How to use BNT for DBNs.docx
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HowtouseBNTforDBNs
HowtouseBNTforDBNs
∙Modelspecification
oHMMs
oKalmanfilters
oCoupledHMMs
oWaternetwork
oBATnetwork
∙Inference
oDiscretehiddennodes
oContinuoushiddennodes
∙Learning
oParameterlearning
oStructurelearning
Modelspecification
DynamicBayesianNetworks（DBNs）aredirectedgraphicalmodelsofstochasticprocesses.TheygeneralisehiddenMarkovmodels（HMMs）andlineardynamicalsystems（LDSs）byrepresentingthehidden（andobserved）stateintermsofstatevariables,whichcanhavecomplexinterdependencies.Thegraphicalstructureprovidesaneasywaytospecifytheseconditionalindependencies,andhencetoprovideacompactparameterizationofthemodel.
Notethat"temporalBayesiannetwork"wouldbeabetternamethan"dynamicBayesiannetwork",sinceitisassumedthatthemodelstructuredoesnotchange,butthetermDBNhasbecomeentrenched.Wealsonormallyassumethattheparametersdonotchange,i.e.,themodelistime-invariant.However,wecanalwaysaddextrahiddennodestorepresentthecurrent"regime",therebycreatingmixturesofmodelstocaptureperiodicnon-stationarities.
Therearesomecaseswherethesizeofthestatespacecanchangeovertime,e.g.,trackingavariable,butunknown,numberofobjects.Inthiscase,weneedtochangethemodelstructureovertime.BNTdoesnotsupportthis.
HiddenMarkovModels（HMMs）
ThesimplestkindofDBNisaHiddenMarkovModel（HMM）,whichhasonediscretehiddennodeandonediscreteorcontinuousobservednodeperslice.Weillustratethisbelow.Asbefore,circlesdenotecontinuousnodes,squaresdenotediscretenodes,clearmeanshidden,shadedmeansobserved.
Wehave"unrolled"themodelforthree"timeslices"--thestructureandparametersareassumedtorepeatasthemodelisunrolledfurther.HencetospecifyaDBN,weneedtodefinetheintra-slicetopology（withinaslice）,theinter-slicetopology（betweentwoslices）,aswellastheparametersforthefirsttwoslices.（Suchatwo-slicetemporalBayesnetisoftencalleda2TBN.）
Wecanspecifythetopologyasfollows.
intra=zeros
（2）;
intra（1,2）=1;%node1inslicetconnectstonode2inslicet
inter=zeros
（2）;
inter（1,1）=1;%node1inslicet-1connectstonode1inslicet
Wecanspecifytheparametersasfollows,whereforsimplicityweassumetheobservednodeisdiscrete.
Q=2;%numhiddenstates
O=2;%numobservablesymbols
ns=[QO];
dnodes=1:
2;
bnet=mk_dbn（intra,inter,ns,'discrete',dnodes）;
fori=1:
4
bnet.CPD{i}=tabular_CPD（bnet,i）;
end
WeassumethedistributionsP（X（t）|X（t-1））andP（Y（t）|X（t））areindependentoftfort>1.HencetheCPDfornodes5,7,...isthesameasfornode3,sowesaytheyareinthesameequivalenceclass,withnode3beingthe"representative"forthisclass.Inotherwords,wehavetiedtheparametersfornodes3,5,7,...Similarly,nodes4,6,8,...aretied.Note,however,that（theparametersfor）nodes1and2arenottiedtosubsequentslices.
AboveweassumedtheobservationmodelP（Y（t）|X（t））isindependentoftfort>1,butitisconventionaltoassumethisistrueforallt.Sowewouldliketoputnodes2,4,6,...allinthesameclass.Wecandothisbyexplicitelydefiningtheequivalenceclasses,asfollows（seehereformoredetailsonparametertying）.
Wedefineeclass1（i）tobetheequivalenceclassthatnodeiinslice1belongsto.Similarly,wedefineeclass2（i）tobetheequivalenceclassthatnodeiinslice2,3,...,belongsto.ForanHMM,wehave
eclass1=[12];
eclass2=[32];
eclass=[eclass1eclass2];
Thistiestheobservationmodelacrossslices,sincee.g.,eclass（4）=eclass
（2）=2.
