPE p (e) = mx(r(e, xu) x u, k(e, xu)). Figure 2 plots the reltion between the MPE probbility MPE p (e) nd the vlue of prmeter x u. According to the figure, if x u > k(e, xu)/r(e, xu), i.e., region A of the plot, then we hve MPE(e) = MPE(e, xu), nd thus the MPE solutions re consistent with xu. Moreover, the MPE identity will remin unchnged s long s the vlue of x u remins greter thn k(e, xu)/r(e, xu). MPE Pr (e) k(e,xu) 0 Region B Region A k(e,xu) / r(e,xu) x u Figure 2: A plot of the reltion between the MPE probbility MPE p (e) nd the vlue of prmeter x u. On the other hnd, if x u < k(e, xu)/r(e, xu), i.e., region B of the plot, then we hve MPE(e) = MPE(e, (xu)), nd thus the MPE solutions re inconsistent with xu. Moreover, the MPE identity nd probbility will remin unchnged s long s the vlue of x u remins less thn k(e, xu)/r(e, xu). Therefore, x u = k(e, xu)/r(e, xu) is the point where there is chnge in the MPE identity if we were to chnge the vlue of prmeter x u. At this point, MPE(e) = MPE(e, xu) MPE(e, (xu)) nd we hve both MPE solutions consistent with xu nd MPE solutions inconsistent with xu. There re no other points where there is chnge in the MPE identity. If we re ble to find the constnts r(e, xu) nd k(e, xu) for the network prmeter x u, we cn then compute robustness conditions for MPE with respect to chnges in this prmeter. 4 Deling with CoVrying Prmeters The bove nlysis ssumed tht we cn chnge prmeter x u without needing to chnge ny other prmeters in the network. This is not relistic though in the context of Byesin networks, where covrying prmeters need to dd up to 1 for the network to induce vlid probbility distribution. For exmple, if vrible X hs two vlues, x nd x, we must lwys hve: x u + x u = 1. We will therefore extend the nlysis conducted in the previous section to ccount for the simultneously chnges in the covrying prmeters. We will restrict our ttention to binry vribles to simplify the discussion, but our results cn be esily extended to multivlued vribles s we will show lter. In prticulr, ssuming tht we re chnging prme
4 ters x u nd x u simultneously for binry vrible X, we cn now ctegorize ll network instntitions which re consistent with evidence e into three groups, depending on whether they re consistent with xu, xu, or u. Moreover, the most likely instntitions in ech group re just MPE(e, xu), MPE(e, xu), nd MPE(e, u) respectively. Therefore, if x MPE(e), then: r(e, xu) x u, if x MPE(e, xu); Pr(x) = r(e, xu) x u, if x MPE(e, xu); k(e, u), if x MPE(e, u); where: r(e, xu) = MPE p(e, xu) x u ; r(e, xu) = MPE p(e, xu) x u ; k(e, u) = MPE p (e, u); nd the MPE probbility is: MPE p (e) = mx(r(e, xu) x u, r(e, xu) x u, k(e, u)). Therefore, chnging the covrying prmeters x u nd x u will not ffect the identity of either MPE(e, xu) or MPE(e, xu), nor will it ffect the identity or probbility of MPE(e, u). The robustness condition of n MPE solution cn now be summrized s follows: If n MPE solution is consistent with xu, it remins solution s long s the following inequlities re true: r(e, xu) x u r(e, xu) x u ; r(e, xu) x u k(e, u). If n MPE solution is consistent with xu, it remins solution s long s the following inequlities re true: r(e, xu) x u r(e, xu) x u ; r(e, xu) x u k(e, u). If n MPE solution is consistent with u, it remins solution s long s the following inequlities re true: k(e, u) r(e, xu) x u ; k(e, u) r(e, xu) x u. We note here tht one cn esily deduce whether n MPE solution is consistent with xu, xu, or u since it is complete vrible instntition. Therefore, ll we need re the constnts r(e, xu) nd k(e, u) for ech network prmeter x u in order to define robustness conditions for MPE. The constnts k(e, u) cn be esily computed from the constnts r(e, xu) by observing the following: k(e, u) = MPE p (e, u) = mx p(e, u ) u :u u = mx p(e, xu ) xu :u u = mx xu :u u xu ) x u. (6) As the lgorithm we will describe lter computes the r(e, xu) constnts for ll fmily instntitions xu, the lgorithm will then llow us to compute ll the k(e, u) constnts s well. As simple exmple, for the Byesin network whose CPTs re shown in Figure 3, the current MPE solution without ny evidence is A =, B = b, nd hs probbility.4. For the prmeters in the CPT of B, we cn compute the corresponding r(e, xu) constnts. In prticulr, we hve r(e, b) = r(e, b) = r(e, bā) = r(e, bā) =.5 in this cse. The k(e, u) constnts