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2 Journal of Multvarate Analyss Extreme naccuracs n Gaussan Bayesan networks Mguel A. Gómez-Vllegas a,, Paloma Maín a, Rosaro Sus b a Departamento de Estadístca e Investgacón Operatva, Unversdad Complutense de Madrd, 8040 Madrd, Span b Departamento de Estadístca e Investgacón Operatva III, Unversdad Complutense de Madrd, 8040 Madrd, Span Receved 9 March 007 Avalable onlne 19 February 008 Abstract To evaluate the mpact of model naccuracs over the network s output, after the evdence propagaton, n a Gaussan Bayesan network, a senstvty measure s ntroduced. Ths senstvty measure s the Kullback Lebler dvergence and ylds dfferent expressons dependng on the type of parameter to be perturbed,.e. on the naccurate parameter. In ths work, the behavor of ths senstvty measure s studd when model naccuracs are extreme,.e. when extreme perturbatons of the parameters can exst. Moreover, the senstvty measure s evaluated for extreme stuatons of dependence between the man varables of the network and ts behavor wth extreme naccuracs. Ths analyss s performed to fnd the effect of extreme uncertanty about the ntal parameters of the model n a Gaussan Bayesan network and about extreme values of evdence. These deas and procedures are llustrated wth an example. c 008 Elsevr Inc. All rghts reserved. AMS 000 subject classfcatons: 6F15; 6F35 Keywords: Gaussan Bayesan network; Senstvty analyss; Kullback Lebler dvergence Introducton A Bayesan network s a probablstc graphcal model where a drected acyclc graph DAG represents a set of varables wth ts dependence structure. The nodes are random varables and the edges gve dependencs between the varables. A set of condtonal dstrbutons of each varable, gven ts parents, completes the jont descrpton of the varables. Correspondng author. E-mal address: ma.gv@mat.ucm.es M.A. Gómez-Vllegas X/$ - see front matter c 008 Elsevr Inc. All rghts reserved. do: /j.jmva

3 1930 M.A. Gómez-Vllegas et al. / Journal of Multvarate Analyss Bayesan networks have been studd by some authors, lke Pearl [1], Laurtzen [], Heckerman [3] and Jensen [4], among others. Dependng on the type of varables n the problem, dscrete, Gaussan and mxed Bayesan networks can be descrbed. When a Bayesan network s consdered, t s necessary to assgn dfferent values to the probablts of the network. In ths step, there can be some uncertanty and then there can exst some naccurate parameters. Therefore, t s necessary to develop a senstvty analyss to study how senstve the network s output s to ntal naccurate parameters. In the last few years, some methods of senstvty analyss for Bayesan networks have been developed; for example, for dscrete Bayesan networks Laskey [5] presents a senstvty analyss based on computng the partal dervatve of a posteror margnal probablty wth respect to a gven parameter, Coupé and van der Gaag [6] developed an effcnt senstvty analyss based on nference algorthms, and Chan and Darwche [7] ntroduced a senstvty analyss based on a dstance measure. For Gaussan Bayesan networks, a senstvty analyss based on symbolc propagaton was developed by Castllo and Kjærulff [8], and on the bass of the Kullback Lebler dvergence, Gómez-Vllegas, Maín and Sus [9] proposed a senstvty measure for performng the senstvty analyss. In ths paper, we work wth the senstvty measure presented by Gómez-Vllegas, Maín and Sus [9] to study ts behavor for extreme naccuracs or perturbatons of the parameters that descrbe the Gaussan Bayesan network. Moreover, we descrbe the senstvty measure and ts behavor for extreme naccuracs, when the dependence between the varable of nterest X and the evdental varable X e s extreme, that s, when the squared coeffcnt of correlaton between X and X e s gven by ρ = 0 and ρ = 1, consderng dfferent types of relatve postonng of these varables n the DAG. Ths paper s organzed as follows. In Secton 1 defntons of Bayesan networks and Gaussan Bayesan networks are ntroduced and the process of evdence propagaton n Gaussan Bayesan networks s revwed. Also, we present the workng example. In Secton, the senstvty measure s defned and, dependng on the parameter to be perturbed, dfferent expressons of the senstvty measure are obtaned, beng able to detect the parameter that perturbs the network s output the most. In Secton 3, we study the behavor of the senstvty measure for extreme perturbatons of the ntal parameters. In Secton 4, we evaluate the senstvty measure consderng extreme stuatons of dependence between X and X e gven by the lnear correlaton coeffcnt ρ and evaluate those cases wth extreme perturbatons of the parameters. In Secton 5, the senstvty analyss wth the workng example s performed for some extreme parameter perturbatons consderng dfferent postons of the varable of nterest X and the evdental varable X e. Fnally, n Secton 6 some conclusons are presented. 1. Gaussan Bayesan networks and evdence propagaton A Bayesan network and a Gaussan Bayesan network are defned n ths secton and the process of evdence propagaton, one of the most mportant results n Bayesan networks, s presented. Moreover, these concepts are llustrated wth an example. Defnton 1. A Bayesan network s a par G, P where G s a DAG, nodes beng random varables X = {X 1,..., X n } and edges probablstc dependencs between varables, and P = {px 1 pax 1,..., px n pax n } s a set of condtonal probablty densts one for each varable wth pax the set of parents of node X n G.

