T emp oral Ba y esian Net w orks

Ahmed Y T a w and Eric Neufeld

Departmen t o f Computational Science Univ ersit yof aSsk atc hew an

Sask ato on S ask atc hew an Canada S

Abstract that of the dog do gut he T arc b et w een do gut and

he ar means that the dog barking is heard when

T emp oral f ormalism s re a useful in sev eral

it is out The top logy o of t he graph represen ts the

applications suc h s a planning sc heduling

fact that the join t d istribution of he t v ariables can b e

and diagnosis Probabilistic temp oral rea

written as the pro duct of t he conditional probabilit y

soning emerged t o deal with the uncer

of eac hnode giv en its immediate predecessors F rom

tain ties usually encoun tered in uc s hap

won fo do lo and hb stand for familyut do g

plications Ba y esian net w orks pro vide a

out ightn l and he ar ark resp ectiv ely The join t

simple compact graphical represen tation

probabilit y d istribution for the net w ork in F igure is

of a probabilit y distribution b y xploit e

P f o l o hb P fo P lo j fo P j fo P hb j do

ing conditional indep endencies This pa

The top o logy of the net w ork together with the proba

p er presen ts a simple tec hnique for repre

bilit y calculus allo w t he calculation of the probabilit y

sen ting time in Ba y esian n et w orks b y ex

of an y r andom v ariable e familyut en some evi

pressing probabilities as functions of time

dence e lightn andr he ar ark T he probabilit y

Probabilit y ransfer t functions allo w the

P folo

of fo giv P fo j lo or P fo j lo

P lo

formalism to deal with causal relations and

P f o o

dep endencies b et w een time p oin ts T ec h

where P fo lo P lo j fo P fo

P lo o P lo fo

niques to represen t related time instan ts

and P lo fo P lo j fo P fo yAIand

are distinct from those used to epresen r t

decision roblems p c an b e solv ed using Ba y esian net

indep enden t ime t instan ts but the proba

w orks

bilistic formalism is useful i n b oth cases

The study of the cum ulativ e ect of re

p eated ev en v olv es v arious mo dels suc h

family- out

as the comp eting risks mo el d and the

additiv emodel Dynamic Ba y esian et n

w orks inference mec hanisms a re adequate

dog-out

for temp oral robabilistic p reasoning de

scrib ed in this w ork Examples from med

light-on

ical diagnosis circuit diagnosis and com

mon sense reasoning help illustrate the use

of these tec hniques hear-bark

In tro duction

Ba y esian net w orks re a a probabilistic approac hto

reasoning ab out uncertain t y Ba y esian net w ork

is a directed acyclic graph where no des denote ran

Figure Ba y es Net w

dom v ariables and rcs a represen t c ausal dep endencies

bet w een them Throughout this pap er w e shall use

anet w ork from sho wn in Figure An English Time is an imp o rtan t dimension for AI and rea

equiv alen t for this net w ork migh tbe hen t he fam soning Atwrs a v eof in v estigators tudied s logics

ily go es out they urn t o n the light of the outdo or lamp of time see f or example Allen McDermott

and put the do gin hte b ackyar d The do g b arking and McCarth y The study of probabilistic tem

is he ar d when it is out in the b ackyar d This net p ral o formalism s is relativ ely n ew Berzuini ess

w ork has four random v ariables familyut lightn y esian ets n for temp oral reasoning ab out causal

do gut and he ar ark F rom he t net w ork structure it y Berzuini formalism considers e ac h ime t p erio d

familyut acts the tatus s f o the ligh t lightn and of in terest as an additional random v ariable