cn lso be computed s k(e, ) =.3 nd k(e, ā) =.4. Given these constnts, we cn esily compute the mount of chnge we cn pply to covrying prmeters, sy b nd b, such tht the MPE solution remins the sme. The conditions we must stisfy re: r(e, b) b r(e, b) b ; r(e, b) b k(e, ). This leds to b b nd b.6. Therefore, the current MPE solution will remin so s long s b.6, which hs current vlue of.8. We close this section by pointing out tht our robustness equtions cn be extended to multivlued vribles s follows. If vrible X hs vlues x 1,..., x j, with j > 2, then ech of the conditions we showed erlier will consist of j inequlities insted of just two. For exmple, if n MPE solution is consistent with x 1 u, it remins solution s long s the following inequlities re true: r(e, x 1 u) x1 u r(e, x u) x u for ll x x 1 ; r(e, x 1 u) x1 u k(e, u). 5 Computing Robustness Conditions In this section, we will develop n lgorithm for computing the constnts r(e, xu) for ll network prmeters x u. In prticulr, we will show tht they cn be computed in time nd spce which is O(n exp(w)), where n is the number of network vribles nd w is its treewidth.
5 A Θ A.5 ā.5 A B Θ B A b.2 b.8 ā b.6 ā b.4 Figure 3: The CPTs for Byesin network A B. with x. Moreover, the term for x evlutes to the probbility vlue Pr(e, x) when the evidence indictors re set ccording to e. Note tht this function is multiliner. Therefore, corresponding rithmetic circuit will hve the property tht two subcircuits tht feed into the sme multipliction node will never contin common vrible. This property is importnt for some of the following developments b b b b b b Figure 4: An rithmetic circuit for the bove Byesin network. The bold lines depict complete subcircuit, corresponding to the term b b. 5.1 Arithmetic Circuits Our lgorithm for computing the r(e, xu) constnts is bsed on n rithmetic circuit representtion of the Byesin network [10]. Figure 4 depicts n rithmetic circuit for smll network consisting of two binry nodes, A nd B, shown in Figure 3. An rithmetic circuit is rooted DAG, where ech internl node corresponds to multipliction ( ) or ddition (+), nd ech lef node corresponds either to network prmeter x u or n evidence indictor x ; see Figure 4. Opertionlly, the circuit cn be used to compute the probbility of ny evidence e by evluting the circuit while setting the evidence indictor x to 0 if x contrdicts e nd setting it to 1 otherwise. Semnticlly though, the rithmetic circuit is simply fctored representtion of n exponentilsize function tht cptures the network distribution. For exmple, the circuit in Figure 4 is simply fctored representtion of the following function: b b + b b + ā b ā b ā + ā bā b ā. This function, clled the network polynomil, includes term for ech instntition x of network vribles, where the term is simply product of the network prmeters nd evidence indictors which re consistent 5.2 Complete SubCircuits nd Their Coefficients Ech term in the network polynomil corresponds to complete subcircuit in the rithmetic circuit. A complete subcircuit cn be constructed recursively from the root, by including ll children of ech multipliction node, nd exctly one child of ech ddition node. The bold lines in Figure 4 depict complete subcircuit, corresponding to the term b b. In fct, it is esy to check tht the circuit in Figure 4 hs four complete subcircuits, corresponding to the four terms in the network polynomil. A key observtion bout complete subcircuits is tht if network prmeter is included in complete subcircuit, there is unique pth from the root to this prmeter in this subcircuit, even though there my be multiple pths from the root to this prmeter in the originl rithmetic circuit. This pth is importnt s one cn relte the vlue of the term corresponding to the subcircuit nd the prmeter vlue by simply trversing the pth s we show next. Consider now complete subcircuit which includes network prmeter x u nd let α be the unique pth in this subcircuit connecting the root to prmeter x u. We will now define the subcircuit coefficient w.r.t. x u, denoted s r, in terms of the pth α such tht r x u is just the vlue of the term corresponding to the subcircuit. Let Σ be the set of ll multipliction nodes on this pth α. The subcircuit coefficient w.r.t. x u is defined s the product of ll children of nodes in Σ which re themselves not on the pth α. Consider for exmple the complete subcircuit highlighted in Figure 4 nd the pth from the root to the network prmeter. The coefficient w.r.t. is r = b b. Moreover, r = b b, which is the term corresponding to the subcircuit. 5.3 Mximizer Circuits An rithmetic circuit cn be esily modified into mximizer circuit to compute the MPE solutions, by simply replcing ech ddition node with mximiztion node; see Figure 5. This corresponds to circuit