4 M.A. Gómez-Vllegas et al. / Journal of Multvarate Analyss The jont probablty densty of X s n px = px pax. =1 1 Defnton. A Gaussan Bayesan network s a Bayesan network over X = {X 1,..., X n } where the jont probablty dstrbuton s{ a multvarate normal dstrbuton } Nµ, Σ, wth densty gven by f x = π n/ Σ 1/ exp 1 x µ Σ 1 x µ where µ s the n-dmensonal mean vector, Σ the postve defnte covarance matrx of dmenson n n and Σ the determnant of Σ. In a Gaussan Bayesan network, the condtonal densty assocated wth X for = 1,..., n n Eq. 1 s the unvarate normal dstrbuton, wth densty f x pax Nµ + 1 j=1 β jx j µ j, ν where β j s the regresson coeffcnt of X j n the regresson of X on the parents of X wth pax {X 1,..., X 1 }, and ν = Σ Σ Pax Σ 1 Pax Σ Pax s the condtonal varance of X gven ts parents n the DAG. It s usual to work wth a varable of nterest X, so the network s output s the nformaton about ths varable of nterest after the evdence propagaton,.e., the posteror margnal densty of X. The evdence propagaton s one of the man results assocated wth Bayesan networks and s the process of updatng the probablty dstrbuton of the varables of the network ntroducng the avalable nformaton about the state of one or more varables, known as evdence varables. Dfferent algorthms have been proposed for propagatng the evdence n Bayesan networks. To perform the evdence propagaton n a Gaussan Bayesan network an ncremental method s presented, updatng one evdental varable at a tme [10]. Ths method s based on computng the condtonal probablty densty of a multvarate normal dstrbuton gven the evdental varable X e. Then, consderng the partton X = Y, E, wth Y the set of non-evdental varables, X Y beng the varable of nterest and E the evdental varable, the condtonal probablty dstrbuton of Y, gven the evdence E = {X e = e}, s a multvarate normal dstrbuton wth parameters µ Y E=e = µ Y + Σ YE Σ 1 E E e µ E Σ Y E=e = Σ YY Σ YE Σ 1 E E Σ EY. Workng wth a varable of nterest X Y, after the evdence propagaton, we obtan that X E = e Nµ Y E=e, σ Y E=e N µ + σ e µ e, σ σ wth the parameters before the evdence propagaton µ and µ e the means of X and X e, σ and the varances of X and X e, and σ the covarance of X and X e. To llustrate these concepts of Gaussan Bayesan networks and evdence propagaton, the followng example s ntroduced. Example 3. The nterest of the problem s n how a machne works. Ths machne s made up of fve elements, the varables of the problem, connected as the DAG presented n Fg. 1. The element of nterest n the machne s the last one X 5.

