%%Creator: %!PS-Adobe-2.0 dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: toftp.dvi %%CreationDate: Wed May 31 08:43:54 1995 %%Pages: 30 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o kjaerulff95b.ps toftp %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1995.05.31:0842 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7
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(w)n(een)g(pairs)f(of)g(sets)h Fs(X)3040 4001 y Fm(A)3094 3989 y Fs(;)14 b(X)3200 4001 y Fm(B)3288 3989 y Fr(\022)31 b Fs(X)3453 4001 y Fm(V)3544 3989 y FF(of)150 4089 y(v)-5 b(ariables)27 b(giv)n(en)h(a)g(third)h(set)g Fs(X)1192 4101 y Fm(C)1247 4089 y FF(.)40 b(W)-7 b(e)29 b(shall)g(use)f(the)h (notation)f Fs(A)d Fr(?)-42 b(?)25 b Fs(B)13 b Fr(j)c Fs(C)35 b FF(for)28 b(an)n(y)g(triple)h(\()p Fs(A;)14 b(B)t(;)g(C)6 b FF(\))30 b(of)150 4188 y(disjoin)n(t)24 b(subsets)f(of)g Fs(V)43 b FF(for)23 b(whic)n(h)g Fs(X)1331 4200 y Fm(A)1409 4188 y FF(and)g Fs(X)1635 4200 y Fm(B)1715 4188 y FF(are)g(conditionally)g(indep)r(enden)n(t)h(giv)n(en)f Fs(X)3090 4200 y Fm(C)3169 4188 y FF(with)h(resp)r(ect)150 4288 y(to)j Fs(p)p FF(.)275 4388 y(All)g(conditional)f(indep)r(endence) h(statemen)n(ts)f(captured)h(b)n(y)f Fr(G)32 b FF(can)26 b(b)r(e)h(found)g(using)g(the)g(d-separation)150 4487 y(criterion)j(of)g(P)n(earl)f(\(1988\))g(or)h(the)h(equiv)-5 b(alen)n(t)30 b(criterion)g(of)g(Lauritzen)g(et)h(al.)f(\(1990\))g (expressed)f(b)n(y)h(the)150 4587 y(follo)n(wing)d(theorem.)150 4786 y FG(Theorem)j(1)42 b([Lauritzen)29 b(et)f(al.)g(\(1990\)])e FD(L)l(et)35 b Fs(p)27 b FD(factorize)i(ac)l(c)l(or)l(ding)f(to)f(a)h (D)n(A)n(G,)f Fr(G)h FF(=)23 b(\()p Fs(V)5 b(;)14 b(E)5 b FF(\))p FD(.)38 b(Then)150 4886 y Fs(A)31 b Fr(?)-42 b(?)30 b Fs(B)13 b Fr(j)c Fs(C)41 b FD(with)35 b(r)l(esp)l(e)l(ct)e(to) h Fs(p)g FD(for)h(any)f(subsets)40 b Fs(A;)14 b(B)t(;)g(C)37 b Fr(\022)30 b Fs(V)53 b FD(whenever)44 b Fs(C)c FD(sep)l(ar)l(ates)h Fs(A)35 b FD(fr)l(om)40 b Fs(B)e FD(in)150 4985 y FF(\()p Fr(G)231 5000 y Fi(A)n(n)r Fk(\()p Fm(A)p Fl([)p Fm(B)s Fl([)p Fm(C)t Fk(\))619 4985 y FF(\))651 4955 y Fm(m)714 4985 y FD(.)150 5151 y FF(See)31 b(Lauritzen)f(et)h(al.)g(\(1990\))e (for)h(a)h(pro)r(of.)46 b(The)31 b(prop)r(ert)n(y)e(describ)r(ed)i(in)g (Theorem)f(1)g(is)h(referred)f(to)g(as)150 5251 y(the)h FD(glob)l(al)j(Markov)h(pr)l(op)l(erty)p FF(.)48 b(W)-7 b(e)31 b(also)f(sa)n(y)g(that)h Fs(p)g FF(is)f(Mark)n(o)n(v)f(with)i (resp)r(ect)g(to)g Fr(G)36 b FF(when)31 b Fs(p)g FF(factorizes)150 5351 y(according)26 b(to)h Fr(G)5 b FF(.)1860 5593 y(6)p eop %%Page: 