6 0.4 mx mx mx Algorithm 1 DMAXC(M: mximizer circuit, e: evidence) 1: evlute the circuit M under evidence e; fterwrds the vlue of ech node v is p[v] 2: r[v] 1 for root v of circuit M 3: r[v] 0 for ll nonroot nodes v in circuit M 4: for nonlef nodes v (prents before children) do 5: if node v is mximiztion node then 6: r[c] mx(r[c], r[v]) for ech child c of node v 7: if node v is multipliction node then 8: r[c] mx (r[c], r[v] c p[c ]) for ech child c of node v, where c re the other children of node v b b b b b b Figure 5: A mximizer circuit for Byesin network, evluted under evidence A =. Given this evidence, the evidence indictors re set to = 1, ā = 0, b = 1, b = 1. The bold lines depict the MPE subcircuit. tht computes the vlue of the mximum term in network polynomil, insted of dding up the vlues of these terms. The vlue of the root will thus be the MPE probbility MPE p (e). The mximizer circuit in Figure 5 is evluted under evidence A =, leding to n MPE probbility of.4. To recover n MPE solution from mximizer circuit, ll we need to do is construct the MPE subcircuit recursively from the root, by including ll children of ech multipliction node, nd one child c for ech mximiztion node v, such tht v nd c hve the sme vlue; see Figure 5. The MPE subcircuit will then correspond to n MPE solution. Moreover, if prmeter x u is in the MPE subcircuit, nd the subcircuit coefficient w.r.t x u is r, then we hve r x u s the probbility of MPE, MPE p (e). Consider Figure 5 nd the highlighted MPE subcircuit, evluted under evidence A =. The term corresponding to this subcircuit is A =, B = b, which is therefore n MPE solution. Moreover, we hve two prmeters in this subcircuit, nd b, with coefficients.8 = (1)(.8) nd.5 = (.5)(1)(1) respectively. Therefore, the MPE probbility cn be obtined by multiplying ny of these coefficients with the corresponding prmeter vlue, s (.8) = (.8)(.5) =.4 nd (.5) b = (.5)(.8) = Computing r(e, xu) Suppose now tht our gol is to compute MPE(e, xu) for some network prmeter x u. Suppose further tht α 1,..., α m re ll the complete subcircuits tht include x u. Moreover, let x 1,..., x m be the instntitions corresponding to these subcircuits nd let r 1,..., r m be their corresponding coefficients w.r.t. x u. It then follows tht the probbilities of these instntitions re r 1 x u,..., r m x u respectively. Moreover, it follows tht: MPE p (e, xu) = mx r 1 x u,..., r m x u, i nd hence, from Eqution 4: MPE p (e, xu) x u = r(e, xu) = mx r 1,..., r m. i Therefore, if we cn compute the mximum of these coefficients, then we hve computed the constnt r(e, xu). Algorithm 1 provides procedure which evlutes the mximizer circuit nd then trverses it topdown, prents before children, computing simultneously the constnts r(e, xu) for ll network prmeters. The procedure mintins n dditionl register vlue r[.] for ech node in the circuit, nd updtes these registers s it visits nodes. When the procedure termintes, it is gurnteed tht the register vlue r[ x u ] will be the constnt r(e, xu). We will lso see lter tht the register vlue r[ x ] is lso constnt which provides vluble informtion for the MPE solutions. Figure 6 depicts n exmple of this procedure. Algorithm 1 cn be modelled s the llpirs shortest pth procedure, with edge v c hving weight 0 = ln 1 if v is mximiztion node, nd weight ln π if v is multipliction node, where π is the product of the vlues of the other children c c of node v. The length of the shortest pth from the root to the network prmeter x u is then ln r(e, xu). It should be cler tht the time nd spce complexity of the bove lgorithm is liner in the number of circuit nodes. 1 It is well known tht we cn compile 1 More precisely, this lgorithm is liner in the number of circuit nodes only if the number of children per multipliction node is bounded. If not, one cn use technique which gives liner complexity by simply storing two dditionl bits with ech multipliction node [10].