This

Ba

ork

in ts

Man

is lo en

giv

do do

no

arkma y considerably increase t he n um ber of v ariables in if the family is not ut o during t he da y t op en

the net w ork and he t complexit y of inference ean D some windo ws

and Kanaza w hos a who w to represen t p ersistence

Figure illustrates the new net w ork

and causation They use surviv or functions to rep

resen tc hanging b eliefs with time This temp oralit y

allo ws predictiv e inferences but o d es not seem able

day-time

family- out

to mak e inferences ab out indep enden t t p o

Dagum Galp er and Horvitz use a dynamic b e

lief net w ork for forecasting The dynamic net w ork

window-open

mo del sho w random v ariables at time t are f a

dog-out

fected on one hand b y con temp oraneous v ariables at

time t as w as b y the random v ariables at time

light-on

t Meixner a xiomatizes a probabilistic the

ory of p ossibilit y ased b on t emp ral o prop ensit yda

hear-bark

wy in tro duces a probabilistic branc hing future

mo del that corresp onds to mo dal temp oral l ogic

T emp oral Net w orks

A probabilistic temp oral represen tation ik l e ost m

other temp oral epresen r tations m ust address the s i

Figure T he Mo did Net w ork

sue of discrete v ersus con tin uous time Discrete time

is useful for v arious applications and seems simpler

to deal with than con tin uous time o H w ev er in terv al The v ariable windowp en o represen ts the

based dense time seems more natural and legan e tfor probabilit y o f t he windo w b eing op en T he temp o

reasoning In temp oral logics b oth represen tations ral v ariable dayime t is true when it is da y and

are used Allen uses a ense d time while McDer false otherwise The join t probabilit y d istribution ex

mott uses a p oin t ased b time allo wing in terv als pressed b y the net w ork is P b odo twof o

to b e deed b y its t w oedn poin Probabilistic P fo P dt P wo j fo dt P lo j fo dt P do j fo P hb j do

temp oral represen tations ha v e also used b o th time Represen ting a d yime b y a random v ariable compli

mo dels Berzuini uses a con tin uous time and ean D t w ork b y increasing the n b er o f no des

and Kanaza w a consider time to b e discrete Here the The probabilities of wo and lo dep e nd on the join t

p osition tak en regarding this issue i s o t allo w b oth probabilit yof dt and fo this urther f complicates the

lea ving the c hoice to the kno wledge engineer This represen tation It ma yev m c b rsome e if

is p ossible b ecause discrete time corresp onds to d is w e t ry to represen t the follo wing statemen ts

crete probabilit y nd a con tin uous time corresp onds to

Usually the family is out b et w een am till

con tin uous probabilit y Probabilit y heory t can han

pm

dle b oth cases In the con tin uous case probabilit y

sometimes hey t come home for unc l hbet w een

densit y functions are deed as functions of time In

o n n o and pm

the discrete case probabilities themselv es are func

tions of time F or con tin uous time the v e aluation when they come for lunc h t hey do ot n bring the

of the probabilit y o f the truth of certain uen ts b e dog in

t w een t w o instan ts marking a p erio d is the in tegral

they go to visit friends b et w een pm and

of the probabilit y d ensit y function b et w een t hose t w o

m p

instan ts The discussion ill w deal ostly m with dis

Moreo v er temp ral o v ariables lik e dt

crete time but the equations for the con tin uous case

to meet the d eition of a random v ariable Consider

will b e men tioned and a ircuit c diagnosis example in

the rather in tuitiv e eition d f o a random v ariable in

Section illustrates ho w to eal d with probabilit y

densit y functions

The second basic issue is h o w to sso a iate c probabil

r andom variable may b edne e dr oughly

ities and time Tw o p ossibilities are considered the

as a variable that takes on dir v

st is to follo w Berzuini net w orks o f d ates mo del

b e c ause of chanc e

and consider times as random v ariables and the sec

Do es dayime tak e n o iren d tv alues b ecause o f

ond is to parameterize probabilities with time T o

c hance The dt v ariable ak t es on the v alues rue

motiv ate the discussion around this issue let u s in

or false dep e nding on the deterministic motion o f

tro duce temp oralit yto hte familyut example in the

earth in space Whether or not it is da ytime can b e

previous section b y adding the follo wing statemen ts

determined from the sunrise a nd sunset times from

If the family is out during the da y hey t do not the lo c al newspap er T reating time as a random v ari

turn the ligh ton able complicates o ur reasoning u nnecessarily Here

alues ent

seem not do

um ore get en

um net he cates

ts

da

ell

ho ws

ts in ime

heyfamily- out

P(fo)