7 8 7 7 bop 275 498 a FF(If)27 b(a)f(probabilit)n(y)g(function)i Fs(p)f FF(is)f(Mark)n(o)n(v)f(with)i(resp)r(ect)g(to)g(a)f(D)n(A)n(G,)i Fr(G)g FF(=)23 b(\()p Fs(V)5 b(;)14 b(E)5 b FF(\),)27 b(then)h(there)f(exists)f(a)150 598 y FD(p)l(otential)k(r)l(epr)l (esentation)1251 709 y Fs(p)p FF(\()p Fs(x)p FF(\))24 b(=)e Fs(z)1558 675 y Fl(\000)p Fk(1)1647 709 y Fs(\036)p FF(\()p Fs(x)p FF(\))i(=)f Fs(z)1962 675 y Fl(\000)p Fk(1)2079 631 y Fn(Y)2064 809 y Fm(C)t Fl(2C)2213 709 y Fs(\036)2262 721 y Fm(C)2319 709 y FF(\()p Fs(x)2398 721 y Fm(C)2454 709 y FF(\))p Fs(;)996 b FF(\(2\))150 919 y(of)31 b Fs(p)p FF(,)i(where)e Fs(\036)639 931 y Fm(C)727 919 y FF(is)g(a)g(non-negativ)n(e)f(function)i(for)f(eac)n(h)g Fs(C)38 b FF(in)31 b(a)g(set)h Fr(C)k FF(of)31 b(complete)h(subsets)f (of)h Fs(V)18 b FF(;)34 b(see,)150 1019 y(for)g(example,)i(Lauritzen)e (&)g(Spiegelhalter)g(\(1988\).)56 b(The)35 b Fs(\036)2161 1031 y Fm(C)2217 1019 y FF('s)g(are)e(called)h FD(b)l(elief)k(p)l (otentials)7 b FF(,)37 b(and)d Fs(z)k FF(a)150 1118 y FD(normalization)31 b(c)l(onstant)p FF(.)36 b(In)28 b(particular,)e (the)i(pro)r(duct)f(in)h(Equation)e(\(1\))i(is)f(a)g(p)r(oten)n(tial)g (represen)n(tation)150 1218 y(with)h(normalization)e(constan)n(t)h(1.) 150 1517 y Fo(2.3)99 b(Junction)35 b(tr)-5 b(e)g(es)150 1670 y FF(By)23 b(exploiting)f(the)h(conditional)g(indep)r(endence)g
(relations)f(among)g(the)h(v)-5 b(ariables)22 b(of)h(a)f(Ba)n(y)n (esian)f(net)n(w)n(ork,)150 1769 y(the)k(underlying)f(join)n(t)h (probabilit)n(y)e(space)h(ma)n(y)g(b)r(e)h(decomp)r(osed)f(in)n(to)g(a) g(set)h(of)f(subspaces)g(corresp)r(onding)150 1869 y(to)29 b(a)f(decomp)r(osable)g(\(h)n(yp)r(ergraph\))g(co)n(v)n(er)f(of)i(the)g (moralized)e(graph)h(suc)n(h)h(that)g(exact)f(inference)h(can)f(b)r(e) 150 1969 y(p)r(erformed)33 b(through)g(a)h(simple)g(message-passing)c (sc)n(heme)k(in)g(a)f(maximal)g(spanning)g(tree)h(of)f(the)i(co)n(v)n (er)150 2068 y([Spiegelhalter)h(\(1986\),)j(Lauritzen)d(&)h (Spiegelhalter)f(\(1988\),)i(Jensen)f(\(1988\),)h(Jensen)f(et)g(al.)g (\(1990\)].)150 2168 y(T)-7 b(ec)n(hnically)g(,)28 b(a)g(decomp)r (osable)f(co)n(v)n(er)f(of)i(a)g(Ba)n(y)n(esian)e(net)n(w)n(ork)h(with) i(underlying)e(D)n(A)n(G)i Fr(G)k FF(is)c(created)e(b)n(y)150 2268 y(triangulating)c Fr(G)699 2237 y Fm(m)787 2268 y FF(\(i.e.,)j(adding)e(undirected)h(links,)g(so-called)e FD(\014l)t(l-ins)p FF(,)j(to)e Fr(G)2632 2237 y Fm(m)2720 2268 y FF(to)h(mak)n(e)e(it)i(triangulated\).)150 2367 y(That)j(is,)f(the)h(set)g(of)f(cliques)h(of)f(the)h(triangulated)f (graph)f(is)i(a)f(decomp)r(osable)f(co)n(v)n(er)g(of)i(the)g(net)n(w)n (ork.)275 2467 y(Jensen)j(\(1988\))f(has)i(sho)n(wn)f(that)h(an)n(y)f (maximal)g(spanning)h(tree)f(of)h(a)g(decomp)r(osable)e(co)n(v)n(er,)h Fr(C)5 b FF(,)33 b(can)150 2566 y(b)r(e)25 b(used)f(as)g(the)g(basis)g (for)g(a)g(simple)g(in)n(w)n(ard/out)n(w)n(ard)d(\(or)j (collect/distribute\))g(message-passing)d(sc)n(heme)150 2666 y(for)k(propagation)d(of)k(evidence)e(\(b)r(elief)j(up)r(dating\)) e(in)g(Ba)n(y)n(esian)f(net)n(w)n(orks,)f(where)i(maximalit)n(y)f(is)h (de\014ned)150 2766 y(in)38 b(terms)g(of)g(the)g(sum)g(of)g (cardinalities)f(of)g(the)i(in)n(tersections)e(b)r(et)n(w)n(een)g (adjacen)n(t)h(no)r(des)g(in)g(the)g(tree.)150 2865 y(Jensen)28 b(\(1988\))f(named)h(these)g(trees)g FD(junction)i(tr)l(e)l(es)p FF(.)38 b(The)29 b(no)r(des)f(of)g(a)g(junction)h(tree)f(are)f(called)h FD(\(b)l(elief)9 b(\))150 2965 y(universes)35 b FF(and)28 b(corresp)r(onds)e(to)i(the)g(cliques)g(of)f(the)i(asso)r(ciated)d (triangulated)h(graph;)g(the)i(in)n(tersections)150 3065 y(b)r(et)n(w)n(een)f(neigh)n(b)r(ouring)e(univ)n(erses)g(of)i(a)f (junction)h(tree)f(are)g(called)g FD(sep)l(ar)l(ators)36 b FF([Jensen)27 b(et)h(al.)f(\(1990\)].)275 3164 y(W)-7 b(e)25 b(shall)h(henceforth)f(refer)g(to)g(a)h(junction)g(tree)f(b)n(y) g(the)h(pair)f(\()p Fr(C)5 b Fs(;)14 b(S)5 b FF(\))25 b(of)h(univ)n(erses)e(and)i(separators.)33 b(It)150 3264 y(can)d(b)r(e)h(sho)n(wn)f(that)h(for)f(eac)n(h)g(path)g Fr(h)p Fs(C)35 b FF(=)27 b Fs(C)1652 3276 y Fk(1)1690 3264 y Fs(;)14 b(:)g(:)g(:)g(;)g(C)1934 3276 y Fm(k)2003 3264 y FF(=)27 b Fs(D)r Fr(i)k FF(in)g(a)f(junction)h(tree,)g Fs(C)c Fr(\\)21 b Fs(D)30 b Fr(\022)d Fs(C)3334 3276 y Fm(i)3393 3264 y FF(for)j(all)150 3364 y(1)f Fr(\024)g Fs(i)g Fr(\024)g Fs(k)s FF(,)k(whic)n(h)e(implies)h(that)g Fs(A)d Fr(?)-42 b(?)30 b Fs(B)13 b Fr(j)c Fs(S)36 b FF(for)31 b(eac)n(h)g(separator)e Fs(S)5 b FF(,)32 b(where)f Fs(A)h FF(and)g Fs(B)j FF(are)c(the)h(sets)f(of)150 3463 y(v)-5 b(ariables)28 b(of)g(the)i(t)n(w)n(o)e(subtrees)g(\(except)h Fs(S)5 b FF(\))29 b(induced)h(b)n(y)e(the)i(remo)n(v)-5 b(al)27 b(of)i(the)g(link)g(corresp)r(onding)e(to)i Fs(S)150 3563 y FF([Jensen)e(\(1988\)].)275 3662 y(T)-7 b(o)21 b(eac)n(h)f(univ)n(erse)h(and)g(eac)n(h)g(separator)e(is)j(asso)r (ciated)e(a)h(\(b)r(elief)6 b(\))23 b(p)r(oten)n(tial,)g Fs(\036)2818 3674 y Fm(A)2873 3662 y FF(.)35 b(The)21 b(join)n(t)h(probabil-)150 3762 y(it)n(y)d(distribution,)j Fs(p)p FF(,)f(of)e(a)g(Ba)n(y)n(esian)f(net)n(w)n(ork)g(with)i(a)f
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