7 0.4 mx mx mx b b b b b b Figure 6: A mximizer circuit for Byesin network, evluted under evidence A =. Next to ech node is the vlue r[.] computed by Algorithm 1. circuit for ny Byesin network in O(n exp(w)) time nd spce, where n is the number of network vribles nd w is its treewidth [10]. Therefore, ll constnts r(e, xu) cn be computed with the sme complexity. We close this section by pointing out tht one cn in principle use the jointree lgorithm to compute MPE p (e, xu) = r(e, xu) x u for ll fmily instntitions xu with the bove complexity. In prticulr, by replcing summtion with mximiztion in the jointree lgorithm, one obtins MPE p (e, c) for ech cluster instntition c. Projecting on the fmilies XU in cluster C, one cn then obtin MPE p (e, xu) for ll fmily instntitions xu, which is ll we need to compute robustness conditions for MPE. 2 Our method bove, however, is more generl for two resons: The rithmetic circuit for Byesin network cn be much smller thn the corresponding jointree by exploiting the locl structures of the Byesin network [12, 13]. The constnts computed by the lgorithm for the evidence indictors cn be used to nswer dditionl MPE queries, which results fter vritions on the current evidence. This will be discussed in Section 7. 6 Exmple We now go bck to the exmple network in Figure 1, nd compute robustness conditions for the current 2 However, in cse some of the prmeters re equl to 0, one needs to use specil jointree [11]. Figure 7: A list of prmeter chnges tht would produce different MPE solution. MPE solution using the inequlities we obtin in Section 4, nd n implementtion of Algorithm 1. After going through the CPT of ech vrible, our procedure found nine possible prmeter chnges tht would produce different MPE solution, s shown in Figure 7. From these nine suggested chnges, only three chnges mke sense from qulittive point of view: Decresing the probbility tht the ignition is working from.9925 to t most (6th row) Decresing the probbility tht the engine is working given both the bttery nd the ignition re working from.97 to t most (1st row) Decresing the flsenegtive rte of the engine test from.09 to t most (9th row) If we pply the first prmeter chnge, we get new MPE solution in which both the ignition nd the engine re not working. If we pply either the second or third prmeter chnge, we get new MPE solution in which the engine is not working. 7 MPE under Evidence Chnge We hve discussed in Section 5.2 the notion of complete subcircuit nd its coefficient with respect to network prmeter x u which is included in the subcircuit. In prticulr, we hve shown how ech subcircuit corresponds to term in the network polynomil, nd tht if complete subcircuit hs coefficient r with respect to prmeter x u, then r x u will be the vlue of the term corresponding to this subcircuit. The notion of subcircuit coefficient cn be extended to evidence indictors s well. In prticulr, if complete subcircuit hs coefficient r with respect to n evidence indictor x which is included in the subcircuit,