5 193 M.A. Gómez-Vllegas et al. / Journal of Multvarate Analyss Fg. 1. DAG of the Gaussan Bayesan network. Let the tme for whch each element s workng be a normal dstrbuton; then X = {X 1, X, X 3, X 4, X 5 } has a multvarate normal dstrbuton gven by X Nµ, Σ, wth the condtonal parameters descrbed by the DAG µ = 3 4 ; Σ = To obtan Σ, the algorthm presented by Shachter and Kenley [11] s used. Consderng the evdence E = {X = 4}, after evdence propagaton the dstrbuton of the varable of nterest s X 5 X = 4 N6, 4 and the jont probablty dstrbuton s a multvarate normal wth parameters µ Y X = 4 4 ; Σ Y X = Senstvty analyss n Gaussan Bayesan networks The senstvty analyss proposed by Gómez-Vllegas, Maín and Sus [9] conssts n comparng the network output of two dfferent models: the orgnal model Nµ, Σ and the perturbed model obtaned after addng a perturbaton δ R to one naccurate parameter of the model. Wth an evdental varable X e, whose value s known: E = {X e = e}, the evdence propagaton s performed n the orgnal model and n the perturbed model, obtanng the network s output as the margnal densts of nterest f x e, for the orgnal model, and f x e, δ, for the perturbed model. Fnally, to evaluate the effect of addng a perturbaton, the senstvty measure s used. Ths measure s the Kullback Lebler dscrepancy [1] used to compare these condtonal densts of the varable of nterest after the evdence propagaton. The Kullback Lebler dvergence for f and f densty functons over the same doman s defned as follows: K L f w, f w = + f w ln f w f w dw.

6 M.A. Gómez-Vllegas et al. / Journal of Multvarate Analyss Ths measure has been used from the begnnng n statstcal nference by Jeffreys, Fsher and Lndley. It takes nto account the whole behavor of the dstrbutons to be consdered. Furthermore, t s very useful when there s no dea about whch properts of the varable of nterest wll be used. The senstvty measure s defned as follows. Defnton 4. Let G, P be a Gaussan Bayesan network Nµ, Σ. Let f x e be the margnal densty of nterest after evdence propagaton and f x e, δ the same densty when the perturbaton δ s added to one parameter of the ntal model. Then, the senstvty measure s defned by S p j f x e, f x e, δ = + f x e ln f x e f x e, δ dx 3 where the subscrpt p j s the naccurate parameter and δ the proposed perturbaton, beng the new value of the parameter p δ j = p j + δ. For small values of the senstvty measure t s possble to conclude that the Gaussan Bayesan network s robust aganst the knd of perturbaton proposed. Consderng the parameters of the mean vector and the parameters of the covarance matrx as dfferent, and havng ρ 0, 1, the followng results are obtaned. These results are generally true for condtonal dstrbutons n the case of a jont multvarate normal dstrbuton; however ther full meanng s reached for Gaussan Bayesan Networks. In fact the condtonal dstrbuton, gven some evdence, s the output n ths knd of representaton. Snce the man pont s to carry out a senstvty analyss, the effect of perturbatons on the condtonal dstrbutons has to be studd..1. Mean vector naccuracy Dependng on the element of µ to be perturbed, the perturbaton can affect the mean of the varable of nterest X Y, the mean of the evdental varable X e E or the mean of any other varable X j Y wth j. Proposton 5. Consderng the perturbaton δ R added to any element of the mean vector µ, and havng ρ 0, 1, the senstvty measure 3 s as follows: When the perturbaton s added to the mean of X, then µ δ = µ + δ; the densty of the varable of nterest after the evdence propagaton s X E = e, δ Nµ Y E=e + δ, σ Y E=e, where S µ f x e, f x e, δ = δ σ Y E=e. If the perturbaton s added to the mean of the evdental varable, µ δ e = µ e + δ; the posteror densty of the varable of nterest after the evdence propagaton s X E = e, δ N µ Y E=e σ δ, σ Y E=e, and δ σ. S µ e f x e, f x e, δ = σ Y E=e

