(original) (raw)

%!PS-Adobe-2.0 %%Creator: dvips 5.516 Copyright 1986, 1993 Radical Eye Software %%Title: AI-memo.dvi %%CreationDate: Fri Feb 2 14:25:43 1996 %%Pages: 32 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -o AI-memo.ps AI-memo %DVIPSSource: TeX output 1996.02.02:1425 %%BeginProcSet: tex.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 4 get round 4 exch put dup dup 5 get round 5 exch put 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 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(endence)i(rela-)37 717 y(tion)d Fk(A)g Fm(?)g Fk(B)r Fm(j)p Fk(C)i Fw(is)d(de\014ned)h(suc)o(h)f(that)g(that)f Fk(A)g Fw(is)i(indep)q(enden)o(t)h(of)d Fk(B)k Fw(giv)o(en)d Fk(C)s Fw(,)h(i.e.,)g Fk(p)p Fw(\()p Fk(A;)8 b(B)r Fm(j)p Fk(C)s Fw(\))20 b(=)37 775 y Fk(p)p Fw(\()p Fk(A)p Fm(j)p Fk(C)s Fw(\))p Fk(p)p Fw(\()p Fk(B)r Fm(j)p Fk(C)s Fw(\).)e (Conditional)d(indep)q(endence)i(is)d(symmetric.)20 b(Note)13 b(also)h(that)f(marginal)h(indep)q(endence)37 834 y(\(no)21 b(conditioning\))h(do)q(es)g(not)e(in)i(general)f(imply)h(conditional)h (indep)q(endence,)i(nor)c(do)q(es)g(conditional)37 892 y(indep)q(enden)q(ce)d(in)e(general)g(imply)g(marginal)g(indep)q (endence)i(\(Whittak)o(er)d(1990\).)108 955 y(With)h(an)o(y)f(set)h(of) f(random)g(v)m(ariables)i Fl(U)f Fw(w)o(e)f(can)h(asso)q(ciate)g(a)f (graph)g Fk(G)h Fw(de\014ned)h(as)e Fk(G)e Fw(=)h(\()p Fk(V)s(;)8 b(E)s Fw(\).)18 b Fk(V)37 1014 y Fw(denotes)12 b(the)f(set)g(of)g(v)o(ertices)h(or)e(no)q(des)i(of)f(the)g(graph)g 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b(means)j(that)f(the)h(edge)g(is)g(directed)h(from)e(no) q(de)i Fk(i)e Fw(to)37 1368 y(no)q(de)e Fk(j)s Fw(,)d Fk(i)i Fw(is)g(a)f Fn(p)n(ar)n(ent)g Fw(of)h(its)f Fn(child)h Fk(j)s Fw(.)k(An)c Fn(anc)n(estor)f Fw(of)g(no)q(de)h Fk(i)g Fw(is)g(a)f(no)q(de)i(whic)o(h)f(has)g(as)f(a)g(c)o(hild)i (either)g Fk(i)37 1426 y Fw(or)d(another)g(ancestor)g(of)g Fk(i)p Fw(.)19 b(A)14 b(subset)f(of)g(no)q(des)i Fk(A)e Fw(is)h(an)f Fn(anc)n(estr)n(al)h(set)f Fw(if)h(it)f(con)o(tains)h(its) f(o)o(wn)g(ancestors.)37 1484 y(A)j(descendan)o(t)g(of)e Fk(i)h Fw(is)h(a)f(either)h(a)f(c)o(hild)h(of)f Fk(i)g Fw(or)g(a)g(c)o(hild)h(of)f(a)g(descendan)o(t)h(of)f Fk(i)p Fw(.)108 1548 y(Tw)o(o)g(no)q(des)h Fk(i)f Fw(and)g Fk(j)j Fw(are)d Fn(adjac)n(ent)g Fw(in)h Fk(G)g Fw(if)f Fk(E)j Fw(con)o(tains)d(the)h(edge)g Fk(e)p Fw(\()p Fk(i;)8 b(j)s Fw(\).)18 b(A)d Fn(p)n(ath)h Fw(is)g(a)f(sequence)i(of)37 1606 y(distinct)f(no)q(des)g Fm(f)p Fw(1)p Fk(;)8 b(:)g(:)g(:)t(;)g(m)p Fm(g)13 b Fw(suc)o(h)i(that)f(there)h(exists)g(an)g(edge)g(for)f(eac)o (h)h(pair)g(of)f(no)q(des)h Fm(f)p Fk(l)q(;)8 b(l)g Fw(+)i(1)p Fm(g)k Fw(on)g(the)37 1664 y(path.)25 b(A)17 b(graph)g(is)g Fn(singly-c)n(onne)n(cte)n(d)d Fw(if)k(there)f(exists)g(only)g(one)g (path)g(b)q(et)o(w)o(een)g(an)o(y)g(t)o(w)o(o)e(no)q(des)j(in)f(the)37 1722 y(graph.)28 b(A)18 b(cycle)h(is)f(a)g(path)f(suc)o(h)i(the)f(b)q (eginning)i(and)e(ending)h(no)q(des)f(on)g(the)g(path)f(are)h(the)g (same.)27 b(A)37 1780 y(directed)17 b(cycle)f(is)g(a)f(cycle)h(of)f (directed)h(edges)g(whic)o(h)g(all)g(p)q(oin)o(t)g(in)g(the)f(same)g (direction.)108 1844 y(If)k Fk(E)i Fw(con)o(tains)e(only)g(undirected)i (edges)e(then)g(the)g(graph)f Fk(G)h Fw(is)g(an)g Fn(undir)n(e)n(cte)n (d)g(gr)n(aph)h(\(UG\))p Fw(.)d(If)i Fk(E)37 1902 y Fw(con)o(tains)14 b(only)g(directed)h(edges)f(and)f(no)h(directed)h(cycles,)f(then)g Fk(G)f Fw(is)h(an)g Fn(acyclic)g(dir)n(e)n(cte)n(d)g(gr)n(aph)h (\(ADG\))p Fw(.)37 1960 y(If)j Fk(E)h Fw(con)o(tains)e(a)g(mixture)g (of)g(directed)h(and)f(undirected)i(edges,)e(then)h(it)f(is)g(referred) g(to)g(as)f(a)h Fn(mixe)n(d)h(or)37 2019 y(chain)f(gr)n(aph)p Fw(.)23 b(W)l(e)16 b(note)g(in)h(passing)f(that)f(there)h(exists)h(a)e (theory)h(for)f(graphical)i(indep)q(endence)j(mo)q(dels)37 2077 y(in)o(v)o(olving)15 b(mixed)e(graphs)g(\(Whittak)o(er)f(1990\))f (but)i(mixed)h(graphs)f(will)h(not)f(b)q(e)g(discussed)i(further)d(in)i (this)37 2135 y(pap)q(er.)108 2199 y(F)l(or)i(an)h(UG)g Fk(G)p Fw(,)f(a)h(subset)g(of)f(no)q(des)i Fk(C)i Fn(sep)n(ar)n(ates)c Fw(t)o(w)o(o)f(other)i(subsets)g(of)f(no)q(des)i Fk(A)f Fw(and)g Fk(B)i Fw(if)f(ev)o(ery)37 2257 y(path)c(joining)h(ev)o(ery)e (pair)h(of)f(no)q(des)i Fk(i)d Fm(2)h Fk(A)h Fw(and)f Fk(j)i Fm(2)e Fk(B)k Fw(con)o(tains)c(at)g(least)h(one)g(no)q(de)g (from)f Fk(C)s Fw(.)19 b(F)l(or)13 b(ADGs)37 2315 y(and)j(mixed)g (graphs)f(analagous,)f(but)i(somewhat)e(more)h(complicated,)h (separation)f(prop)q(erties)h(exist.)108 2379 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g(passing)g(\015o)o(ws)f(out)g(from)g(this)h(ro)q(ot)f(in)37 249 y(the)h(rev)o(erse)f(direction)i(along)e(the)h(same)f(edges.)20 b(There)c(are)f(at)g(most)f(t)o(w)o(o)g(\015o)o(ws)h(along)g(an)o(y)g (edge)h(in)g(the)37 307 y(tree)f(in)h(a)e(non-redundan)o(t)i(sc)o (hedule.)21 b(Note)15 b(that)f(the)h(directionalit)o(y)i(of)d(the)h (\015o)o(ws)f(in)i(the)f(junction)h(tree)37 365 y(need)h(ha)o(v)o(e)d (nothing)i(to)f(do)g(with)g(an)o(y)g(directed)h(edges)g(in)g(the)f (original)i(DPIN)e(structure.)37 498 y Ff(5.4)56 b(The)19 b(JLO)f(Algorithm)f(for)h(Inference)g(giv)n(en)g(Observ)n(ed)g (Evidence)37 589 y Fw(The)f(particular)g(case)f(of)g(calculating)i(the) e(e\013ect)g(of)f(observ)o(ed)i(evidence)h(\(inference\))f(is)g (handled)g(in)g(the)37 647 y(follo)o(wing)g(manner.)23 b(Consider)17 b(that)f(w)o(e)g(observ)o(e)g(evidence)i(of)e(the)g(form) f Fk(e)g Fw(=)g Fm(f)p Fk(X)1508 654 y Fi(i)1535 647 y Fw(=)g Fk(x)1611 630 y Fd(\003)1611 659 y Fi(i)1631 647 y Fk(;)8 b(X)1690 654 y Fi(j)1721 647 y Fw(=)15 b Fk(x)1797 630 y Fd(\003)1797 659 y Fi(j)1817 647 y Fk(;)8 b(:)g(:)g(:)m Fm(g)37 705 y Fw(and)13 b Fl(U)163 689 y Fi(e)194 705 y Fw(=)g Fm(f)p Fk(X)303 712 y Fi(i)317 705 y Fk(;)8 b(X)376 712 y Fi(j)393 705 y Fk(;)g(:)g(:)g(:)m Fm(g)k Fw(denotes)h(the)g(set)f(of)g(v)m(ariables)i(whic)o(h)f(ha)o(v)o (e)f(b)q(een)i(observ)o(ed.)19 b(Let)12 b Fl(U)1698 689 y