8 then r x will be the vlue of the term corresponding to this subcircuit. Suppose now tht α 1,..., α m re ll the complete subcircuits tht include x. Moreover, let x 1,..., x m be the terms corresponding to these subcircuits nd let r 1,..., r m be their corresponding coefficients with respect to x. It then follows tht the vlues of these terms re r 1 x,..., r m x respectively. Moreover, it follows tht: MPE p (e X, x) = mx r 1,..., r m, i where e X denotes the new evidence fter hving retrcted the vlue of vrible X from e (if X E, otherwise e X = e). Therefore, if we cn compute the mximum of these coefficients, then we hve computed MPE p (e X, x). Note, however, tht Algorithm 1 lredy computes the mximum of these coefficients for ech x s the evidence indictors re nodes in the mximizer circuit s well, nd therefore the register vlue r[ x ] gives us MPE p (e X, x) for every vrible X nd vlue x. Consider for exmple the circuit in Figure 6, nd the coefficients computed by Algorithm 1 for the four evidence indictors. According to the bove nlysis, these coefficients hve the following menings: x e X, x r[ x ] = MPE p (e X, x).4 ā ā.3 b, b.1 b, b.4 For exmple, the second row bove tells us tht the MPE probbility would be.3 if the evidence ws A = ā insted of A =. In generl, if we hve n vribles, we then hve O(n) vritions on the current evidence of the form e X, x. The MPE probbility of ll of these vritions re immeditely vilble from the coefficients with respect to the evidence indictors. The computtion of these coefficients llows us to deduce the MPE identity fter evidence retrction. In prticulr, suppose tht vrible X is set s x in evidence e, nd MPE p (e) MPE p (e X, x ) for ll other vlues x x. We cn then conclude tht MPE p (e) = MPE p (e X). Moreover, MPE(e) = MPE(e X) if MPE p (e) > MPE p (e X, x ) for ll other vlues x x, or MPE(e) MPE(e X) if there exists some x x such tht MPE p (e) = MPE p (e X, x ). Therefore, the current MPE solutions will remin so even fter we retrct X = x from the evidence. This mens tht X = x is not integrl in the determintion of the current MPE solutions given the other evidence, i.e., e X. The result bove lso hs implictions on the identifiction of multiple MPE solutions given evidence e. In prticulr, suppose tht vrible X is not set in evidence e, then: If the coefficients for the evidence indictors x nd x re equl, we must hve both MPE solutions with X = x nd MPE solutions with X = x. In fct, the coefficients must both equl the MPE probbility MPE p (e) in this cse. If the coefficient for the evidence indictor x is lrger thn the coefficient for the evidence indictor x, then every MPE solution must hve X = x. In the exmple bove, we hve r[ b] > r[ b ], suggesting tht every MPE solution must hve b in this cse. 8 Conclusion We considered in this pper the problem of finding robustness conditions for MPE solutions of Byesin network under single prmeter chnges. We were ble to solve this problem by identifying some interesting reltionships between n MPE solution nd the network prmeters. In prticulr, we found tht the robustness condition of n MPE solution under single prmeter chnge depends on two constnts tht re independent of the prmeter vlue. We lso proposed method for computing such constnts nd, therefore, the robustness conditions of MPE in O(n exp(w)) time nd spce, where n is the number of network vribles nd w is the network treewidth. Our lgorithm is the first of its kind for ensuring the robustness of MPE solutions under prmeter chnges in Byesin network. Acknowledgments This work hs been prtilly supported by Air Force grnt #FA P00002 nd JPL/NASA grnt # We would lso like to thnk Jmes Prk for reviewing this pper nd mking the observtion on how to compute k(e, u) in Eqution 6. References [1] Jude Perl. Probbilistic Resoning in Intelligent Systems: Networks of Plusible Inference. Morgn Kufmnn Publishers, Sn Frncisco, Cliforni, [2] Hei Chn nd Adnn Drwiche. When do numbers relly mtter? Journl of Artificil Intelligence Reserch, 17: , 2002.