7 1934 M.A. Gómez-Vllegas et al. / Journal of Multvarate Analyss The perturbaton δ added to the mean of any other non-evdental varable, dfferent from the varable of nterest, has no nfluence over X ; then f x e, δ = f x e and the senstvty measure s zero. When the evdence about X e s naccurate, wth e δ = e + δ, the senstvty measure obtaned s smlar to S µ e f x e, f x e, δ. Therefore, ths case s studd when the mean of the evdental varable s naccurate... Covarance matrx naccuracy When the covarance matrx s perturbed, the structure of the network can change. These changes are n the precson matrx of the perturbed network,.e., n the nverse covarance matrx consderng the perturbaton δ. To guarantee the normalty of the network, Σ δ and Σ Y E=e,δ must be postve defnte matrces; ths restrcton, mposed n the followng proposton, ylds dfferent constrants for the perturbaton δ. Proposton 6. Addng the perturbaton δ R to the covarance matrx Σ, wth ρ senstvty measure 3 s as follows: 0, 1, the If the perturbaton s added to the varance of the varable of nterest, then σ δ = σ +δ wth δ > σ + σ and after the evdence propagaton X E = e, δ N µ Y E=e, σ Y E=e,δ where σ Y E=e,δ = σ + δ σ and [ ] S σ f x e, f x e, δ = 1 δ δ ln 1 +. σ Y E=e σ Y E=e,δ When the perturbaton s added to the varance of the evdental varable, wth σee δ = +δ and δ > 1 max X j Y ρ je wth ρ je the correspondng correlaton coeffcnt, the posteror densty of nterest s X E = e, δ N µ Y E=e,δ, σ Y E=e,δ wth µ Y E=e,δ = µ + σ +δ e µ e and σ Y E=e,δ = σ σ +δ and the senstvty measure s gven by S f x e, f x e, δ = 1 Y E=e,δ σ ln σ Y E=e + σ δ +δ 1 + e µ e δ σ Y E=e,δ +δ The perturbaton δ added to the varance of any other non-evdental varable X j Y wth j, wth σ j δ j = σ j j + δ, has no nfluence over X ; then f x e, δ = f x e and the senstvty measure s zero. v When the covarance of the varable of nterest X and the evdental varable X e s perturbed, σ δ = σ + δ = σe δ and, consderng the restrcton over δ gven as σ σ < δ < σ + σ, then X E = e, δ N µ Y E=e,δ µ Y E=e,δ., σ Y E=e,δ wth = µ + σ +δ e µ e and σ Y E=e,δ = σ σ +δ. The senstvty measure

8 M.A. Gómez-Vllegas et al. / Journal of Multvarate Analyss obtaned s S σ f x e, f x e, δ = 1 ln 1 δ + σ δ σ Y E=e + σ Y E=e + δ e µ e σ Y E=e,δ 1. v The perturbaton δ added to any other covarance,.e., of X and any other non-evdental varable X j, or of the evdence varable X e and X j Y wth j, has no nfluence over the varable of nterest; then f x e, δ = f x e and the senstvty measure s zero. The proof and more detals about the senstvty analyss proposed wth an example can be seen n Gómez-Vllegas, Maín and Sus [9]. 3. Extreme behavor of the senstvty measure To know the effect of extreme uncertanty about the ntal parameters of the network, we study the behavor of the senstvty measure for extreme perturbatons wth the lmt of the senstvty measure. In ths case, the squared correlaton coeffcnt of X and X e consdered s ρ 0, 1. Proposton 7. When the perturbaton added to the mean vector s extreme and ρ senstvty measure s extreme too and t s: a lm δ ± S µ f x e, f x e, δ = +, b lm δ 0 S µ f x e, f x e, δ = 0; a lm δ ± S µ e f x e, f x e, δ = +, b lm δ 0 S µ e f x e, f x e, δ = 0. 0, 1, the Proof. If follows drectly. Therefore the lmt for an extreme value of the evdence e corresponds to the case. Proposton 8. When the extreme perturbaton s added to the elements of the covarance matrx and ρ 0, 1, the senstvty measure s as follows: a lm δ + S σ f x e, f x e, δ = + but S σ f x e, f x e, δ = oδ, b lm δ M S σ f x e, f x e, δ = + wth M = σ + σ = σ 1 ρ, c lm δ 0 S σ f x e, f x e, δ = 0; [ a lm δ + S f x e, f x e, δ = 1 ln 1 ρ ρ 1 e µ e ], b lm S f x e, f x e, δ δ M ee [ = 1 M ln ee ρ Mee 1 ρ + ρ 1 M ee 1 Mee + e µ e 1 M ] ee ρ σ ee Mee where M ee = 1 Mee and M ee = max X j Y ρ je ; f M ee = ρ then lm δ M ee S f x e, f x e, δ = +, c lm δ 0 S f x e, f x e, δ = 0; a lm δ M 1 S σ f x e, f x e, δ = + wth M 1 = σ σ,