Bydefault,eclass1=1:
ss,andeclass2=（1:
ss）+ss,wheress=slicesize=thenumberofnodesperslice.Butbyusingtheabovetieingpattern,wenowonlyhave3CPDstospecify,insteadof4:
bnet=mk_dbn（intra,inter,ns,'discrete',dnodes,'eclass1',eclass1,'eclass2',eclass2）;
prior0=normalise（rand（Q,1））;
transmat0=mk_stochastic（rand（Q,Q））;
obsmat0=mk_stochastic（rand（Q,O））;
bnet.CPD{1}=tabular_CPD（bnet,1,prior0）;
bnet.CPD{2}=tabular_CPD（bnet,2,obsmat0）;
bnet.CPD{3}=tabular_CPD（bnet,3,transmat0）;
Wediscusshowtodoinferenceandlearningonthismodelbelow.（SeealsomyHMMtoolbox,whichisincludedwithBNT.）
SomecommonvariantsonHMMsareshownbelow.BNTcanhandleallofthese.
LinearDynamicalSystems（LDSs）andKalmanfilters
ALinearDynamicalSystem（LDS）hasthesametopologyasanHMM,butallthenodesareassumedtohavelinear-Gaussiandistributions,i.e.,
x（t+1）=A*x（t）+w（t）,w~N（0,Q）,x（0）~N（init_x,init_V）
y（t）=C*x（t）+v（t）,v~N（0,R）
Somesimplevariantsareshownbelow.
WecancreatearegularLDSinBNTasfollows.
intra=zeros
（2）;
intra（1,2）=1;
inter=zeros
（2）;
inter（1,1）=1;
n=2;
X=2;%sizeofhiddenstate
Y=2;%sizeofobservablestate
ns=[XY];
dnodes=[];
onodes=[2];
eclass1=[12];
eclass2=[32];
bnet=mk_dbn（intra,inter,ns,'discrete',dnodes,'eclass1',eclass1,'eclass2',eclass2）;
x0=rand（X,1）;
V0=eye（X）;%mustbepositivesemidefinite!
C0=rand（Y,X）;
R0=eye（Y）;
A0=rand（X,X）;
Q0=eye（X）;
bnet.CPD{1}=gaussian_CPD（bnet,1,'mean',x0,'cov',V0,'cov_prior_weight',0）;
bnet.CPD{2}=gaussian_CPD（bnet,2,'mean',zeros（Y,1）,'cov',R0,'weights',C0,...
'clamp_mean',1,'cov_prior_weight',0）;
bnet.CPD{3}=gaussian_CPD（bnet,3,'mean',zeros（X,1）,'cov',Q0,'weights',A0,...
'clamp_mean',1,'cov_prior_weight',0）;
Wediscusshowtodoinferenceandlearningonthismodelbelow.（SeealsomyKalmanfiltertoolbox,whichisincludedwithBNT.）
CoupledHMMs
HereisanexampleofacoupledHMMwithN=5chains,unrolledforT=3slices.EachhiddendiscretenodehasaprivateobservedGaussianchild.
Wecanmakethisusingthefunction
Q=2;%binaryhiddennodes
discrete_obs=0;%ctsobservednodes
Y=1;%scalarobservednodes
bnet=mk_chmm（N,Q,Y,discrete_obs）;
Waternetwork
Considerthefollowingmodelofawaterpurificationplant,developedbyFinnV.Jensen,UffeKj鎟ulff,KristianG.Olesen,andJanPedersen.（Clickhereformoredetailsonthismodel.FollowingBoyenandKoller,wehaveaddeddiscreteevidencenodes.）
WenowshowhowtospecifythismodelinBNT.
ss=12;%slicesize
intra=zeros（ss）;
intra（1,9）=1;
intra（3,10）=1;
intra（4,11）=1;
intra（8,12）=1;
inter=zeros（ss）;
inter（1,[13]）=1;%node1inslice1connectstonodes1and3inslice2
inter（2,[237]）=1;
inter（3,[345]）=1;
inter（4,[346]）=1;
inter（5,[356]）=1;
inter（6,[456]）=1;
inter（7,[78]）=1;
inter（8,[678]）=1;
onodes=9:
12;%observed
dnodes=1:
ss;%discrete
ns=2*ones（1,ss）;%binarynodes
eclass1=1:
12;
eclass2=[13:
209:
12];
eclass=[eclass1eclass2];
bnet=mk_dbn（intra,inter,ns,'discrete',dnodes,'eclass1',eclass1,'eclass2',eclass2）;
fore=1:
max（eclass）
bnet.CPD{e}=tabular_CPD（bnet,e）;
end
Wehavetiedtheobservationparametersacrossallslices.Clickhereforamorecomplexexampleofparametertieing.
BATnet
AsanexampleofamorecomplicatedDBN,considerthefollowingexample,whichisamodelofacar'shighlevelstate,asmightbeusedbyanautomatedcar.（ThemodelisfromForbes,Huang,KanazawaandRussell,"TheBATmobile:
TowardsaBayesianAutomatedTaxi",IJCAI95.ThefigureisfromBoyenandKoller,"TractableInferenceforComplexStochasticProcesses",UAI98.Forsimplicity,weonlyshowtheobservednodesforslice2.）
Sincethistopologyissocomplicated,itisusefultobeabletorefertothenodesbyname,insteadofnumber.
names={'LeftClr','RightClr','LatAct',...'Bclr','BYdotDiff'};
ss=length（names）;
Wecanspecifytheintra-slicetopologyusingacellarrayasfollows,whereeachrowspecifiesaconnectionbetweentwonamednodes:
intrac={...
'LeftClr','LeftClrSens';
'RightClr','RightClrSens';
...
'BYdotDiff','BcloseFast'};
Finally,wecanconvertthiscellarraytoanadjacencymatrixusingthefollowingfunction:
[intra,names]=mk_adj_mat（intrac,names,1）;
Thisfunctionalsopermutesthenamessothattheyareintopologicalorder.Giventhisorderingofthenames,wecanmaketheinter-sliceconnectivitymatrixasfollows:
interc={...
'LeftClr','LeftClr';
'LeftClr','LatAct';
...
'FBStatus','LatAct'};
inter=mk_adj_mat（interc,names,0）;
Torefertoanode,wemustknowitsnumber,whichcanbecomputedasinthefollowingexample:
obs={'LeftClrSens','RightClrSens','TurnSignalSens','XdotSens','YdotSens','FYdotDiffSens',...
'FclrSens','BXdotSens','BclrSens','BYdotDiffSens'};
fori=1:
length（obs）
onodes（i）=stringmatch（obs{i},names）;
end
onodes=sort（onodes）;
（WesorttheonodessincemostBNTroutinesassumethatset-valuedargumentsareinsortedorder.）WecannowmaketheDBN:
dnodes=1:
ss;
ns=2*ones（1,ss）;%binarynodes
bnet=mk_dbn（intra,inter,ns,'iscrete',dnodes）;
Tospecifytheparameters,wemustknowtheorderoftheparents.SeethefunctionBNT/general/mk_named_CPTforawaytodothisinthecaseoftabularnodes.Forsimplicity,wejustgeneraterandomparameters:
fori=1:
2*ss
bnet.CPD{i}=tabular_CPD（bnet,i）;
end
AcompleteversionofthisexampleisavailableinBNT/examples/dynamic/bat1.m.
Inference
ThegeneralinferenceproblemforDBNsistocomputeP（X（i,t0）|Y（:
t1:
t2））,whereX（i,t）representsthei'thhiddenvariableattimetandY（:
t1:
t2）representsalltheevidencebetweentimest1andt2.Thereareseveralspecialcasesofinterest,illustratedbelow.Thearrowindicatest0:
itisX（t0）thatwearetryingtoestimate.
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