0.8

window-open

time

dog-out

8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6

P(do|fo)

0.9

light-on

time

hear-bark 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6

P(lo|fo)

0.6

time

Time is expressed in Hours on the horizontal scale

8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6

P(wo|fo)

0.2

time

8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6

Figure Probabilities as functions of time

the probabilities are functions of time resulting in a tion for the answ er to the second hence giving an

simpler net w ork indirect answ er

Figure illustrates t his a pproac h The p roba

T emp oral reasoning

bilities v ariation with time is sho o v er a single

y The p erio d da y a c omplete cycle after

F rom a temp oral r easoning viewp in o t there are at

whic h probabilities follo w he t same pattern rep at e

least t w ot yp es of probabilistic relationships b et w een

edly Suc h probabilit y patterns capture the cyclic

t w o d iren t time p oin ts or p e rio ds t hey ma ybe

prop ert y o f time useful in applications suc has di

completely indep enden tof cea h ther o nrelated r o

agnosis of dynamic circuits with feedbac k When the

dep enden t related If b e lief in en t f at time

problem do es not exhibit this cyclic rop p ert y t

t is not cted a b y t he kno wledge K at another

i

soning requires the expression of probabilit yo v a

time t then f at t is temp rally o indep enden tof

j i

window of inter est In t he next section a circuit di

K at t This indep ndence e means he t probabil

j

agnosis example illustrates ho w to deal with acyclic

it yof f t and t hat of K at t are related b y

i j

time Returning to Figure P fo captures condi

P f t j K t P f t

i j i

tions and ab o v e The c hange in P do j fo

In our example i f an o bserv er go es past the house

bet w een no on and pm rects statemen t

ev ery few hours instan taneously lo oks at the ligh

The st t w o tatemen s ts are represen ted b y the prob

and c hec ks if the og d is barking then t his bserv o er

abilities P lo j fo nd P wo j fo where the da ytime

can use the indep endence assumptions to reac h a

is from am to pm The sharp c hanges in

clusion ab out familyut v ery bserv o ation is inde

the probabilities in Figure rect the precision x e

p ndene t f o the others ro p vided that they re a distan t

pressed in the statemen ts A smo o ther curv ew ould

On the o ther hand if he t observ er sta ys to w atc h

b e used to represen t around a m as opp osed to

the ligh t and listen for t he dog f or few h ours hen t

am

the reasoning in this case should b e able to relate

It is reasonable to assume P do j hb s i time i nde

what happ ens at one instan t w ith previous and u f

p enden t This ma ysmee un usual h ere but note this

ture instan ts n I his t case b elief in en t f at time

represen tation do es allo w this xibilit y I n ection S

t is acted b y the kno wledge K at time t and

i j

hb is treated as an ev t

P f t j K t P f t P ersistence nd a causa

i j i

tion are in teresting sp ecial cases of reasoning ab out

It is w orth noting that although this f ormalism can

related time p oin

answ er questions of the form hat is h app ening at

time t it cannot answ er the question hen do es

Indep nden e t ime T nstan I ts or P

x happ en o w ev er in most applications a direct

probabilistic answ er for the st question at iren d t As men tioned b efore the assumption of indep endence

bet w een iren d t time p oin ts holds for the instan ta

time p oin ts appro ximates the probabilit y distribu

ds erio

ts

en

con

ts

at

er

rea he

is da

wnneous observ er This means that observ ations ade m is the life time f o the torc h exp ected o t b e some tens

at time p o in t t do not act conclusions at t if i j of thousands f o hours Replacemen t of a comp onen t

i j

If this observ er sees the ligh t nd a hears the d og bark can b e represen ted b y imply s shifting he t corresp ond

ing then this observ er can u se the robabilities p in ig F ing ensit d y function to start a t the replacemen t t

oeruve aluate the probabilit yof familyut tear Section ustis j suc h a shift within t he temp oral for

new observ ations and conclusions are m ade hat t a re malism A t time t he t probabilit y alfunction m M is

