T e m p o r a l B a y e s i a n N e t w o r k s

reverandrunAI and Robotics

Nov 7, 2013 (3 years and 8 months ago)

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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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