Fi(h)1734 705 y Fw(=)h Fl(U)5 b Fm(n)g Fl(U)1895 689 y Fi(e)37 763 y Fw(denote)16 b(the)f(set)g(of)g(hidden)i(or)e(unobserv) o(ed)g(v)m(ariables)i(and)e Fl(u)1133 747 y Fi(h)1171 763 y Fw(a)g(v)m(alue)h(assignmen)o(t)f(for)g Fl(U)1670 747 y Fi(h)1692 763 y Fw(.)108 823 y(Consider)h(the)f(calculation)i(of) e Fk(p)p Fw(\()p Fl(U)739 807 y Fi(h)761 823 y Fm(j)p Fk(e)p Fw(\).)k(De\014ne)d(an)f(evidence)i(function)f Fk(g)1434 807 y Fi(e)1452 823 y Fw(\()p Fk(x)1496 830 y Fi(i)1510 823 y Fw(\))f(suc)o(h)g(that)714 964 y Fk(g)738 945 y Fi(e)756 964 y Fw(\()p Fk(x)800 971 y Fi(i)814 964 y Fw(\))d(=)892 892 y Fh(\()946 935 y Fw(1)41 b(if)16 b Fk(x)1078 942 y Fi(i)1105 935 y Fw(=)d Fk(x)1179 919 y Fd(\003)1179 948 y Fi(i)946 992 y Fw(0)41 b(otherwise.)1832 964 y(\(12\))37 1101 y(Let)753 1165 y Fk(f)780 1146 y Fd(\003)800 1165 y Fw(\()p Fl(u)p Fw(\))12 b(=)h Fk(p)p Fw(\()p Fl(u)p Fw(\))1021 1125 y Fh(Y)1021 1219 y Fc(U)1055 1210 y Fe(e)1080 1165 y Fk(g)1104 1146 y Fi(e)1122 1165 y Fw(\()p Fk(x)1166 1172 y Fi(i)1180 1165 y Fw(\))634 b(\(13\))37 1301 y(Th)o(us,)17 b(w)o(e)g(ha)o(v)o(e)f(that)g Fk(f)469 1284 y Fd(\003)489 1301 y Fw(\()p Fl(u)p Fw(\))f Fm(/)g Fk(p)p Fw(\()p Fl(u)689 1284 y Fi(h)711 1301 y Fm(j)p Fk(e)p Fw(\).)24 b(T)l(o)17 b(obtain)g Fk(f)1039 1284 y Fd(\003)1059 1301 y Fw(\()p Fl(u)p Fw(\))f(b)o(y)h(op)q (erations)g(on)f(the)h(junction)h(tree)e(one)37 1359 y(pro)q(ceeds)f(as)f(follo)o(ws.)20 b(First)13 b(assign)i(eac)o(h)f (observ)o(ed)g(v)m(ariable)h Fk(X)1181 1366 y Fi(i)1208 1359 y Fm(2)e Fl(U)1291 1343 y Fi(e)1323 1359 y Fw(to)h(one)g (particular)h(clique)h(whic)o(h)37 1417 y(con)o(tains)i(it)f(\(this)g (is)g(termed)g(\\en)o(tering)h(the)f(evidence)i(in)o(to)d(the)i (clique"\).)26 b(Let)17 b Fk(C)1528 1401 y Fi(E)1575 1417 y Fw(denote)g(the)g(set)g(of)37 1475 y(all)g(cliques)g(in)o(to)e (whic)o(h)h(evidence)h(is)f(en)o(tered)f(in)h(this)g(manner.)k(F)l(or) 14 b(eac)o(h)i Fk(C)f Fm(2)e Fk(C)1495 1459 y Fi(E)1540 1475 y Fw(let)582 1585 y Fk(g)604 1592 y Fi(C)633 1585 y Fw(\()p Fk(x)677 1592 y Fi(C)707 1585 y Fw(\))f(=)987 1544 y Fh(Y)785 1645 y Fd(f)p Fi(i)p Fj(:)p Fk(X)863 1652 y Fi(i)891 1645 y Fw(is)k(en)o(tered)f(in)o(to)h Fk(C)r Fd(g)1251 1585 y Fk(g)1275 1566 y Fi(e)1292 1585 y Fw(\()p Fk(x)1336 1592 y Fi(i)1350 1585 y Fw(\))464 b(\(14\))37 1745 y(Th)o(us,)684 1809 y Fk(f)711 1790 y Fd(\003)731 1809 y Fw(\()p Fl(u)p Fw(\))12 b(=)h Fk(p)p Fw(\()p Fl(u)p Fw(\))d Fm(\002)1025 1768 y Fh(Y)999 1864 y Fi(C)r Fd(2)p Fi(C)1077 1854 y Fe(E)1111 1809 y Fk(g)1133 1816 y Fi(C)1162 1809 y Fw(\()p Fk(x)1206 1816 y Fi(C)1235 1809 y Fw(\))p Fk(:)566 b Fw(\(15\))37 1939 y(One)16 b(can)f(no)o(w)f(propagate)g(the)h(e\013ects)g(of)f(these)h(mo)q (di\014cations)h(throughout)e(the)h(tree)g(using)g(the)g(collect)37 1998 y(and)22 b(distribute)h(sc)o(hedule)g(describ)q(ed)h(in)e(5.3.)38 b(Let)22 b Fk(x)1024 1981 y Fi(h)1024 2010 y(C)1075 1998 y Fw(denote)g(a)f(v)m(alue)i(assignmen)o(t)e(of)g(the)h(hidden)37 2056 y(\(unobserv)o(ed\))e(v)m(ariables)h(in)f(clique)i Fk(C)s Fw(.)33 b(When)20 b(the)f(sc)o(hedule)i(of)f(\015o)o(ws)f(is)h (complete)g(one)g(gets)f(a)g(new)37 2114 y(represen)o(tation)g Fk(K)380 2097 y Fd(\003)377 2127 y Fi(f)418 2114 y Fw(suc)o(h)f(that)g (the)g(lo)q(cal)h(p)q(oten)o(tial)g(on)f(eac)o(h)h(clique)h(is)f Fk(f)1392 2097 y Fd(\003)1411 2114 y Fw(\()p Fk(x)1455 2121 y Fi(C)1485 2114 y Fw(\))e(=)h Fk(p)p Fw(\()p Fk(x)1640 2097 y Fi(h)1640 2127 y(C)1669 2114 y Fk(;)8 b(e)p Fw(\),)18 b(i.e.,)g(the)37 2172 y(join)o(t)e(probabilit)o(y)g(of)f(the)h(lo)q (cal)h(unobserv)o(ed)e(clique)j(v)m(ariables)e(and)g(the)g(observ)o(ed) f(evidence)i(\(Jensen)g(et)37 2230 y(al.)j(1990\))12 b(\(similarly)i(for)f(the)g(separator)f(p)q(oten)o(tial)h(functions\).) 20 b(If)13 b(one)g(marginalizes)i(at)d(the)h(clique)i(o)o(v)o(er)37 2288 y(the)h(unobserv)o(ed)g(lo)q(cal)g(clique)h(v)m(ariables,)791 2357 y Fh(X)794 2455 y Fi(X)826 2444 y Fe(h)823 2467 y(C)859 2398 y Fk(p)p Fw(\()p Fk(x)926 2379 y Fi(h)926 2409 y(C)955 2398 y Fk(;)8 b(e)p Fw(\))j(=)i Fk(p)p Fw(\()p Fk(e)p Fw(\))p Fk(;)665 b Fw(\(16\))37 2559 y(one)22 b(gets)f(the)g(probabilit)o(y)h(of)f(the)h(observ)o(ed)f(evidence)i (directly)l(.)39 b(Similarly)l(,)25 b(if)d(one)f(normalizes)h(the)37 2617 y(p)q(oten)o(tial)17 b(function)g(at)e(a)h(clique)i(to)d(sum)h(to) g(1,)f(one)h(obtains)h(the)f(conditional)h(probabilit)o(y)h(of)d(the)h (lo)q(cal)37 2675 y(unobserv)o(ed)g(clique)h(v)m(ariables)g(giv)o(en)f (the)f(evidence,)h Fk(p)p Fw(\()p Fk(x)1053 2658 y Fi(h)1053 2688 y(C)1083 2675 y Fm(j)p Fk(e)p Fw(\).)952 2799 y(14)p eop %%Page: 15 16 15 15 bop 37 16 a Ff(5.5)56 b(Complexit)n(y)16 b(of)j(the)f (Propagation)h(Step)g(of)f(the)h(JLO)f(Algorithm)37 104 y Fw(In)c(general,)g(the)f(time)h(complexit)o(y)g Fk(T)19 b Fw(of)13 b(propagation)g(within)h(a)f(junction)h(tree)f(is)h Fk(O)q Fw(\()1552 72 y Fh(P)1595 83 y Fi(N)1623 89 y Fe(C)1595 117 y Fi(i)p Fj(=1)1662 104 y Fk(s)p Fw(\()p Fk(C)1734 111 y Fi(i)1748 104 y Fw(\)\))e(where)37 162 y Fk(N)74 169 y Fi(C)119 162 y Fw(is)k(the)f(n)o(um)o(b)q(er)h(of)f (cliques)i(in)g(the)e(junction)i(tree)e(and)g Fk(s)p Fw(\()p Fk(C)1171 169 y Fi(i)1185 162 y Fw(\))g(is)h(the)f(n)o(um)o(b)q (er)h(of)f(states)g(in)h(the)g(join)o(t)37 220 y(state-space)21 b(of)e Fk(C)365 227 y Fi(i)399 220 y Fw(\(equal)i(to)f(the)g(pro)q (duct)h(o)o(v)o(er)f(eac)o(h)g(v)m(ariable)i(in)f Fk(C)1339 227 y Fi(i)1373 220 y Fw(of)f(the)h(n)o(um)o(b)q(er)f(of)g(states)g(of) 37 278 y(eac)o(h)g(v)m(ariable\).)35 b(Th)o(us,)21 b(for)e(inference)j (to)d(b)q(e)h(e\016cien)o(t,)i(w)o(e)d(need)i(to)e(construct)h (junction)g(trees)g(with)37 336 y(small)15 b(clique)h(sizes.)k (Problems)15 b(of)e(\014nding)i(optimally)g(small)g(junction)g(trees)f (\(e.g.,)e(\014nding)k(the)e(junction)37 394 y(tree)j(with)h(the)f (smallest)h(maximal)f(clique\))i(are)e(NP-hard.)26 b(Nonetheless,)18 b(the)f(heuristic)i(algorithm)e(for)37 452 y(triangulation)k(describ)q (ed)h(earlier)e(has)g(b)q(een)h(found)f(to)f(w)o(ork)g(w)o(ell)h(in)h (practice)f(\(Jensen)h(et)e(al.)34 b(1990;)37 511 y(Jensen)17 b(1995\).)37 652 y Fo(6)70 b(Inference)21 b(and)j(MAP)f(Calculations)e (in)h(HMM\(1,1\))37 755 y Ff(6.1)56 b(The)21 b(F-B)h(Algorithm)d(for)j (HMM\(1,1\))f(is)g(a)h(Sp)r(ecial)e(Case)i(of)g(the)f(JLO)g(Algo-)165 814 y(rithm)37 901 y Fw(Figure)13 b(5b)g(sho)o(ws)f(the)g(junction)i (tree)e(for)g(HMM\(1,1\).)17 b(In)c(this)g(section)g(w)o(e)g(apply)g (the)g(JLO)h(algorithm)e(to)37 959 y(the)h(HMM\(1,1\))e(junction)j (tree)e(structure)h(to)f(obtain)h(a)f(particular)i(inference)g (algorithm)f(for)f(HMM\(1,1\).)37 1017 y(As)17 b(men)o(tioned)h (earlier,)f(the)g(HMM\(1,1\))e(inference)j(problem)g(consists)f(of)f(b) q(eing)i(giv)o(en)f(a)f(set)h(of)f(v)m(alues)37 1075 y(for)f(the)g(observ)m(able)i(v)m(ariables,)602 1169 y Fk(e)c Fw(=)g Fm(f)p Fk(O)742 1176 y Fj(1)773 1169 y Fw(=)g Fk(o)843 1176 y Fj(1)863 1169 y Fk(;)8 b(O)919 1176 y Fj(2)950 1169 y Fw(=)13 b Fk(o)1020 1176 y Fj(2)1040 1169 y Fk(;)8 b(:)g(:)g(:)d(;)j(O)1177 1176 y Fi(N)1222 1169 y Fw(=)13 b Fk(o)1292 1176 y Fi(N)1326 1169 y Fm(g)483 b Fw(\(17\))37 1263 y(and)20 b(inferring)h(the)f(lik)o(eliho)q(o)q(d)i (of)d Fk(e)h Fw(giv)o(en)g(the)g(mo)q(del.)33 b(As)20 b(describ)q(ed)i(in)e(the)g(previous)g(section)g(this)37 1321 y(problem)13 b(can)g(b)q(e)f(solv)o(ed)h(exactly)f(b)o(y)g(lo)q (cal)i(propagation)d(in)i(an)o(y)f(junction)h(tree)f(using)h(the)f(JLO) h(inference)37 1379 y(algorithm.)108 1437 y(Let)18 b(the)g(\014nal)g (clique)i(in)f(the)e(c)o(hain)i(con)o(taining)f(\()p Fk(H)1047 1444 y Fi(N)t Fd(\000)p Fj(1)1125 1437 y Fk(;)8 b(H)1184 1444 y Fi(N)1217 1437 y Fw(\))17 b(b)q(e)i(the)f(ro)q(ot)e (clique.)30 b(Th)o(us,)18 b(a)f(non-)37 1495 y(redundan)o(t)11 b(sc)o(hedule)i(consists)e(of)f(\014rst)g(recursiv)o(ely)i(passing)f (\015o)o(ws)f(from)g(eac)o(h)g(\()p Fk(O)1465 1502 y Fi(i)1479 1495 y Fk(;)e(H)1538 1502 y Fi(i)1551 1495 y Fw(\))i(and)h(\()p Fk(H)1719 1502 y Fi(i)p Fd(\000)p Fj(2)1778 1495 y Fk(;)d(H)1837 1502 y Fi(i)p Fd(\000)p Fj(1)1895 1495 y Fw(\))37 1554 y(to)k(eac)o(h)f(\()p Fk(H)244 1561 y Fi(i)p Fd(\000)p Fj(1)303 1554 y Fk(;)d(H)362 1561 y Fi(i)375 1554 y Fw(\))k(in)g(the)g(appropriate)g(sequence)h (\(the)e(\\collect")i(phase\),)f(and)g(then)g(distributing)h(\015o)o (ws)37 1612 y(out)18 b(in)h(the)f(rev)o(erse)f(direction)i(from)f(the)g (ro)q(ot)f(clique.)29 b(If)18 b(w)o(e)g(are)f(only)i(in)o(terested)f (in)h(calculating)g(the)37 1670 y(lik)o(eliho)q(o)q(d)g(of)14 b Fk(e)i Fw(giv)o(en)f(the)g(mo)q(del,)h(then)g(the)f(distribute)h (phase)g(is)g(not)e(necessary)i(since)g(w)o(e)f(can)g(simply)37 1728 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Fw(on)h(the)g(separators)f(are)h(exactly)g(the)g Fk(\014)i Fw(v)m(ariables.)24 b(Th)o(us,)15 b(w)o(e)h(ha)o(v)o(e)g(sho) o(wn)f(that)37 1091 y(the)g(JLO)h(inference)h(algorithm)e(recreates)f (the)h(F-B)g(algorithm)g(for)g(the)g(sp)q(ecial)h(case)f(of)g(the)g (HMM\(1,1\))37 1149 y(probabilit)o(y)i(mo)q(del.)37 1289 y Ff(6.2)56 b(Equiv)m(alence)12 b(of)j(Da)n(wid's)g(Propagation)g (Algorithm)e(for)h(Iden)n(tifying)g(MAP)h(As-)165 1347 y(signmen)n(ts)i(and)j(the)e(Viterbi)f(Algorithm)37 1441 y Fw(Consider)12 b(that)e(one)h(wishes)g(to)f(calculate)779 1429 y(^)769 1441 y Fk(f)5 b Fw(\()p Fl(u)843 1425 y Fc(h)870 1441 y Fk(;)j(e)p Fw(\))j(=)i(max)1074 1448 y Fi(x)1094 1453 y Fg(1)1111 1448 y Fi(;:::)o(;x)1180 1454 y Fe(K)1219 1441 y Fk(p)p Fw(\()p Fk(x)1286 1448 y Fj(1)1305 1441 y Fk(;)8 b(:)g(:)g(:)d(;)j(x)1433 1448 y Fi(K)1466 1441 y Fk(;)g(e)p Fw(\))i(and)h(one)g(also)g(wishes)37 1499 y(to)17 b(iden)o(tify)h(a)f(set)f(of)h(v)m(alues)h(of)f(the)g 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Fw(\()p Fk(x)1143 1913 y Fi(C)1172 1906 y Fw(\))642 b(\(37\))37 2044 y(where)16 b Fk(C)i Fw(is)e(the)f(originating)h (clique)h(for)e(the)g(\015o)o(w.)20 b(The)15 b(de\014nition)i(for)e Fk(\025)1357 2051 y Fi(S)1382 2044 y Fw(\()p Fk(x)1426 2051 y Fi(S)1451 2044 y Fw(\))g(is)g(also)h(c)o(hanged)f(in)h(the)37 2102 y(ob)o(vious)g(manner.)108 2164 y(Secondly)l(,)h(marginalization)f (within)g(a)f(clique)i(is)f(replaced)g(b)o(y)g(maximization:)831 2265 y(^)821 2277 y Fk(f)843 2284 y Fi(C)886 2277 y Fw(=)e(max)934 2309 y Fc(u)p Fd(n)p Fi(x)997 2315 y Fe(C)1029 2277 y Fk(p)p Fw(\()p Fl(u)p Fw(\))p Fk(:)702 b Fw(\(38\))108 2416 y(Giv)o(en)13 b(these)f(t)o(w)o(o)f(c)o(hanges)h(it)h(can)f(b)q(e) h(sho)o(wn)f(that)f(if)i(the)f(same)g(propagation)g(op)q(erations)g (are)g(carried)37 2474 y(out)g(as)f(describ)q(ed)i(earlier,)g(the)e (resulting)i(represen)o(tation)1076 2462 y(^)1064 2474 y Fk(K)1103 2481 y Fi(f)1137 2474 y Fw(at)d(equilibriu)q(m)k(is)e(suc)o (h)g(that)e(the)i(p)q(oten)o(tial)37 2532 y(function)17 b(on)e(eac)o(h)g(clique)i Fk(C)h Fw(is)652 2633 y(^)643 2645 y Fk(f)5 b Fw(\()p Fk(x)714 2652 y Fi(C)743 2645 y Fw(\))12 b(=)25 b(max)821 2679 y Fc(u)846 2670 y Fe(h)866 2679 y Fd(n)p Fi(x)904 2685 y Fe(C)937 2645 y Fk(p)p Fw(\()p Fk(x)1004 2627 y Fi(h)1004 2657 y(C)1033 2645 y Fk(;)8 b(e;)g Fm(f)p Fl(u)1148 2627 y Fi(h)1179 2645 y Fm(n)i Fk(x)1238 2652 y Fi(C)1267 2645 y Fm(g)p Fw(\))524 b(\(39\))952 2799 y(18)p eop %%Page: 19 20 19 19 bop 37 16 a Fw(where)17 b Fk(x)196 0 y Fi(h)196 29 y(C)241 16 y Fw(denotes)g(a)e(v)m(alue)j(assignmen)o(t)e(of)f(the)i (hidden)g(\(unobserv)o(ed\))f(v)m(ariables)i(in)f(clique)h Fk(C)s Fw(.)k(Th)o(us,)37 74 y(once)11 b(the)222 63 y(^)210 74 y Fk(K)249 81 y Fi(f)282 74 y Fw(represen)o(tation)g(is)g(obtained,) h(one)e(can)h(lo)q(cally)i(iden)o(tify)e(the)g(v)m(alues)h(of)e Fk(X)1563 58 y Fi(h)1559 87 y(C)1599 74 y Fw(whic)o(h)h(maximize)37 133 y(the)16 b(full)g(join)o(t)f(probabilit)o(y)i(as)795 195 y(^)-26 b Fk(x)818 176 y Fi(h)818 206 y(C)860 195 y Fw(=)13 b(arg)972 208 y Fi(x)992 197 y Fe(h)992 220 y(C)1037 183 y Fw(^)1027 195 y Fk(f)5 b Fw(\()p Fk(x)1098 202 y Fi(C)1128 195 y Fw(\))p Fk(:)673 b Fw(\(40\))37 285 y(In)17 b(the)e(probabilistic)j(exp)q(ert)e(systems)f(literature)h (this)g(pro)q(cedure)h(is)f(kno)o(wn)f(as)g(generating)h(the)g(\\most) 37 343 y(probable)g(explanation")g(\(MPE\))e(giv)o(en)i(the)f(observ)o (ed)h(evidence.)108 403 y(The)21 b(HMM\(1,1\))d(MAP)j(problem)g (consists)g(of)f(b)q(eing)i(giv)o(en)f(a)f(set)h(of)f(v)m(alues)i(for)e (the)g(observ)m(able)37 461 y(v)m(ariables,)d Fk(e)12 b Fw(=)h Fm(f)p Fk(O)378 468 y Fj(1)410 461 y Fw(=)g Fk(o)480 468 y Fj(1)500 461 y Fk(;)8 b(O)556 468 y Fj(2)587 461 y Fw(=)13 b Fk(o)657 468 y Fj(2)677 461 y Fk(;)8 b(:)g(:)g(:)d(;)j(O)814 468 y Fi(N)859 461 y Fw(=)13 b Fk(o)929 468 y Fi(N)962 461 y Fm(g)i Fw(and)h(inferring)773 569 y(max)748 599 y Fi(h)768 604 y Fg(1)785 599 y Fi(;:::)o(;h)854 605 y Fe(N)891 569 y Fk(p)p Fw(\()p Fk(h)958 576 y Fj(1)977 569 y 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(in)o(tractable.)35 b(Recen)o(t)20 b(researc)o(h)g(has)g(therefore)f (fo)q(cused)i(on)f(appro)o(ximate)f(algorithms)h(for)37 1004 y(inference)d(in)f(suc)o(h)f(structures|see)g(Saul)h(and)f(Jordan) g(\(1996\))e(for)h(HMM\(K,1\))f(and)j(Ghahramani)e(and)37 1062 y(Jordan)i(\(1996\))e(and)h(Williams)j(and)d(Hin)o(ton)h(\(1990\)) e(for)h(HMM\(1,K\).)e(These)j(authors)f(ha)o(v)o(e)g(dev)o(elop)q(ed)37 1121 y(an)22 b(appro)o(ximation)g(metho)q(dology)g(based)g(on)g (mean-\014eld)h(theory)e(from)g(statistical)i(ph)o(ysics.)40 b(While)37 1179 y(discussion)15 b(of)d(mean-\014eld)h(algorithms)g(is)g (b)q(ey)o(ond)g(the)f(scop)q(e)h(of)f(this)h(pap)q(er,)g(it)g(is)g(w)o (orth)e(noting)i(that)e(the)37 1237 y(graphical)17 b(mo)q(deling)g (framew)o(ork)d(pla)o(ys)i(a)f(useful)h(role)g(in)g(the)g(dev)o (elopmen)o(t)g(of)f(these)h(appro)o(ximations.)37 1295 y(Essen)o(tially)22 b(the)f(mean-\014eld)h(approac)o(h)e(in)o(v)o(olv)o (es)i(creating)e(a)h(simpli\014ed)i(graph)d(for)g(whic)o(h)i(tractable) 37 1353 y(algorithms)17 b(are)f(a)o(v)m(ailable,)j(and)e(minimizing)i (a)d(probabilistic)j(distance)f(b)q(et)o(w)o(een)f(the)f(tractable)h (graph)37 1411 y(and)e(the)g(in)o(tractable)g(graph.)k(The)c(JLO)g (algorithm)g(is)g(called)h(as)e(a)g(subroutine)h(on)g(the)f(tractable)h (graph)37 1469 y(during)i(the)e(minimization)i(pro)q(cess.)37 1612 y Fo(9)70 b(Learning)23 b(and)h(PINs)37 1716 y Ff(9.1)56 b(P)n(arameter)17 b(Estimation)g(for)i(PINs)37 1803 y Fw(The)28 b(parameters)e(of)h(a)g(graphical)h(mo)q(del)g(can)f(b)q(e)h (estimated)f(with)g(maxim)o(um-lik)o(eliho)q(o)q(d)j(\(ML\),)37 1861 y(maxim)o(um-a-p)q(osteriori)18 b(\(MAP\),)e(or)h(full)h(Ba)o(y)o (esian)g(metho)q(ds,)f(using)h(traditional)g(tec)o(hniques)g(suc)o(h)g (as)37 1919 y(gradien)o(t)j(descen)o(t,)g(exp)q(ectation-maximization)h (\(EM\))e(\(e.g.,)g(Dempster)g(et)g(al.,)h(1977\),)f(and)g(Mon)o(te-)37 1977 y(Carlo)c(sampling)h(\(e.g.,)e(Neal,)h(1993\).)k(F)l(or)c(the)g (standard)f(HMM\(1,1\))f(mo)q(del)j(discussed)g(in)g(this)f(pap)q(er,) 37 2036 y(where)h(either)f(discrete,)h(Gaussian,)e(or)h (Gaussian-mixture)g(co)q(deb)q(o)q(oks)h(are)e(used,)i(a)e(ML)h(or)f (MAP)h(esti-)37 2094 y(mate)g(using)g(EM)g(is)g(a)g(w)o(ell-kno)o(wn)h (e\016cien)o(t)f(approac)o(h)g(\(P)o(oritz)f(1988;)g(Rabiner)i(1989\).) k(An)16 b(imp)q(ortan)o(t)37 2152 y(asp)q(ect)g(of)g(the)g(application) h(of)e(the)h(EM)f(algorithm)h(to)f(PINs)h(is)g(that)f(the)h(JLO)h (algorithm)f(can)f(b)q(e)i(used)37 2210 y(to)e(p)q(erform)g(the)g(E)g (step.)108 2268 y(F)l(or)21 b(purp)q(oses)i(of)e(illustration,)k(and)d (in)h(k)o(eeping)g(with)f(the)g(rest)g(of)f(the)h(pap)q(er,)i(let)e(us) h(consider)37 2326 y(the)e(case)g(where)g(all)g(v)m(ariables)h(in)g Fl(U)f Fw(are)f(discrete.)37 b(Let)21 b Fk(x)1137 2310 y Fi(k)1179 2326 y Fw(and)g Fk(pa)p Fw(\()p Fk(X)t Fw(\))1398 2310 y Fi(j)1435 2326 y Fw(denote)g(the)f Fk(k)q Fw(th)h(state)f(of)37 2384 y(v)m(ariable)e Fk(X)i Fw(and)d Fk(j)s Fw(th)e(state)h(of)g(v)m (ariables)i Fk(pa)p Fw(\()p Fk(X)t Fw(\),)d(resp)q(ectiv)o(ely)l(.)25 b(Supp)q(ose)18 b(w)o(e)e(ha)o(v)o(e)g(a)g(directed)i(HMM-)37 2442 y(lik)o(e)j(mo)q(del)f Fk(M)25 b Fw(\(a)19 b(DPIN\))g(with)h(m)o (utually)g(indep)q(enden)o(t)h(parameters)e Fa(\022)i Fw(=)f Fm([)1483 2449 y Fi(j)r(k)1521 2442 y Fm(f)p Fk(\022)1565 2449 y Fi(H)s(j)r(k)1634 2442 y Fk(;)8 b(\022)1676 2449 y Fi(O)q(j)r(k)1740 2442 y Fm(g)p Fw(,)20 b(where)37 2500 y Fk(\022)58 2507 y Fi(H)s(j)r(k)143 2500 y Fw(=)15 b Fk(p)p Fw(\()p Fk(h)260 2484 y Fi(k)260 2513 y(i)281 2500 y Fm(j)p Fk(pa)p Fw(\()p Fk(H)397 2507 y Fi(i)411 2500 y Fw(\))429 2484 y Fi(j)446 2500 y Fk(;)8 b(M)d Fw(\))16 b(and)h Fk(\022)661 2507 y Fi(O)q(j)r(k)742 2500 y Fw(=)e Fk(p)p Fw(\()p Fk(o)855 2484 y Fi(k)855 2513 y(i)876 2500 y Fm(j)p Fk(pa)p Fw(\()p Fk(O)989 2507 y Fi(i)1002 2500 y Fw(\))1020 2484 y Fi(j)1038 2500 y Fk(;)8 b(M)d Fw(\))15 b(for)h(all)i Fk(i)p Fw(.)24 b(In)17 b(addition,)h(supp)q(ose)f(w)o(e)g(ha)o(v)o(e)37 2559 y(observ)o(ed)f(data)e Fk(D)g Fw(=)f Fm(f)p Fk(e)472 2566 y Fj(1)491 2559 y Fk(;)8 b(:)g(:)g(:)d(;)j(e)614 2566 y Fi(S)639 2559 y Fm(g)p Fw(,)14 b(an)i(\(iid\))g(random)e(sample) i(from)f(the)g(true)g(distribution.)108 2617 y(The)k(EM)f(algorithm)g (\014nds)h(a)f(lo)q(cal)h(maxim)o(um)f(of)g(the)h(lik)o(eliho)q(o)q(d)i Fk(p)p Fw(\()p Fk(D)q Fm(j)p Fa(\022)p Fk(;)8 b(M)d Fw(\))17 b(b)o(y)h(initializi)q(ng)j(the)37 2675 y(parameters)14 b Fa(\022)i Fw(\(e.g.,)d(at)h(random)h(or)f(via)h(some)f(clustering)i (algorithm\))f(and)g(rep)q(eating)g(E)g(and)g(M)f(steps.)952 2799 y(24)p eop %%Page: 25 26 25 25 bop 37 16 a Fw(In)20 b(the)e(E)g(step,)h(w)o(e)f(compute)h(the)f (exp)q(ected)i(su\016cien)o(t)f(statistic)f(for)g(eac)o(h)h(of)f(the)g (parameters,)g(giv)o(en)37 74 y Fk(D)i Fw(and)f(the)g(curren)o(t)g(v)m (alues)h(for)e Fa(\022)q Fw(.)31 b(Let)20 b Fk(S)828 81 y Fi(H)s(j)r(k)915 74 y Fw(b)q(e)g(the)f(su\016cien)o(t)h(statistic) e(for)h Fk(\022)1528 81 y Fi(H)s(j)r(k)1597 74 y Fw(.)31 b(The)19 b(exp)q(ected)37 133 y(su\016cien)o(t)d(statistic)g Fk(E)s Fw(\()p Fk(S)484 140 y Fi(H)s(j)r(k)551 133 y Fm(j)p Fk(D)q(;)8 b Fa(\022)p Fk(;)g(M)d Fw(\))14 b(is)i(giv)o(en)g(b)o (y)488 277 y Fk(E)s Fw(\()p Fk(S)571 284 y Fi(H)s(j)r(k)639 277 y Fm(j)p Fk(D)q(;)8 b Fa(\022)p Fk(;)g(M)d Fw(\))11 b(=)904 224 y Fi(S)885 236 y Fh(X)887 328 y Fi(l)p Fj(=1)953 236 y Fh(X)977 327 y Fi(i)1020 277 y Fk(p)p Fw(\()p Fk(h)1087 258 y Fi(k)1087 288 y(i)1108 277 y Fk(;)d(pa)p Fw(\()p Fk(H)1232 284 y Fi(i)1245 277 y Fw(\))1263 258 y Fi(j)1281 277 y Fm(j)p Fk(e)1315 284 y Fi(l)1328 277 y Fk(;)g Fa(\022)p Fk(;)g(M)d Fw(\))37 415 y(As)15 b(men)o(tioned,)h(an)f(imp)q(ortan)o(t) f(feature)h(of)f(the)h(EM)g(algorithm)g(applied)i(to)d(PINs)h(is)h (that)e(eac)o(h)h(term)g(in)37 473 y(the)i(sum)f(can)h(b)q(e)g (computed)g(using)g(the)f(JLO)i(algorithm.)23 b(The)17 b(exp)q(ected)h(su\016cien)o(t)f(statistic)f(for)g Fk(\022)1847 480 y Fi(O)q(j)r(k)37 531 y Fw(is)i(computed)g(similarly)l(.)27 b(In)18 b(the)f(M)g(step,)g(w)o(e)g(use)h(the)f(exp)q(ected)h (su\016cien)o(t)g(statistics)f(as)g(if)h(they)f(w)o(ere)37 589 y(actual)d(su\016cien)o(t)h(statistics,)e(and)h(set)g(the)f(new)h (v)m(alues)h(of)f Fa(\022)g Fw(to)f(b)q(e)i(those)e(that)g(maximize)i (the)f(lik)o(eliho)q(o)q(d)37 647 y(of)h(these)h(statistics:)371 763 y Fk(\022)392 770 y Fi(H)s(j)r(k)474 763 y Fw(=)563 731 y Fk(E)s Fw(\()p Fk(S)646 738 y Fi(H)s(j)r(k)714 731 y Fm(j)p Fk(D)q(;)8 b Fa(\022)p Fk(;)g(M)d Fw(\))p 527 752 410 2 v 527 762 a Fh(P)571 805 y Fi(k)600 794 y Fk(E)s Fw(\()p Fk(S)683 801 y Fi(H)s(j)r(k)750 794 y Fm(j)p Fk(D)q(;)j Fa(\022)p Fk(;)g(M)d Fw(\))1017 763 y Fk(\022)1038 770 y Fi(O)q(j)r(k)1116 763 y Fw(=)1205 731 y Fk(E)s Fw(\()p Fk(S)1288 738 y Fi(O)q(j)r(k)1352 731 y Fm(j)p Fk(D)q(;)j Fa(\022)p Fk(;)g(M)d Fw(\))p 1169 752 406 2 v 1169 762 a Fh(P)1213 805 y Fi(k)1242 794 y Fk(E)s Fw(\()p Fk(S)1325 801 y Fi(O)q(j)r(k)1389 794 y Fm(j)p Fk(D)q(;)j Fa(\022)o Fk(;)g(M)d Fw(\))108 890 y(The)18 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1843 y Fk(;)c Fa(\022)p Fk(;)g(M)d Fw(\))37 2002 y Ff(9.2)56 b(Mo)r(del)18 b(Selection)f(and)i(Av)n(eraging)f(for)h(PINs)37 2090 y Fw(In)e(some)f(situations)h(it)f(is)h(useful)g(to)f(use)g(data)g (to)f(guide)i(the)g(selection)g(of)f(an)g(appropriate)g(mo)q(del.)24 b(F)l(or)37 2148 y(example,)13 b(the)f(presence)g(of)f(some)g(arcs)g (or)g(the)g(n)o(um)o(b)q(er)h(of)f(states)g(of)g(a)g(hidden)i(v)m (ariable)g(ma)o(y)d(b)q(e)i(in)h(doubt.)37 2206 y(One)18 b(solution)g(to)e(this)h(problem)g(is)h(the)e(Ba)o(y)o(esian)h(approac) o(h,)g(in)g(whic)o(h)h(w)o(e)e(assign)h(prior)g(probabilities)37 2264 y Fk(p)p Fw(\()p Fk(M)5 b Fw(\))15 b(to)f(di\013eren)o(t)i(mo)q (dels,)g(and)f(compute)g(their)h(relativ)o(e)g(p)q(osterior)f (probabilities)i(giv)o(en)f(data:)395 2384 y Fk(p)p Fw(\()p Fk(M)5 b Fm(j)p Fk(D)q Fw(\))12 b Fm(/)h Fk(p)p Fw(\()p Fk(M)5 b Fw(\))14 b Fk(p)p Fw(\()p Fk(D)q Fm(j)p Fk(M)5 b Fw(\))12 b(=)h Fk(p)p Fw(\()p Fk(M)5 b Fw(\))1073 2326 y Fh(Z)1121 2384 y Fk(p)p Fw(\()p Fk(D)q Fm(j)p Fa(\022)p Fk(;)j(M)d Fw(\))14 b Fk(p)p Fw(\()p Fa(\022)p Fm(j)p Fk(M)5 b Fw(\))15 b Fk(d)p Fa(\022)278 b Fw(\(44\))37 2500 y(W)l(e)17 b(then)g(select)g(the)g(mo)q(del)g(with)g(the)g (highest)g(p)q(osterior)f(probabilit)o(y)l(,)i(or)e(a)o(v)o(erage)f (the)i(predictions)h(of)37 2558 y(t)o(w)o(o)c(or)h(more)g(mo)q(dels)h (w)o(eigh)o(ted)f(b)o(y)g(their)h(relativ)o(e)g(p)q(osterior)f (probabilities.)108 2617 y(When)c(data)e(is)i(missing|for)g(example,)g (when)g(some)f(v)m(ariables)h(are)f(hidden|the)j(exact)c(computation)37 2675 y(of)k(the)f(in)o(tegral)h(in)h(Equation)e(44)g(is)h(usually)h(in) o(tractable.)20 b(Nonetheless,)14 b(simple)g(appro)o(ximations)e(to)g (this)952 2799 y(25)p eop %%Page: 26 27 26 26 bop 37 16 a Fw(in)o(tegral)16 b(exist,)f(suc)o(h)h(as)f(the)g(Ba) o(y)o(esian)g(Information)g(Criterion)h(\(BIC\))f(describ)q(ed)i(b)o(y) e(Sc)o(h)o(w)o(arz)g(\(1978\):)599 142 y(log)9 b Fk(p)p Fw(\()p Fk(D)q Fm(j)p Fk(M)c Fw(\))11 b Fm(\031)i Fw(log)8 b Fk(p)p Fw(\()p Fk(D)q Fm(j)1046 130 y Fw(^)1044 142 y Fa(\022)p Fk(;)g(M)d Fw(\))k Fm(\000)1218 112 y Fk(d)p 1218 132 24 2 v 1218 174 a Fw(2)1254 142 y(log)f Fk(S)37 269 y Fw(where)174 257 y(^)172 269 y Fa(\022)19 b Fw(is)g(the)g(ML)f (estimate,)h Fk(S)h Fw(is)f(the)g(n)o(um)o(b)q(er)f(of)g(cases)h(in)g Fk(D)q Fw(,)f(and)h Fk(d)f Fw(is)h(the)f(dimension)i(of)e Fk(M)5 b Fw(|)37 328 y(t)o(ypically)l(,)20 b(the)d(n)o(um)o(b)q(er)h (of)e(parameters)h(of)g Fk(M)5 b Fw(.)26 b(The)18 b(\014rst)e(term)h (of)g(this)h(\\score")e(for)h Fk(M)22 b Fw(rew)o(ards)17 b(ho)o(w)37 386 y(w)o(ell)e(the)f(data)g(\014ts)f Fk(M)5 b Fw(,)14 b(whereas)g(the)g(second)g(term)g(punishes)h(mo)q(del)g (complexit)o(y)l(.)21 b(Note)13 b(that)g(this)i(score)37 444 y(do)q(es)21 b(not)f(dep)q(end)i(on)e(the)g(parameter)f(prior,)i (and)g(th)o(us)f(can)g(b)q(e)h(applied)h(easily)l(.)1533 427 y Fj(1)1588 444 y Fw(F)l(or)e(examples)h(of)37 502 y(applications)c(of)e(BIC)h(in)g(the)f(con)o(text)g(of)f(PINs)i(and)f (other)g(statistical)h(mo)q(dels,)f(see)h(Raftery)f(\(1995\).)108 561 y(The)22 b(BIC)g(score)f(is)i(the)e(additiv)o(e)i(in)o(v)o(erse)f (of)f(Rissanen's)i(\(1987\))d(minim)o(um)j(description)g(length)37 619 y(\(MDL\).)16 b(Other)i(scores,)f(whic)o(h)h(can)f(b)q(e)h(view)o (ed)g(as)e(appro)o(ximations)h(to)g(the)g(marginal)h(lik)o(eliho)q(o)q (d,)i(are)37 677 y(h)o(yp)q(othesis)c(testing)f(\(Raftery)f(1995\))f (and)i(cross)g(v)m(alidation)h(\(F)l(ung)e(and)h(Cra)o(wford)f(1990\).) k(Bun)o(tine)e(\(in)37 735 y(press\))f(pro)o(vides)h(a)f(comprehensiv)o (e)h(review)g(of)f(the)g(literature)h(on)f(learning)i(PINs.)108 794 y(In)g(the)g(con)o(text)f(of)g(HMM\()p Fk(K)q(;)8 b(J)t Fw(\))14 b(t)o(yp)q(e)j(structures,)f(an)h(ob)o(vious)f(question) i(is)f(ho)o(w)f(one)g(could)i(learn)37 852 y(suc)o(h)g(structure)f (from)f(data,)g(where)i Fk(K)h Fw(and)f Fk(J)j Fw(are)c(unkno)o(wn)g(a) g(priori.)26 b(F)l(rom)16 b(a)h(mo)q(del)h(iden)o(ti\014cation)37 910 y(viewp)q(oin)o(t,)c(this)f(is)f(an)g(easier)h(problem)g(than)e (that)h(of)g(learning)h(an)f(arbitrary)f(PIN,)h(b)q(ecause)i(the)e(p)q (ossible)37 968 