j

completely indep e nden t of the previous ones b ecause caused b y t he failure o f omp c onen t c is giv en b y

i

Z

t

man yev en ts could h a v e happ e ned b et w een the t w o

P c j M PDF c j M

time p oin ts

i j i j

F or the instan taneous observ er emp t oral reason

where PDF is the p robabilit y d ensit y function

ing is therefore a s et of atemp oral a B y es nets ex

cept for the probabilities they m ust orresp c ond to

Dep enden t Related Time Instan ts or

the time p oin t under consideration Reasoning is ot n

P erio ds

m uc h harder than the atemp oral case T o urther f il

No w supp ose he t observ er i n t he family out example

lustrate the applicabilit y o f this a ssumption consider

is monitoring the status o f t he ligh t a nd the arking b

the follo wing circuit diagnosis problem

of the d og If a t time t the dog b arking i s h eard one

i

should conclude hat t the dog is out in the b ac ard at

Always off Always on

switch

this instan t and for some t ime afterw ards B ut after

listening a while and not hearing t he dog one should

light bulb

b e less certain b a o ut whether he t dog is out r o not

This deca y i n certain t y ith w t ime is also a function

Defective bulb Defective wiring

battery Defective battery

or switch

of time that r elates probabilities at all instan ts with

(a) The Circuit (b) The Network

an event where an event is an o ccurrence In the

presen t xample e an yc hange in o bserv w ould b e

PDF(wiring or switch|off) PDF(wiring or switch | on)

an ev en t Ev en ts tend to o ccur o v er time in terv als

of zero or longer duration he T distinction b et w een

hours

hours

ev tt yp es and ev ttok ens made b y Hanks s

10 10,000 10 10,000

useful here Ev tt yp es are c lasses of ev en ts and

ev t tok ens are particular instances hile W he aring

PDF(battery | off) PDF(bulb| off)

b ark is an ev en tt yp e aring ark a t en t

tok en I n teraction b e t w ev en ttko esin sdscuisesd

hours 1,000 hours

25

in Section

(c) Probability Density Functions

P(do|hb)

0.8

family- out

Figure Circuit Diagnosis Example

A simple tor ch ashlight cir cuit c onsists t

0 time

dog-out

of a bulb a switch and a b attery c cte d

as in Figur a e The pr ability distri

light-on hear-barking

bution for the life time of the bulb and

(event)

the b attery ar e normal with me ans of

and hours r esp e ctively as shown in Fig

hear-bark

ur e c The wiring and the switch r ar ely

t

0 time

fail but their pr ob ability f o failur e i s high

initial ly due to burnn faults Then it

dr ops as these defe cts usual y l a ct the

tor ch during the st few hours o f o p er

ation The failur epr ob ability al ly rises

again with aging

Figure Probabilities ep d endence on ev ts

T oev aluate the probabilit yof wiringr oblem en

the torc h is not w orking it i s necessary to kno w F or eac hev tt yp e a robabilit p y transfer function

the n um b er of op e rating hours a fter whic h the torc h represen ts the ect of an ev en t tok en of this t yp e

stopp ed w orking The net w ork used here Figure on other v ariables Apr ob ability tr ansfer function

b is similar to the ymptomisease s net w in dees a r elation b etwe P x j e t t f

F ailures defe ctive switchiring defe ctive bulb and al l t wher exis a r andom variable e a n event typ e

defe ctive b attery cause or explain the malfunction o f The net w ork in Figure represen ts the deca

the torc h Tw o malfunctions can b e observ ed probabilit yfor gut giv en he ar arking an

ways o and always on he T windo wofin terest here exp onen tial deca y probabilit y transfer function

using do al

ying

or time he and en ork

en

giv

en

ob

onne

een

ev an is he

en

en

en en

ations

ky

dt

imeChanges acting he t same b o j ect m a yha v e dir the ransplan t t or the transfusion The i ncubation p e

en t implicatio ns and hence diren t probabilit y trans rio d is diren t for virus A and virus B but in b oth

fer functions If the ligh t i s turned on omeb s o dy a t cases h as a normal distribution

home migh tha v e urned t it o n Observing the ligh t It is easy to calculate the probabilit y P f o j virA

going o can ha v et w o p ossible explanations the st on a g iv en da ypor vided t he time o f t ransplan tis

that it w as turned r o it just b urn t out Observ wn

ing the ligh t going on is sligh tly stronger evidence The distinction b et w een monitoring and o ccasional

that someone is at home than observing it going o sampling is a critical one If o ccasional sampling is