9 1936 M.A. Gómez-Vllegas et al. / Journal of Multvarate Analyss b lm δ M S σ f x e, f x e, δ = + wth M = σ + σ, c lm δ 0 S σ f x e, f x e, δ = 0. Proof. a and c follow drectly. b When σ δ = σ + δ the new varance of X s σ Y E=e,δ = σ Y E=e + δ. Consderng σ Y E=e,δ > 0 then δ > σ Y E=e. Defnng M = σ Y E=e and wth x = σ Y E=e + δ, we have lm S σ f x e, f x e, δ δ M 1 = lm x 0 [ ln x ln σ Y E=e x σ Y E=e x a lm δ + S f x e, f x e, δ = 1 ln σ Y E=e = σ 1 ρ and ρ = σ σ ; then lm δ + S f x e, f x e, δ = 1 [ ln ] = +. σ σ Y E=e + 1 ρ ρ σ 1 e µ e σee σee σ wth 1 e µ e ] b When σ δ ee = + δ, the new condtonal varance for all non-evdental varables s σ Y E=e,δ j j = σ j j σ je Y E=e,δ +δ. If t s mposed that σ j j > 0 for all X j Y then δ must satsfy followng condton: δ > 1 max X j Y ρ je. Defnng Mee = max X j Y ρ je and M ee = 1 Mee, lm S f x e, f x e, δ δ M ee 1 = lm ln σ σ +δ + δ M ee = 1 = 1 ln [ ln σ σ σ M ee σ M ee σ σ M ee ρ M ee 1 ρ + σ δ +δ 1 + e µ e δ σ 1 M ee Mee σ Mee + ρ 1 M ee M ee ρ σ σ 1 + e µ e +δ 1 M ee Mee +δ Mee ρ 1 + e µ e 1 M ] ee. M ee If Mee = ρ 0 then M ee = 1 ρ = + σ σ ; t follows that lm S f x e, f x e, δ δ M ee [ M ee ρ 1 = lm ln δ M ee Mee 1 ρ 1 + e µ e 1 M ee M ee + ρ 1 M ee Mee ρ ] = +

10 M.A. Gómez-Vllegas et al. / Journal of Multvarate Analyss because for K, M > 0 [ ln x M + K ] x lm x 0 x>0 [ 1 = lm [x ln x ] x 0 x M + K 1 ln x = lm M x 0 x 1 x c It follows drectly for the case M ee = 1. a For σ δ = σ + δ, the new condtonal varance s σ Y E=e,δ + K ] = +. = σ σ +δ ; then, f σ Y E=e,δ > 0 s mposed, δ must satsfy the followng condton: σ σ < δ < σ + σ. Frst, defnng M = σ + σ, t s possble to calculate lm δ M S σ f x e, f x e, δ. But δ M s equvalent to δ + σ δ σ Y E=e and gven that S σ f x e, f x e, δ = 1 σee σ Y E=e δ + σ δ σ Y E=e + δ e µ e ln σ Y E=e + σ Y E=e 1 δ + σ δ [ ] and usng lm x 0 ln x + k x = + for every k, then S σ f x e, f x e, δ = +. lm δ M b Analogously to a. c It follows drectly. The results obtaned are the expected ones, except when the extreme perturbaton s added to the evdental varance havng a fnte lmt. Ths s because the state of the evdental varable s known and the varance of ths varable has a lmted effect on the varable of nterest. In ths case, the posteror densty of nterest wth the perturbaton n the model f x e, δ s not so dfferent to the posteror densty of nterest wthout the perturbaton f x e. Therefore, although an extreme perturbaton added to the evdental varance can exst, the senstvty measure tends to a fnte value. 4. Extreme dependence between the varable of nterest and the evdental varable In ths secton we evaluate some partcular cases of the senstvty measure and ts extreme behavor, dependng on the relatons between the varable of nterest X and the evdental varable X e. Then, consderng extreme values for the squared lnear correlaton coeffcnt, for example ρ = 0, X and X e are ndependent and the connecton between these varables n the DAG must be a convergng connecton only ths connecton consders ndependence between the varables. On the other hand, wth ρ = 1, the connecton n the DAG