y(mo)q(dels)i(under)g(consideration)g(are)f(highly)i (constrained.)k(Th)o(us,)14 b(using)f(b)q(oth)h(the)f(estimation)g(tec) o(hniques)37 1027 y(for)h(a)f(particular)h(mo)q(del)h(describ)q(ed)h (in)e(the)g(previous)g(section)h(\(and)e(the)h(JLO)h(algorithm)f(for)f (solving)h(the)37 1085 y(E-step)19 b(as)e(describ)q(ed)j(in)f(detail)g (earlier)g(in)g(the)f(pap)q(er\),)g(and)h(the)f(Ba)o(y)o(esian)g(\(and) g(alternativ)o(e\))g(mo)q(del)37 1143 y(selection)h(pro)q(cedures)f (outlined)g(ab)q(o)o(v)o(e,)f(the)f(algorithmic)i(prescriptions)g(for)f (learning)h(suc)o(h)f(mo)q(dels)h(in)37 1201 y(a)d(principled)k (fashion)c(are)g(already)h(in)g(place.)37 1349 y Fo(10)70 b(Summary)37 1454 y Fw(Probabilistic)18 b(indep)q(endence)h(net)o(w)o (orks)14 b(pro)o(vide)j(a)e(useful)i(framew)o(ork)d(for)h(b)q(oth)h (the)f(analysis)i(and)e(ap-)37 1512 y(plication)f(of)e(m)o(ultiv)m (ariate)h(probabilit)o(y)g(mo)q(dels)g(when)f(there)g(is)h (considerable)g(structure)f(in)h(the)f(mo)q(del)h(in)37 1570 y(the)f(form)f(of)g(conditional)j(indep)q(endence.)22 b(The)12 b(graphical)g(mo)q(delling)i(approac)o(h)d(b)q(oth)h (clari\014es)h(the)f(inde-)37 1628 y(p)q(endence)k(seman)o(tics)e(of)f (the)g(mo)q(del)h(and)g(yields)h(e\016cien)o(t)f(computational)g (algorithms)f(for)g(probabilistic)37 1686 y(inference.)22 b(This)15 b(pap)q(er)g(has)g(sho)o(wn)g(that)f(it)h(is)g(useful)h(to)e (cast)g(HMM)g(structures)h(in)g(a)g(graphical)h(mo)q(del)37 1744 y(framew)o(ork.)j(In)14 b(particular,)h(the)f(w)o(ell)h(kno)o(wn)f (F-B)g(and)g(Viterbi)i(algorithms)e(w)o(ere)g(sho)o(wn)f(to)h(b)q(e)h (sp)q(ecial)37 1802 y(cases)j(of)f(more)g(general)h(algorithms)f(from)g (the)h(graphical)g(mo)q(delling)i(literature.)27 b(F)l(urthermore,)17 b(more)37 1861 y(complex)f(HMM)d(structures,)h(b)q(ey)o(ond)h(the)g 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Fw(.)h Fl(19)p Fw(,)d(177{189.)60 2255 y(Ken)o(t,)d(R.)g(D.)f(&)h (Mini\014e,)i(F.)d(D.)g(1977.)18 b(Coarticulation)12 b(in)h(recen)o(t)e(sp)q(eec)o(h)i(pro)q(duction)g(mo)q(dels.)19 b Fn(Journal)151 2313 y(of)e(Phonetics)p Fw(.)i Fl(5)p Fw(,)c(115-117.)60 2407 y(Lauritzen,)f(S.)e(L.)h(and)f(Spiegelhalter)j (D.)d(J.)g(1988.)18 b(Lo)q(cal)13 b(computations)g(with)f (probabilities)j(on)e(graph-)151 2465 y(ical)j(structures)e(and)h (their)g(application)h(to)e(exp)q(ert)g(systems)g(\(with)h (discussion\).)21 b Fn(J.)15 b(R)n(oy.)21 b(Statist.)151 2523 y(So)n(c.)f(Ser.)h(B)p Fw(.)15 b Fl(50)g Fw(157{224.)60 2617 y(Lauritzen,)20 b(S.,)e(and)g(W)l(erm)o(uth,)g(N.)f(1989.)27 b(Graphical)19 b(mo)q(dels)g(for)e(asso)q(ciations)h(b)q(et)o(w)o(een)h (v)m(ariables,)151 2675 y(some)c(of)g(whic)o(h)h(are)f(qualitativ)o(e)h (and)f(some)g(quan)o(titativ)o(e.)20 b Fn(A)o(nnals)14 b(of)j(Statistics)p Fw(.)h Fl(17)p Fw(,)e(31{57.)952 2799 y(27)p eop %%Page: 28 29 28 28 bop 60 16 a Fw(Lauritzen,)14 b(S.)e(L.,)g(Da)o(wid,)g(A.)g(P)l (.,)g(Larsen,)h(B.)f(N.,)g(and)g(Leimer,)h(H.)f(G)g(1990.)17 b(Indep)q(endence)f(prop)q(erties)151 74 y(of)f(directed)h(Mark)o(o)o (v)e(\014elds.)21 b Fn(Networks)p Fw(.)f Fl(20)p Fw(,)15 b(491{505.)60 173 y(Lindblom,)21 b(B.)c(1990.)28 b(Explaining)20 b(phonetic)f(v)m(ariation:)26 b(A)18 b(sk)o(etc)o(h)g(of)f(the)h(H&H)h (theory)l(.)28 b(In)19 b Fn(Sp)n(e)n(e)n(ch)151 231 y(Pr)n(o)n(duction) d(and)g(Sp)n(e)n(e)n(ch)e(Mo)n(deling)p Fw(,)g(W.J.)g(Hardcastle)h(and) g(A.)f(Marc)o(hal,)g(\(Eds.\).)19 b(Klu)o(w)o(er:)h(Dor-)151 289 y(drec)o(h)o(t.)60 388 y(Luc)o(k)o(e,)c(H.)g(1995.)k(Ba)o(y)o (esian)c(Belief)h(Net)o(w)o(orks)e(as)g(a)g(to)q(ol)h(for)f(sto)q(c)o (hastic)h(parsing.)22 b Fn(Sp)n(e)n(e)n(ch)15 b(Communi-)151 446 y(c)n(ation)p Fw(.)20 b Fl(16)p Fw(,)15 b(89{118.)60 544 y(Mo)q(destino,)k(J.)f(and)g(Zhang,)g(J.)f(1992.)27 b(A)18 b(Mark)o(o)o(v)e(random)i(\014eld)h(mo)q(del-based)h(approac)o (h)d(to)g(image)151 603 y(segmen)o(tation.)j Fn(IEEE)15 b(T)m(r)n(ans.)k(Patt.)i(A)o(nal.)f(Mach.)h(Int.)e Fl(14)p Fw(\(6\),)14 b(606{615.)60 701 y(Neal,)k(R.)f(1993.)23 b(Probabilistic)c(inference)f(using)g(Mark)o(o)o(v)d(c)o(hain)j(Mon)o (te)e(Carlo)g(metho)q(ds.)25 b(CR)o(G-TR-)151 759 y(93-1,)14 b(Departmen)o(t)h(of)f(Computer)h(Science,)i(Univ)o(ersit)o(y)f(of)e(T) l(oron)o(to.)60 858 y(P)o(earl,)e(J.)g(1988.)18 b Fn(Pr)n(ob)n (abilistic)11 b(R)n(e)n(asoning)h(in)g(Intel)r(ligent)g(Systems:)18 b(Networks)13 b(of)g(Plausible)f(Infer)n(enc)n(e)p Fw(,)151 916 y(San)k(Mateo,)d(CA:)i(Morgan)f(Kaufmann)i(Publishers.)60 1015 y(P)o(earl,)11 b(J.,)g(Geiger,)g(D.,)g(and)f(V)l(erma,)h(T.)f (1990.)17 b(The)11 b(logic)g(of)f(in\015uence)i(diagrams.)18 b Fn(In\015uenc)n(e)10 b(Diagr)n(ams,)151 1073 y(Belief)18 b(Nets,)h(and)f(De)n(cision)g(A)o(nalysis)p Fw(.)26 b(Oliv)o(er,)20 b(R.)e(M.)f(and)h(Smith,)g(J.)g(Q.)g(\(eds.\).)27 b(Chic)o(hester,)151 1131 y(U.K.:)20 b(John)15 b(Wiley)i(and)e(Sons.)20 b(67{83.)60 1229 y(P)o(erk)o(ell,)i(J.)d(S.,)i(Matthies,)f(M.)f(L.,)h(Svirsky)l(,)i (M.)d(A.,)h(and)g(Jordan,)h(M.)e(I.)g(1993.)33 b(T)l(rading)20 b(relations)151 1287 y(b)q(et)o(w)o(een)f(tongue-b)q(o)q(dy)g(raising)h (and)e(lip)i(rounding)g(in)f(pro)q(duction)h(of)e(the)h(v)o(o)o(w)o(el) f(/u/:)26 b(A)19 b(pilot)151 1346 y(motor)14 b(equiv)m(alence)k(study)l (.)i Fn(Journal)c(of)h(the)f(A)n(c)n(oustic)n(al)f(So)n(ciety)h(of)g(A) o(meric)n(a)p Fw(.)k Fl(93)p Fw(,)15 b(2948-2961.)60 1444 y(P)o(oritz,)i(A.)h(M.)e(1988.)25 b(Hidden)19 b(Mark)o(o)o(v)d(mo) q(dels:)25 b(a)17 b(guided)i(tour.)26 b Fn(Pr)n(o)n(c)n(e)n(e)n(dings) 16 b(of)i(the)h(IEEE)e(Inter-)151 1502 y(national)e(Confer)n(enc)n(e)e (on)i(A)n(c)n(oustics,)f(Sp)n(e)n(e)n(ch)g(and)h(Signal)f(Pr)n(o)n(c)n (essing)p Fw(.)j(New)d(Y)l(ork:)19 b(IEEE)14 b(Press.)151 1560 y(v)o(ol.1,)g(7-13.)60 1659 y(Rabiner,)19 b(L.,)f(1989.)26 b(A)18 b(tutorial)g(on)f(hidden)j(Mark)o(o)o(v)c(mo)q(dels)i(and)g (selected)h(applications)g(in)f(sp)q(eec)o(h)151 1717 y(recognition.)j Fn(Pr)n(o)n(c)n(e)n(e)n(dings)14 b(of)i(the)h(IEEE)p Fw(,)c Fl(77)p Fw(,)i(257-285.)60 1816 y(Raftery)l(,)20 b(A.)e(1995.)30 b(Ba)o(y)o(esian)19 b(mo)q(del)h(selection)g(in)g(so)q (cial)g(researc)o(h.)31 b(In)20 b(Marsden,)f(P)l(.,)g Fn(So)n(ciolo)n(gic)n(al)151 1874 y(Metho)n(dolo)n(gy)p Fw(.)h(Blac)o(kw)o(ells,)c(Cam)o(bridge,)f(MA.)60 1972 y(Rissanen,)20 b(J.)d(1987.)27 b(Sto)q(c)o(hastic)18 b(complexit)o(y)h(\(with)e(discussion\).)29 b Fn(Journal)19 b(of)g(the)g(R)n(oyal)g(Statistic)n(al)151 2030 y(So)n(ciety,)d(Series) f(B.)g Fl(49)p Fw(,)g(223{239)e(and)j(253{265.)60 2129 y(Saul,)k(L.)f(K.,)g(and)g(Jordan,)g(M.)f(I.)g(1995.)29 b(Boltzmann)19 b(c)o(hains)h(and)f(hidden)h(Mark)o(o)o(v)d(mo)q(dels.) 31 b(In)19 b(G.)151 2187 y(T)l(esauro,)11 b(D.)e(S.)h(T)l(ouretzky)g(&) h(T.)e(K.)h(Leen,)i(\(Eds.