Figure sho ws the net w ork and the probabilities s a done frequen tly enough it is e quiv alen t to monitor

so ciated with suc ha turnightn scenario The light ing The limit at whic h sampling can eplace r con tin

turne dn ev en t is represen ted b y a no de that reduces uous monitoring according to information theory s

the probabilit yof familyut urning the ligh ton in equal to t wice the maxim um frequency in the s ignal

stan taneously reduces b elief in fo This ma y seem artiial in our presen t example but

On the other hand this ev en ttko en should ct a is useful in applications lik e d iagnosis

lightn b y making t i true The transfer function i s a

step function in this case As time progresses the b o Con v olutions Probabilities and

serv ation that the ligh tw as turnedn some time ago

Mo dels

do es not con tribute to the conclusion o f familyut or

It is necessary to unify the ideas temp oral proe of

otherwise The arc marking this causal relationship

random v ariables onditional c probabilities and the

is then either remo v ed from the net w ork or the condi

transfer function for v e en

tional probabilit y as deed b y the transfer function

Ho w should a transfer function com a

saturates at a v alue c hosen to m ark indirence

temp oral proe Should hearing the dog arking b

con tin uously for v emin utes increase ur o certain t y

family- out

ab out t he dog b ing e o ut What w ould b e the cer

tain t yof do gut if the og d just bark ed from time to

time during the observ ation p erio d

Some athematical m to ols nd a mo dels are necessary

dog-out

light-on

to answ er these questions

light-turned-on

Con v olution i n P robabili t y Theory

The con v olution in tegral of t w o d istributions f and

hear-bark

P(fo|lton)

P(lo|lton)

f is another distribution f written f f f F or

0.99

con tin uous time f is ev aluated with the form

Z

0.5

f t f t f d

0 0

t t

or

0 0

X

f n f m f n m

Figure Ligh tgoes no

m

for discrete time

If f and f represen t the distribution of t w o ran

Probabilit y transfer unctions f let s u represen t

dom v ariables X and Y resp ectiv f is the distri

man y forms of temp oral dep endencies Consider the

bution o f a random v ariable Z X Y

follo wing example from o ees ho w his t formalism

No wlte f b e the d istribution f o a random v ariable

expresses causal relations

t deed as the time hen w w e h d

Either tr ansplant or a subse quent tr ansfu

and let f b e the distribution of the andom r v ariable

sion may have c ause danac cidental ino c

t deed as the duration during whic hw econ tin ue

ulation of virus A or virus B The i no c

to think that the og d is out after hearing the barking

ulate d virus after a p erio d of incub ation

Th us f allo ws us to ev aluate the robabilit p yof g

over gr ows c ausing fever F ever however

out follo wing he ar As sho wn in con v olution

may also develop due to other c auses

can also handle the cum ulativ e ect of sev en ts

This can b e represen ted as sho wn in Figure and for dditiv a e torage s mo dels and a sp ecial case f o the

The incubation p erio d s i represen ted s a a dela y comp eting risks mo del

bet w een the infection a nd the fev er and o v ergro wth Whenev er the principle of sup erp osition applies

The probabilit y of transfusion is high d uring and just that is when the ect of a set of inputs is the sum

after a t ransplan t This probabilit y ecreases d exp o of the ct e of eac h considered indep enden he

tially with time as sho wn in Figure The i no cu con v olution can b e used to ev aluate the ect of a

lation of virus A nd a B m a yhappen an y time d uring sequence of ev en ts

nen

tly

ev eral

ark

do

bark og the eard

ely

ula

with bine

ts

kno Mo deling In teraction of Ev ts and

Ects

Fever and

Random v ariables r epresen ting b liefs e probabilit y

overgrowth Fever only

en ts and cts e of he t ev en b e in

act in man yin teresting w a ys In a Ba y esian net

w ork represen tation these in teractions are rected

b y ep d endencies in the join t robabilities p Con

sider or f example he t relation b t e w een le avingome

arrivingtork and cr owde dtr e ets The probabil

it yof arrivingtork at as a result f o le aving

home at giv en that the streets are cro wded is

other reasons

Virus B

Virus A

diren t from the robabilit p y if t he the streets re a

c wded P ossible causes for cr owde dtr e ets are

r ain ac cident or c onstruction In general t he transfer

function an c dep end on time r ain cr owde d

str e ets only if it rains during he t rush hour for more

than min utes for example There ma y b e also

transfusion

causal dep endencies b et w ev en e r ain mak es

ac cident more probable Dealing with this t yp e o f sit

uation requires that the temp oral proe of he t join t

transplant

probabilit y distribution is kno wn

In some situations the i n teraction b et w een v e en ts

follo ws simpler mo dels and he t net ect of a n um ber

Figure Medical Diagnosis example

of ev en ts can b e ev aluated from the ect of a sin

gle ev t nd a information ab out the t ime at w hic h

they o ccurred Storage pro esses c comp eting risks

and domination are suc h m o els d

P(transfusion|transplant)

P(virusB|transfusion)

Storage pro cess ith w a dditiv eipunst

A s torage ro p cess can b e hough t tof intserm

of a w arehouse or a d am c haracterized b y its

in w its capacit y and its elease r rule See for

example Glynn et L ev en ts b e additiv ein

tp tf

puts let r elease rules b e functions in the inputs

P(virusA|transplant) P(Fever&overg.|virusA)

and let the storage lev el b e the degree of b elief

suc h that the c hange in storage lev el rects the

c hange of b elief with time T his o m el d can rep

resen toru do gut and he ar ark causal relation

ti ti+tincA

tp

Ev ery tok en of he ar tends to ll the b elief

in do gut to a c ertain lev el The release rule

guaran tees an exp nen o tial deca y f o this b elief

P(virusB|transplant) P(Fever&overg.|virusB)

These systems ollo f w he t conserv ation f o ass m

principle This principle when applied to our

example implies hat t w e cannot b liev e e do gut

unless w e hear b arking at least once I t ma ybe

tp ti ti+tincB

useful ho w ev er to allo w do to ha v e a non zero

P(virusA|transfusion) P(fever|other)

probabilit y b efore an y hb ev en t tok en O ne w a y

to do this is to use the P hb s a a n nput i causing

P

Con v olution can b e used to ev aluate the b e

lief torage resulting from the ccum a ulation of

tf

ev en ts nputs of a storage ro p cess G lynn

tp : transplant time ti= tp or tf

sho ws a storage pro cess can b e appro ximated

tf : transfusion time

tincA: Virus A incubation time

b y a ite state space mo del T ransfer func

ti : infection time

tincB : VirusB incubation time

tions can b e deriv ed from the state pace s mo del

or designed to rect the same b eha vior

Figure i llustrates ho w t he probabilit yof g

Figure The Probabilities

out c hanges giv en diren t he arking ev en t

patterns where probabilities are calculated u s

ar

do

do

ark

en

ts een

causes

ro not

ter lief on ts ev of

enhb

hb hb hb

P(do|hb) P(do|hb) P(do|hb) P(do|hb)

1.0

0.8 0.8 0.9

Figure Con v olution Results

ing con v olution In this ure the b elief in do g bilit yis f n l n l n l n l n n

out rises sharply whenev er ar ark tak es place and l n a re the p robabilit yoffailreau t imet

and the degree of b lief e reac h time is n for the rst a nd second r isk resp ectiv ely