9 [3] Hei Chn nd Adnn Drwiche. Sensitivity nlysis in Byesin networks: From single to multiple prmeters. In Proceedings of the Twentieth Conference on Uncertinty in Artificil Intelligence (UAI), pges 67 75, Arlington, Virgini, AUAI Press. [13] Mrk Chvir, Adnn Drwiche, nd Mnfred Jeger. Compiling reltionl Byesin networks for exct inference. Interntionl Journl of Approximte Resoning, 42:4 20, [4] Enrique Cstillo, José Mnuel Gutiérrez, nd Ali S. Hdi. Sensitivity nlysis in discrete Byesin networks. IEEE Trnsctions on Systems, Mn, nd Cybernetics, Prt A (Systems nd Humns), 27: , [5] Veerle M. H. Coupé, Niels Peek, Jp Ottenkmp, nd J. Dik F. Hbbem. Using sensitivity nlysis for efficient quntifiction of belief network. Artificil Intelligence in Medicine, 17: , [6] Uffe Kjærulff nd Lind C. vn der Gg. Mking sensitivity nlysis computtionlly efficient. In Proceedings of the Sixteenth Conference on Uncertinty in Artificil Intelligence (UAI), pges , Sn Frncisco, Cliforni, Morgn Kufmnn Publishers. [7] Kthryn B. Lskey. Sensitivity nlysis for probbility ssessments in Byesin networks. IEEE Trnsctions on Systems, Mn, nd Cybernetics, 25: , [8] Mlcolm Prdhn, Mx Henrion, Gregory Provn, Brendn Del Fvero, nd Kurt Hung. The sensitivity of belief networks to imprecise probbilities: An experimentl investigtion. Artificil Intelligence, 85: , [9] Lind C. vn der Gg nd Silj Renooij. Anlysing sensitivity dt from probbilistic networks. In Proceedings of the Seventeenth Conference on Uncertinty in Artificil Intelligence (UAI), pges , Sn Frncisco, Cliforni, Morgn Kufmnn Publishers. [10] Adnn Drwiche. A differentil pproch to inference in Byesin networks. Journl of the ACM, 50: , [11] Jmes D. Prk nd Adnn Drwiche. A differentil semntics for jointree lgorithms. Artificil Intelligence, 156: , [12] Mrk Chvir nd Adnn Drwiche. Compiling Byesin networks with locl structure. In Proceedings of the Nineteenth Interntionl Joint Conference on Artificil Intelligence (IJCAI), pges , Denver, Colordo, Professionl Book Center.
Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More informationSection A4 Rational Expressions: Basic Operations
A Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr opentopped bo is to be constructed out of 9 by 6inch sheets of thin crdbord by cutting inch squres out of ech corner nd bending the
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationBabylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity
Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationAlgebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationThe Velocity Factor of an Insulated TwoWire Transmission Line
The Velocity Fctor of n Insulted TwoWire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationProtocol Analysis. 17654/17764 Analysis of Software Artifacts Kevin Bierhoff
Protocol Anlysis 17654/17764 Anlysis of Softwre Artifcts Kevin Bierhoff TkeAwys Protocols define temporl ordering of events Cn often be cptured with stte mchines Protocol nlysis needs to py ttention
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationNetwork Configuration Independence Mechanism
3GPP TSG SA WG3 Security S3#19 S3010323 36 July, 2001 Newbury, UK Source: Title: Document for: AT&T Wireless Network Configurtion Independence Mechnism Approvl 1 Introduction During the lst S3 meeting
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More informationRedistributing the Gains from Trade through Nonlinear. Lumpsum Transfers
Redistributing the Gins from Trde through Nonliner Lumpsum Trnsfers Ysukzu Ichino Fculty of Economics, Konn University April 21, 214 Abstrct I exmine lumpsum trnsfer rules to redistribute the gins from
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More informationCHAPTER 11 Numerical Differentiation and Integration
CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods
More informationDlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report
DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of
More informationP.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
More informationA Note on Complement of Trapezoidal Fuzzy Numbers Using the αcut Method
Interntionl Journl of Applictions of Fuzzy Sets nd Artificil Intelligence ISSN  Vol.  A Note on Complement of Trpezoidl Fuzzy Numers Using the αcut Method D. Stephen Dingr K. Jivgn PG nd Reserch Deprtment
More informationHealth insurance marketplace What to expect in 2014
Helth insurnce mrketplce Wht to expect in 2014 33096VAEENBVA 06/13 The bsics of the mrketplce As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum
More informationEconomics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999
Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,
More informationicbs: Incremental Cost based Scheduling under Piecewise Linear SLAs
i: Incrementl Cost bsed Scheduling under Piecewise Liner SLAs Yun Chi NEC Lbortories Americ 18 N. Wolfe Rd., SW3 35 Cupertino, CA 9514, USA ychi@sv.nec lbs.com Hyun Jin Moon NEC Lbortories Americ 18 N.