\),)e Fn(A)n(dvanc)n(es)g(in)h(Neur)n(al)h (Information)f(Pr)n(o)n(c)n(essing)151 2245 y(Systems)k(7)p Fw(,)g(MIT)g(Press,)g(Cam)o(bridge)g(MA.)60 2344 y(Saul,)21 b(L.)e(K.,)g(and)h(Jordan,)f(M.)g(I.)g(1996.)30 b(Exploiting)21 b(tractable)e(substructures)g(in)h(in)o(tractable)f(net-)151 2402 y(w)o(orks.)k(In)17 b(D.)e(S.)i(T)l(ouretzky)l(,)f(M.)g(C.)g (Mozer,)g(&)g(M.)g(E.)g(Hasselmo)h(\(Eds.\),)e Fn(A)n(dvanc)n(es)h(in)h (Neur)n(al)151 2460 y(Information)f(Pr)n(o)n(c)n(essing)e(Systems)h(8)p Fw(,)g(MIT)g(Press,)g(Cam)o(bridge)g(MA.)60 2559 y(Shac)o(h)o(ter,)f (R.)h(D.,)f(Anderson,)g(S.)h(K.)f(and)h(Szolo)o(vits,)g(P)l(.)f(1994.) 19 b(Global)c(conditioning)h(for)e(prababilistic)151 2617 y(inference)k(in)f(b)q(elief)h(net)o(w)o(orks.)k Fn(Pr)n(o)n(c)n(e)n(e)n(dings)15 b(of)i(the)h(Unc)n(ertainty)e(in)h(AI) f(Confer)n(enc)n(e)f(1994)p Fw(,)i(San)151 2675 y(F)l(rancisco,)e(CA:)g (Morgan)f(Kaufmann,)h(514{522.)952 2799 y(28)p eop %%Page: 29 30 29 29 bop 60 16 a Fw(Sc)o(h)o(w)o(arz,)14 b(G.)h(1978.)k(Estimating)c (the)g(dimension)i(of)e(a)g(mo)q(del.)20 b Fn(A)o(nnals)15 b(of)h(Statistics)p Fw(.)j Fl(6)p Fw(,)c(461{464.)60 107 y(Sm)o(yth,)22 b(P)l(.)f(1994.)36 b(Hidden)23 b(Mark)o(o)o(v)c(mo)q (dels)j(for)e(fault)h(detection)h(in)g(dynamic)g(systems.)37 b Fn(Pattern)151 165 y(R)n(e)n(c)n(o)n(gnition)p Fw(,)13 b Fl(27)p Fw(,)i(149-164.)60 256 y(Spiegelhalter,)k(D.)d(J.,)g(Da)o (wid,)h(A.)f(P)l(.,)g(Hutc)o(hinson,)i(T.)e(A.,)g(and)g(Co)o(w)o(ell,)h (R.)g(G.)f(1991.)22 b(Probabilistic)151 315 y(exp)q(ert)d(systems)f (and)h(graphical)h(mo)q(delling:)29 b(a)18 b(case)g(study)h(in)h(drug)e (safet)o(y)l(.)30 b Fn(Phil.)g(T)m(r)n(ans.)e(R.)151 373 y(So)n(c.)20 b(L)n(ond.)g(A)p Fw(.)15 b(337,)f(387{405.)60 464 y(Spirtes,)25 b(P)l(.)d(and)g(Meek,)i(C.)e(1995.)40 b(Learning)23 b(Ba)o(y)o(esian)g(net)o(w)o(orks)e(with)h(discrete)i(v)m (ariables)f(from)151 522 y(data.)c(In)13 b Fn(Pr)n(o)n(c)n(e)n(e)n (dings)f(of)i(First)g(International)f(Confer)n(enc)n(e)f(on)i(Know)r (le)n(dge)g(Disc)n(overy)f(and)i(Data)151 580 y(Mining,)g Fw(Mon)o(treal,)f(QU.)h(Morgan)f(Kaufmann.)151 654 y(Stolorz,)h(P)l(.)f (1994.)19 b(Recursiv)o(e)d(approac)o(hes)f(to)f(the)h(statistical)h(ph) o(ysics)g(of)e(lattice)i(proteins.)k(In)15 b(L.)151 713 y(Hun)o(ter,)g(ed.)20 b Fn(Pr)n(o)n(c.)h(27th)c(Hawaii)f(Intl.)f(Conf.) h(on)g(System)f(Scienc)n(es)p Fw(,)e Fl(V)p Fw(,)h(316-325.)60 803 y(T)l(ao,)23 b(C.,)f(1992.)38 b(A)22 b(generalization)h(of)f(the)f (discrete)i(hidden)g(Mark)o(o)o(v)e(mo)q(del)h(and)g(of)f(the)h (Viterbi)151 862 y(algorithm.)e Fn(Pattern)c(R)n(e)n(c)n(o)n(gnition)p Fw(,)d Fl(25)p Fw(\(11\),)h(1381{1387.)60 953 y(V)l(andermeulen,)19 b(D.,)e(V)l(erb)q(eec)o(k,)h(R.,)g(Berb)q(en,)g(L.)f(Delaere,)h(D.,)e (Suetens,)i(P)l(.)f(and)h(Marc)o(hal,)f(G.)f(1994.)151 1011 y(Con)o(tin)o(uous)j(v)o(o)o(xel)f(classi\014cation)j(b)o(y)d(sto) q(c)o(hastic)h(relaxation:)27 b(theory)18 b(and)h(application)h(to)e (MR)151 1069 y(imaging)e(and)f(MR)h(angiograph)o(y)l(.)j Fn(Image)d(and)g(Vision)g(Computing)p Fw(.)k Fl(12)p Fw(\(9\))14 b(559{572.)60 1160 y(Whittak)o(er,)e(J.)g(1990.)18 b 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Fj(1)1415 445 y Fw(\))p 830 470 603 2 v 1059 512 a Fk(p)p Fw(\()p Fk(h)1126 519 y Fi(i)p Fd(\000)p Fj(1)1185 512 y Fw(\))1832 481 y(\(49\))37 604 y(and)16 b(this)g(gets)e(absorb)q(ed)i(in)o(to)f (clique)i(\()p Fk(H)780 611 y Fi(i)p Fd(\000)p Fj(1)839 604 y Fk(;)8 b(H)898 611 y Fi(i)911 604 y Fw(\))15 b(to)f(pro)q(duce) 300 693 y(^)291 705 y Fk(f)313 712 y Fj(\010)338 717 y Fg(1)p Fe(;i)378 705 y Fw(\()p Fk(h)422 712 y Fi(i)p Fd(\000)p Fj(1)481 705 y Fk(;)8 b(h)528 712 y Fi(i)542 705 y Fw(\))41 b(=)688 693 y(^)678 705 y Fk(f)700 712 y Fi(O)727 717 y Fe(i)742 705 y Fw(\()p Fk(h)786 712 y Fi(i)p Fd(\000)p Fj(1)845 705 y Fk(;)8 b(h)892 712 y Fi(i)906 705 y Fw(\))p Fk(\025)951 712 y Fj(\010)976 717 y Fg(1)p Fe(;i)p Fb(\000)p Fg(1)1054 705 y Fw(\()p Fk(h)1098 712 y Fi(i)p Fd(\000)p Fj(1)1157 705 y Fw(\))657 b(\(50\))601 808 y(=)42 b Fk(p)p Fw(\()p Fk(h)745 815 y Fi(i)p Fd(\000)p Fj(1)804 808 y Fk(;)8 b(h)851 815 y Fi(i)864 808 y Fw(\))p Fk(p)p Fw(\()p Fk(o)945 790 y Fd(\003)945 820 y Fi(i)964 808 y Fm(j)p Fk(h)1003 815 y Fi(i)1017 808 y Fw(\))1040 773 y(max)1124 780 y Fi(h)1144 785 y Fg(1)p Fe(;i)p Fb(\000)p Fg(2)1231 773 y Fk(p)p Fw(\()p Fk(h)1298 780 y Fi(i)p Fd(\000)p Fj(1)1357 773 y Fk(;)g(h)1404 780 y Fj(1)p Fi(;i)p Fd(\000)p Fj(2)1490 773 y Fk(;)g(\036)1538 756 y Fd(\003)1538 785 y Fj(1)p Fi(;i)p Fd(\000)p Fj(1)1624 773 y Fw(\))p 1040 798 V 1269 839 a Fk(p)p Fw(\()p Fk(h)1336 846 y Fi(i)p Fd(\000)p Fj(1)1395 839 y Fw(\))1647 808 y Fk(:)172 b Fw(\(51\))108 932 y(W)l(e)16 b(can)h(no)o(w)f(obtain)g(the)h(new)f(p)q(oten)o(tial)h (on)g(the)f(separator)f(for)h(the)g(\015o)o(w)g(from)g(clique)i(\()p Fk(H)1764 939 y Fi(i)p Fd(\000)p Fj(1)1823 932 y Fk(;)8 b(H)1882 939 y Fi(i)1895 932 y Fw(\))37 990 y(to)15 b(\()p Fk(H)149 997 y Fi(i)163 990 y Fk(;)8 b(H)222 997 y Fi(i)p Fj(+1)280 990 y Fw(\),)339 1079 y(^)329 1091 y Fk(f)351 1098 y Fj(\010)376 1103 y Fg(1)p Fe(;i)417 1091 y Fw(\()p Fk(h)461 1098 y Fi(i)475 1091 y Fw(\))41 b(=)h(max)617 1121 y Fi(h)637 1126 y Fe(i)p Fb(\000)p Fg(1)713 1079 y Fw(^)703 1091 y Fk(f)725 1098 y Fj(\010)750 1103 