between X and X e can be a seral or dvergng connecton, wth a lnear dependence relaton between X and X e. If X and X e are ndependent, when the evdence propagaton s done, the nformaton about = µ and σ Y E=e = σ. In ths case, only when the perturbaton s added to the parameters of X s the senstvty measure not zero. Moreover, when the covarance between X and X e s perturbed, the relaton between those varables changes to σ = δ and therefore the structure of the network changes. In those cases the senstvty measure s obtaned consderng σ = 0 n Propostons 5 and 6. Any other perturbaton of the parameters of the network does not dsturb the results about X. X e does not affect the varable of nterest; then after the evdence propagaton µ Y E=e

11 1938 M.A. Gómez-Vllegas et al. / Journal of Multvarate Analyss The other extreme case for the squared correlaton coeffcnt s ρ = 1. In ths case, any perturbaton assocated wth the ntal parameters of X or X e changes sgnfcantly the results concernng X ; then the senstvty measure computed n all cases s nfnty. Therefore, for a lnear relaton between the evdental varable and the varable of nterest, and then a maxmum value of ρ, the senstvty measure s also extreme. When X and X e are ndependent, only perturbatons assocated wth the parameters of X can dsturb the network s output, and f the perturbatons are extreme, the senstvty measure s also extreme. However, for ρ = 1, any perturbaton added to the parameters of X or X e greatly affects the network s output, the senstvty measure beng nfnty. 5. Workng example Example 9. Consder the Gaussan Bayesan network n Example 3. Experts dsagree wth the parameters assgned to the jont probablty dstrbuton and they want to quantfy the effect of naccuracy when some extreme perturbatons of the parameters are proposed. Let the mean and the varance of the varable of nterest X 5 be µ δ 5 5 = 0 = µ 5 + δ 5 wth δ 5 = 5 and σ δ = 3 wth δ 55 = 3. Let the mean and the varance of the evdental varable X be gven by µ δ = 30 = µ +δ wth δ = 7 and σ δ = 0.7 wth δ = Fnally, let σ δ 5 5 = 3 wth δ 5 = 1. The senstvty measure ylded for these naccuracy parameters s S µ 5 f x 5 X = 4, f x 5 X = 4, δ 5 = 13.0 S σ 55 f x 5 X = 4, f x 5 X = 4, δ 55 = 9.91 S µ f x 5 X = 4, f x 5 X = 4, δ = S σ f x 5 X = 4, f x 5 X = 4, δ =.03 where.113 s the lmt of the senstvty measure when δ tends to M ee ; S σ 5 f x 5 X = 4, f x 5 X = 4, δ 5 = It can be ponted out that although there can exst more perturbed parameters, those naccuracs do not dsturb the network s output. To show the behavor of the measure, we present the senstvty measures as a functon of the perturbaton δ n Fg.. In ths case, t can be noted that the senstvty measure of the mean of X s the same as the senstvty measure of the mean of X e, because of the ntal parameters that have been chosen. As can be seen n the example, the senstvty measure grows when the perturbaton s large. When the evdental varance s perturbed t s necessary to compute the lmt of the measure to know whether the perturbaton proposed s also large. Example 10. Consder the Gaussan Bayesan network gven n Fg. 1, the varable of nterest beng X 1, wth the same evdental varable X. The varables X 1 and X are n a convergng connecton n the graph, beng ndependent varables. In ths case, f any parameter dfferent to µ 1, σ 11 or σ 1 s perturbed, ths naccuracy does not affect the posteror densty of X 1 ; then the senstvty measure s zero. However, f µ 1, σ 11 or σ 1 are perturbed, the senstvty measure s dfferent to zero. Therefore, n ths case t s mportant to be very careful n assgnng the parameters of the varable of nterest X 1.