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More informationAssuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;
B26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndomnumer genertor supplied s stndrd with ll computer systems Stn KellyBootle,
More informationDecision Rule Extraction from Trained Neural Networks Using Rough Sets
Decision Rule Extrction from Trined Neurl Networks Using Rough Sets Alin Lzr nd Ishwr K. Sethi Vision nd Neurl Networks Lbortory Deprtment of Computer Science Wyne Stte University Detroit, MI 48 ABSTRACT
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More informationMath Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.
Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while
More informationMODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
More informationClearPeaks Customer Care Guide. Business as Usual (BaU) Services Peace of mind for your BI Investment
ClerPeks Customer Cre Guide Business s Usul (BU) Services Pece of mind for your BI Investment ClerPeks Customer Cre Business s Usul Services Tble of Contents 1. Overview...3 Benefits of Choosing ClerPeks
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More informationHealth insurance exchanges What to expect in 2014
Helth insurnce exchnges Wht to expect in 2014 33096CAEENABC 02/13 The bsics of exchnges As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum mount
More informationUNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics STRATEGIC SECOND SOURCING IN A VERTICAL STRUCTURE
UNVERSTY OF NOTTNGHAM Discussion Ppers in Economics Discussion Pper No. 04/15 STRATEGC SECOND SOURCNG N A VERTCAL STRUCTURE By Arijit Mukherjee September 004 DP 04/15 SSN 10438 UNVERSTY OF NOTTNGHAM Discussion
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationHelicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationEuler Euler Everywhere Using the EulerLagrange Equation to Solve Calculus of Variation Problems
Euler Euler Everywhere Using the EulerLgrnge Eqution to Solve Clculus of Vrition Problems Jenine Smllwood Principles of Anlysis Professor Flschk My 12, 1998 1 1. Introduction Clculus of vritions is brnch
More information2. Transaction Cost Economics
3 2. Trnsction Cost Economics Trnsctions Trnsctions Cn Cn Be Be Internl Internl or or Externl Externl n n Orgniztion Orgniztion Trnsctions Trnsctions occur occur whenever whenever good good or or service
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationWeek 11  Inductance
Week  Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationCOMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT
COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, crossclssified
More informationTRUST and reputation are crucial requirements for most
IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, VOL. 9, NO. 3, MAY/JUNE 2012 375 Itertive Trust nd Reputtion Mngement Using Belief Propgtion Ermn Aydy, Student Member, IEEE, nd Frmrz Feri, Senior
More informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More informationg(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany
Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required
More informationLower Bound for EnvyFree and Truthful Makespan Approximation on Related Machines
Lower Bound for EnvyFree nd Truthful Mespn Approximtion on Relted Mchines Lis Fleischer Zhenghui Wng July 14, 211 Abstrct We study problems of scheduling jobs on relted mchines so s to minimize the mespn
More informationNovel Methods of Generating SelfInvertible Matrix for Hill Cipher Algorithm
Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy Novel Methods of Generting SelfInvertible Mtrix for Hill Cipher lgorithm Bibhudendr chry Deprtment of Electronics & Communiction Engineering
More informationVirtual Machine. Part II: Program Control. Building a Modern Computer From First Principles. www.nand2tetris.org
Virtul Mchine Prt II: Progrm Control Building Modern Computer From First Principles www.nnd2tetris.org Elements of Computing Systems, Nisn & Schocken, MIT Press, www.nnd2tetris.org, Chpter 8: Virtul Mchine,
More informationSolving BAMO Problems
Solving BAMO Problems Tom Dvis tomrdvis@erthlink.net http://www.geometer.org/mthcircles Februry 20, 2000 Abstrct Strtegies for solving problems in the BAMO contest (the By Are Mthemticl Olympid). Only
More informationVersion 001 Summer Review #03 tubman (IBII20142015) 1
Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This printout should he 35 questions. Multiplechoice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03
More informationEnterprise Risk Management Software Buyer s Guide
Enterprise Risk Mngement Softwre Buyer s Guide 1. Wht is Enterprise Risk Mngement? 2. Gols of n ERM Progrm 3. Why Implement ERM 4. Steps to Implementing Successful ERM Progrm 5. Key Performnce Indictors
More informationData replication in mobile computing
Technicl Report, My 2010 Dt repliction in mobile computing Bchelor s Thesis in Electricl Engineering Rodrigo Christovm Pmplon HALMSTAD UNIVERSITY, IDE SCHOOL OF INFORMATION SCIENCE, COMPUTER AND ELECTRICAL
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More information4 Approximations. 4.1 Background. D. Levy
D. Levy 4 Approximtions 4.1 Bckground In this chpter we re interested in pproximtion problems. Generlly speking, strting from function f(x) we would like to find different function g(x) tht belongs to
More informationDouble Integrals over General Regions
Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More informationThe Definite Integral
Chpter 4 The Definite Integrl 4. Determining distnce trveled from velocity Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: If we know
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationFactoring Trinomials of the Form. x 2 b x c. Example 1 Factoring Trinomials. The product of 4 and 2 is 8. The sum of 3 and 2 is 5.
Section P.6 Fctoring Trinomils 6 P.6 Fctoring Trinomils Wht you should lern: Fctor trinomils of the form 2 c Fctor trinomils of the form 2 c Fctor trinomils y grouping Fctor perfect squre trinomils Select
More informationSecondDegree Equations as Object of Learning
Pper presented t the EARLI SIG 9 Biennil Workshop on Phenomenogrphy nd Vrition Theory, Kristinstd, Sweden, My 22 24, 2008. Abstrct SecondDegree Equtions s Object of Lerning Constnt Oltenu, Ingemr Holgersson,
More informationPlotting and Graphing
Plotting nd Grphing Much of the dt nd informtion used by engineers is presented in the form of grphs. The vlues to be plotted cn come from theoreticl or empiricl (observed) reltionships, or from mesured
More informationWeek 7  Perfect Competition and Monopoly
Week 7  Perfect Competition nd Monopoly Our im here is to compre the industrywide response to chnges in demnd nd costs by monopolized industry nd by perfectly competitive one. We distinguish between
More informationModeling POMDPs for Generating and Simulating Stock Investment Policies
Modeling POMDPs for Generting nd Simulting Stock Investment Policies Augusto Cesr Espíndol Bff UNIRIO  Dep. Informátic Aplicd Av. Psteur, 458  Térreo Rio de Jneiro  Brzil ugusto.bff@uniriotec.br Angelo
More informationRational Functions. Rational functions are the ratio of two polynomial functions. Qx bx b x bx b. x x x. ( x) ( ) ( ) ( ) and
Rtionl Functions Rtionl unctions re the rtio o two polynomil unctions. They cn be written in expnded orm s ( ( P x x + x + + x+ Qx bx b x bx b n n 1 n n 1 1 0 m m 1 m + m 1 + + m + 0 Exmples o rtionl unctions
More information2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
More informationOstrowski Type Inequalities and Applications in Numerical Integration. Edited By: Sever S. Dragomir. and. Themistocles M. Rassias
Ostrowski Type Inequlities nd Applictions in Numericl Integrtion Edited By: Sever S Drgomir nd Themistocles M Rssis SS Drgomir) School nd Communictions nd Informtics, Victori University of Technology,
More information3 The Utility Maximization Problem
3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best
More informationHomework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule.
Text questions, Chpter 5, problems 15: Homework #4: Answers 1. Drw the rry of world outputs tht free trde llows by mking use of ech country s trnsformtion schedule.. Drw it. This digrm is constructed
More informationCOMPONENTS: COMBINED LOADING
LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of
More informationSmall Business Networking
Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer
More informationPHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
More informationDiscovering General Logical Network Topologies
Discovering Generl Logicl Network Topologies Mrk otes McGill University, Montrel, Quebec Emil: cotes@ece.mcgill.c Michel Rbbt nd Robert Nowk Rice University, Houston, TX Emil: {rbbt, nowk}@rice.edu Technicl
More information