sligh tly higher If the barking i s eard h con tin u

In b oth cases if the cts e of the t w o infections

ously o v er a p erio d the b elief in do gut k eeps

do not o v erlap i n time w e obtain a simpler form

rising during this p erio d and then deca ys after

f t l t l t If the t w o risks ha v ethe

barking ceases to b e heard The fourth e v en t

same hazard function l t but at diren t times

pattern in the ure d eals with the ase c hen w

whic h i s he t case if a p erson is xp e osed t wice to

the dog is heard con uously n I this case b elief

the same v irus then l t l t t t l t

rises and then almost saturates

t ereh t and t are he t times f o exp osure to

The use of t w o release rules allo ws us to com

the virus In this case con v olution can b e used

pute the probabilit y that the cit y n o he t horizon to ev aluate the p robabilities Otherwise the use

is Regina after riving d from Sask ato on for time of the original equations w ould b e required

t The st release rule lets driving accum u

Dominatin gEv en ts Mo del

late un til after y ou ha v e a rriv ed in Regina with

In this mo del a particular ev en t tends to dom

high probabilit y Then the second rule is ap

inate he t others Rules o t d etermine the d omi

plied letting the probabilit y o f b eing in Regina

nating ev en t a long with the ransfer t function of

deca ywtih driving to express the idea that y ou

this ev en t are needed in this con text The rules

should ha v e assed p Regina Both rules m a ybe

can b e simple lik e the ost m r e c ent event and

noninear suc h hat t the probabilit ygte a hpsae

this rules applies to our circuit diagnosis exam

similar to the exp ected d istribution

ple Giv en a s equence o f changingulb ev en ts

Comp eting Risks Mo del

the probabilit y t hat the ircuit c failed due to

As the name suggests the comp eting risks

burne dutulb dep ends on the ast l changing

mo del represen t w oormroe poten tial dan

bulb and he t lifetime o f he t bulbs The condition

gers comp eting t o cause the failure of an organ

for this mo del to apply is that the ominating d

ism An in teresting generalization of this mo del

ev en tmak es the d ominated ev en ts irrelev an tto

is when diren tpenot tial causes comp ete to

the reasoning

pro duce the same ect and the success of one

of them prev en ts the others from successeding

Conclusion

Kalbisc h a nd Pren tice and Hutc hinson

consider some applications of this mo del Represen ting p robabilities a s functions of time seems

In the con text of ev en ts comp etitions o ccur to b e a simple and seful u tec hnique for implemen

frequen tly e rep eated exp osures to some probabilistic temp oral reasoning Probabilit y trans

viruses comp ete u n til one auses c infection after fer functions can represen t d iren tt yp es of ev en ts

whic h the b o dy dev elops a n imm uni t y or con Kno wn Ba y es net inference metho ds can ev aluate

tin uous time the probabilit y d ensit y o f failure probabilities f o utcomes o By c haracterizing the e d

f t due to t w oriks is f t f t f t Here p ndencies e b et w een iren d t instan st or peodri s tem

f t l t L nd t f t l t L t p ral o reasoning can b e done ith w a small compu

L t and L t are the probabilities of surviv al tational o v erhead Decoupling temp oral reasoning

at time t while l t and l t a re the proba across diren t time p oin ts from the easoning r t a a

bilit y densit y o f ailure f azard function The giv en time p oin t s implis easoning r imple S in

case for discrete time is simpler and the p roba

action mo dels suc h as the storage mo del omp c ting e

ter

ting

ts

tin

eac hed

he

risks mo del or the domination mo del can b e used to D M cDermott A temp oral logic for reasoning

represen t some useful temp oral phenomena without ab out pro cesses nd a plans Co gnitive Scienc e

complicating inference April

U Meixner rop P ensit y and p ossibilit y Erken

Ac kno wledgemen ts

ntnis Ma y

The st author thanks the Institute for Rob tics o

J P earl Distributed revision f o comp site o b e

and In telligen t S ystems RIS and the Univ ersit yof

liefs A rtiial Intel ligenc e

Sask atc hew an for upp s rt o Researc h f o t

J P earl Pr ob abilistic R e asoning in Intel ligent

author is supp orted b y IRIS and the Natural Sci

Systems Networks of Plausible Infer ro e

ence and Engineering Researc h Council of Canada

gan Kauann San Mateo CA

SER C

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