12 M.A. Gómez-Vllegas et al. / Journal of Multvarate Analyss Fg.. Senstvty measures obtaned n the example for any perturbaton. 6. Conclusons In ths paper the behavor of the senstvty measure ntroduced by Gómez-Vllegas, Maín and Sus [9] has been studd. Ths measure s useful for evaluatng the mpact of parameter naccuracs n a Gaussan Bayesan network over the densty of nterest after the evdence propagaton, when the naccuracs are extreme. Wth ths analyss t s possble to prove that ths s a well-defned measure for developng a senstvty analyss n a Gaussan Bayesan network even f the proposed perturbatons are extreme. The results obtaned are the expected ones. When the evdental varance s naccurate wth a large perturbaton assocated wth ths value, there exsts a fnte value as the lmt of the senstvty measure. Ths s because the evdence about ths varable explans the behavor of the varable of nterest regardless of ts naccurate varance. Moreover, n all possble cases for the senstvty measure, f the perturbaton added to a parameter s very small, tendng to zero, the senstvty measure s also zero. Furthermore, t s possble to evaluate the senstvty measure n some partcular cases dependng on the dependence relaton between the varable of nterest X and the evdental varable X e. In that way, ρ 0, 1 and also extreme values of the lnear squared correlaton coeffcnt ρ = 0 and ρ = 1 are consdered, havng dfferent types of connectons n the DAG. If there s no relaton between X and X e, the connecton n the graph s a convergng connecton. Therefore the evdence and any perturbatons added to parameters of the evdental varable X e do not affect the nformaton about the varable of nterest X. In ths case, the senstvty measure s dfferent to zero only when the parameters of X are naccurate. For a lnear relaton between X and X e, wth ρ = 1, havng a seral or dvergng connecton n the DAG, the senstvty measure s nfnty because any perturbaton about X or X e produces an mportant effect over the network s output. Therefore, f X e s so related to X, then any perturbaton added to the parameters that descrbe these varables makes the senstvty measure extreme.

13 1940 M.A. Gómez-Vllegas et al. / Journal of Multvarate Analyss Acknowledgments Ths research was supported by the MEC from Span, Grant MTM , and the Unversdad Complutense-Comundad de Madrd, Grant UCM References [1] J. Pearl, Probablstc reasonng n ntellgent systems, n: Networks of Plausble Inference, Morgan Kaufmann, Palo Alto, [] S.L. Laurtzen, Graphcal Models, Clarendon Press, Oxford, [3] D. Heckerman, A tutoral on learnng wth Bayesan networks, n: M.I. Jordan Ed., Learnng n Graphcal Models, MIT Press, Cambrdge, MA, [4] F.V. Jensen, Bayesan Networks and Decson Graphs, Sprnger, New York, 001. [5] K.B. Laskey, Senstvty analyss for probablty assessments n Bayesan networks, IEEE Transactons on Systems, Man, and Cybernetcs [6] V.M.H. Coupé, L.C. van der Gaag, Properts of senstvty analyss of Bayesan belf networks, Annals of Mathematcs and Artfcal Intellgence [7] H. Chan, A. Darwche, A dstance measure for boundng probablstc belf change, Internatonal Journal of Approxmate Reasonng [8] E. Castllo, U. Kjærulff, Senstvty analyss n Gaussan Bayesan networks usng a symbolc numercal technque, Relablty Engneerng and System Safety [9] M.A. Gómez-Vllegas, P. Maín, R. Sus, Senstvty analyss n Gaussan Bayesan networks usng a dvergence measure, Communcatons n Statstcs Theory and Methods [10] E. Castllo, J.M. Gutérrez, A.S. Had, Expert Systems and Probablstc Network Models, Sprnger-Verlag, New York, [11] R. Shachter, C. Kenley, Gaussan nfluence dagrams, Management Scnce [1] S. Kullback, R.A. Lebler, On nformaton and suffcncy, Annals of Mathematcal Statstcs