y Fg(1)p Fe(;i)791 1091 y Fw(\()p Fk(h)835 1098 y Fi(i)p Fd(\000)p Fj(1)894 1091 y Fk(;)8 b(h)941 1098 y Fi(i)954 1091 y Fw(\))860 b(\(52\))534 1184 y(=)42 b Fk(p)p Fw(\()p Fk(o)674 1165 y Fd(\003)674 1195 y Fi(i)693 1184 y Fm(j)p Fk(h)732 1191 y Fi(i)746 1184 y Fw(\))8 b(max)777 1214 y Fi(h)797 1219 y Fe(i)p Fb(\000)p Fg(1)856 1184 y Fm(f)p Fk(p)p Fw(\()p Fk(h)946 1191 y Fi(i)959 1184 y Fm(j)p Fk(h)998 1191 y Fi(i)p Fd(\000)p Fj(1)1057 1184 y Fw(\))14 b(max)1083 1214 y Fi(h)1103 1219 y Fg(1)p Fe(;i)p Fb(\000)p Fg(2)1188 1184 y Fk(p)p Fw(\()p Fk(h)1255 1191 y Fi(i)p Fd(\000)p Fj(1)1314 1184 y Fk(;)8 b(h)1361 1191 y Fj(1)p Fi(;i)p Fd(\000)p Fj(2)1447 1184 y Fk(;)g(\036)1495 1165 y Fd(\003)1495 1195 y Fj(1)p Fi(;i)p Fd(\000)p Fj(1)1580 1184 y Fw(\))p Fm(g)211 b Fw(\(53\))534 1279 y(=)42 b Fk(p)p Fw(\()p Fk(o)674 1260 y Fd(\003)674 1290 y Fi(i)693 1279 y Fm(j)p Fk(h)732 1286 y Fi(i)746 1279 y Fw(\))14 b(max)772 1309 y Fi(h)792 1314 y Fg(1)p Fe(;i)p Fb(\000)p Fg(1)869 1279 y Fm(f)p Fk(p)p Fw(\()p Fk(h)959 1286 y Fi(i)972 1279 y Fm(j)p Fk(h)1011 1286 y Fi(i)p Fd(\000)p Fj(1)1070 1279 y Fw(\))p Fk(p)p Fw(\()p Fk(h)1155 1286 y Fi(i)p Fd(\000)p Fj(1)1214 1279 y Fk(;)8 b(h)1261 1286 y Fj(1)p Fi(;i)p Fd(\000)p Fj(2)1347 1279 y Fk(;)g(\036)1395 1260 y Fd(\003)1395 1290 y Fj(1)p Fi(;i)p Fd(\000)p Fj(1)1481 1279 y Fw(\))p Fm(g)310 b Fw(\(54\))534 1374 y(=)48 b(max)611 1404 y Fi(h)631 1409 y Fg(1)p Fe(;i)p Fb(\000)p Fg(1)716 1374 y Fk(p)p Fw(\()p Fk(h)783 1381 y Fi(i)797 1374 y Fk(;)8 b(h)844 1381 y Fj(1)p Fi(;i)p Fd(\000)p Fj(1)930 1374 y Fk(;)g(\036)978 1355 y Fd(\003)978 1385 y Fj(1)p Fi(;i)1018 1374 y Fw(\))796 b(\(55\))37 1496 y(whic)o(h)18 b(is)e(the)h(result)f(one)g(exp)q(ects)h(for)f(the)g(up)q(dated)h(p)q (oten)o(tial)g(at)f(this)g(clique.)25 b(Th)o(us,)16 b(w)o(e)g(can)g (express)37 1554 y(the)g(separator)e(p)q(oten)o(tial)517 1542 y(^)508 1554 y Fk(f)530 1561 y Fj(\010)555 1566 y Fg(1)p Fe(;i)595 1554 y Fw(\()p Fk(h)639 1561 y Fi(i)653 1554 y Fw(\))h(recursiv)o(ely)i(\(via)e(Equation)g(54\))f(as)499 1649 y(^)489 1661 y Fk(f)511 1668 y Fj(\010)536 1673 y Fg(1)p Fe(;i)577 1661 y Fw(\()p Fk(h)621 1668 y Fi(i)635 1661 y Fw(\))e(=)h Fk(p)p Fw(\()p Fk(o)776 1643 y Fd(\003)776 1673 y Fi(i)796 1661 y Fm(j)p Fk(h)835 1668 y Fi(i)849 1661 y Fw(\))8 b(max)880 1691 y Fi(h)900 1696 y Fe(i)p Fb(\000)p Fg(1)958 1661 y Fm(f)p Fk(p)p Fw(\()p Fk(h)1048 1668 y Fi(i)1062 1661 y Fm(j)p Fk(h)1101 1668 y Fi(i)p Fd(\000)p Fj(1)1160 1661 y Fw(\))1187 1649 y(^)1178 1661 y Fk(f)1200 1668 y Fj(\010)1225 1673 y Fg(1)p Fe(;i)p Fb(\000)p Fg(1)1304 1661 y Fw(\()p Fk(h)1348 1668 y Fi(i)p Fd(\000)p Fj(1)1408 1661 y Fw(\))p Fm(g)p Fk(:)370 b Fw(\(56\))37 1785 y(This)13 b(is)g(the)f(same)f(recursiv)o(e)i (equation)f(as)g(used)h(in)f(the)g Fk(\016)i Fw(v)m(ariables)g(in)e (the)g(Viterbi)i(algorithm)e(\(Equation)37 1843 y(33a)21 b(in)h(Rabiner)h(\(1990\)\):)31 b(the)21 b(separator)f(p)q(oten)o (tials)j(in)f(Da)o(wid's)f(algorithm)g(using)h(a)g(left-to-righ)o(t)37 1901 y(sc)o(hedule)i(are)d(exactly)h(the)g(same)f(as)g(the)h Fk(\016)844 1885 y Fd(0)855 1901 y Fk(s)g Fw(used)g(in)h(the)e(Viterbi) i(metho)q(d)f(for)f(solving)h(the)g(MAP)37 1959 y(problem)16 b(in)g(HMM\(1,1\).)108 2017 y(Pro)q(ceeding)g(recursiv)o(ely)h(in)f (this)g(manner)f(one)g(\014nally)i(obtains)e(at)g(the)g(ro)q(ot)f (clique)494 2106 y(^)484 2118 y Fk(f)506 2125 y Fj(\010)531 2131 y Fg(1)p Fe(;N)588 2118 y Fw(\()p Fk(h)632 2125 y Fi(N)t Fd(\000)p Fj(1)710 2118 y Fk(;)8 b(h)757 2125 y Fi(N)790 2118 y Fw(\))13 b(=)27 b(max)868 2148 y Fi(h)888 2154 y Fg(1)p Fe(;N)s Fb(\000)p Fg(2)989 2118 y Fk(p)p Fw(\()p Fk(h)1056 2125 y Fi(N)t Fd(\000)p Fj(1)1135 2118 y Fk(;)8 b(h)1182 2125 y Fi(N)1215 2118 y Fk(;)g(h)1262 2125 y Fi(N)t Fd(\000)p Fj(2)1340 2118 y Fk(;)g(\036)1388 2099 y Fd(\003)1388 2129 y Fj(1)p Fi(;N)1448 2118 y Fw(\))366 b(\(57\))37 2241 y(from)15 b(whic)o(h)g(one)g(can)g(get)f(the)h(lik)o (eliho)q(o)q(d)j(of)c(the)h(evidence)i(giv)o(en)e(the)g(most)f(lik)o (ely)i(state)e(of)h(the)g(hidden)37 2299 y(v)m(ariables:)644 2388 y(^)635 2400 y Fk(f)5 b Fw(\()p Fk(e)p Fw(\))41 b(=)73 b(max)837 2430 y Fi(h)857 2436 y Fe(N)s Fb(\000)p Fg(1)925 2430 y Fi(;h)955 2436 y Fe(N)1001 2388 y Fw(^)992 2400 y Fk(f)1014 2407 y Fj(\010)1039 2413 y Fg(1)p Fe(;N)1095 2400 y Fw(\()p Fk(h)1139 2407 y Fi(N)t Fd(\000)p Fj(1)1218 2400 y Fk(;)8 b(h)1265 2407 y Fi(N)1298 2400 y Fw(\))516 b(\(58\))760 2494 y(=)42 b(max)842 2524 y Fi(h)862 2530 y Fg(1)p Fe(;N)929 2494 y Fk(p)p Fw(\()p Fk(h)996 2501 y Fj(1)p Fi(;N)1057 2494 y Fk(;)8 b(\036)1105 2475 y Fd(\003)1105 2505 y Fj(1)p Fi(;N)1165 2494 y Fw(\))649 b(\(59\))108 2617 y(Iden)o(ti\014cation)17 b(of)f(the)g(v)m(alues)h(of) e(the)h(hidden)h(v)m(ariables)h(whic)o(h)e(maximize)h(the)f(evidence)i (lik)o(eliho)q(o)q(d)37 2675 y(can)j(b)q(e)g(carried)g(out)f(in)h(the)f (standard)g(manner)h(as)f(in)h(the)f(Viterbi)i(metho)q(d,)f(namely)g(b) o(y)f(k)o(eeping)i(a)952 2799 y(30)p eop %%Page: 31 32 31 31 bop 37 16 a Fw(p)q(oin)o(ter)18 b(at)e(eac)o(h)h(clique)h(along)f (the)g(\015o)o(w)f(in)h(the)g(forw)o(ard)f(direction)i(bac)o(k)e(to)g (the)h(previous)h(clique)g(and)37 74 y(then)g(bac)o(ktrac)o(king)f (along)h(this)g(list)g(of)f(p)q(oin)o(ters)h(from)e(the)i(ro)q(ot)e (clique)k(after)c(the)i(collection)h(phase)f(is)37 133 y(complete.)k(An)16 b(alternativ)o(e)g(approac)o(h)f(is)h(to)f(use)g (the)h(distribute)h(phase)f(of)f(the)g(Da)o(wid)h(algorithm:)k(this)37 191 y(has)f(the)f(same)g(e\013ect,)h(namely)l(,)g(once)g(the)g (distribution)h(\015o)o(ws)d(are)h(completed,)i(eac)o(h)f(lo)q(cal)g (clique)h(can)37 249 y(calculate)g(b)q(oth)f(the)f(maxim)o(um)h(v)m (alue)g(of)f(the)h(evidence)h(lik)o(eliho)q(o)q(d)i(giv)o(en)d(the)f (hidden)i(v)m(ariables)g(and)37 307 y(the)c(v)m(alues)g(of)f(the)g (hidden)i(v)m(ariables)g(in)f(this)f(maxim)o(um)g(whic)o(h)i(are)d(lo)q (cal)j(to)d(that)h(particular)h(clique.)952 2799 y(31)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF