Supplementary Material of "LUME-DBN: Full Bayesian Learning of DBNs from Incomplete data in Intensive Care"

1
Supplemen a y Ma e ial
A Full Condi ional Dis ibu ions o Missing Da a
We conside h ee con igu a ions whe e exac ly one alue is missing a ime
= 2:x2
1[MIS],x2
2[MIS],o x2
3[MIS]. The DBN is ep esen ed in Figu e A.1a and
he depic ed ela ionships a e mean o ep esen empo al ela ionship wi h a
single lag ( om ime −1 o ime ). These se ings e lec s uc u ally di e en
oles in he DBN:
–X2
1has no pa en s and one child,
–X2
2has one pa en and one child,
–X2
3has one pa en and no child en.
Fig. A.1. Two examples o DBNs. a) A DBN wi h 3 empo al nodes and 2 a cs.
b) A mo e complex DBN wi h 5 empo al nodes, wi h a node X
3wi h 2 pa en s
{X −1
1, X −1
2}, 2 child en {X +1
4, X +1
5}and one node wi h a common child en {X
4}.
Assuming a Uni o m p io o e he domain o each a iable, he FCD o a
missing alue only depends on he likelihood e ms in which ha a iable ap-
pea s. Since he samples a e independen , missing alues o di e en samples
bu o he same a iable a he same ime jus lead o he same o m o he
Gaussian.
Consequen ly:
– he FCD o X2
3depends only on i s own likelihood: no demons a ion is
needed because he condi ional likelihood o X2
3|X1
2is a p ope ly de ined
Gaussian dis ibu ion wi h pa ame e s µ3=β0(3) +β1(3)X1
2and σ2(3);
– he FCD o X2
1depends on i s ma ginal likelihood and he condi ional
likelihood o i s child X3
2;
– he FCD o X2
2depends on bo h he condi ional likelihood gi en i s pa en
X1
1and he condi ional likelihood o i s child X3
3.
2
P(x2
1[MIS]| · )∝P(X2
1)·P(X3
2|X2
1)
∝ N(x2
1[MIS]|µ1=β(1)
0, σ2(1))· N (x3
2|µ2=β(2)
0+β(2)
1x2
1[MIS], σ2(2))
∝exp −(x2
1[MIS]−β(1)
0)2
2σ2(1) !·exp −(x3
2−β(2)
0−β(2)
1x2
1[MIS])2
2σ2(2) !
∝exp −1
2"(x2
1[MIS])2−2x2
1[MIS]β(1)
0
σ2(1) +(x3
2)2
σ2(2)
−2x3
2(β(2)
0+β(2)
1x2
1[MIS])
σ2(2) +(β(2)
0+β(2)
1x2
1[MIS])2
σ2(2) #!
∝exp −1
2"(x2
1[MIS])2 1
σ2(1) +(β(2)
1)2
σ2(2) !
+2x2
1[MIS] −β(1)
0
σ2(1) +−β(2)
1(x3
2−β(2)
0)
σ2(2) !+C#!
∝ N x2
1[MIS]
µ∗=σ2∗· β(1)
0
σ2(1) +β(2)
1(x3
2−β(2)
0)
σ2(2) !,
σ2∗= 1
σ2(1) +(β(2)
1)2
σ2(2) !−1

P(x2
2[MIS]| · )∝P(X2
2|X1
1)·P(X3
3|X2
2)
∝ N(x2
2[MIS]|µ2=β(2)
0+β(2)
1x1
1, σ2(2))
· N(x3
3|µ3=β(3)
0+β(3)
2x2
2[MIS], σ2(3))
∝exp −(x2
2[MIS]−µ2)2
2σ2(2) !·exp −(x3
3−µ3)2
2σ2(3) 
∝exp −1
2"(x2
2[MIS])2 1
σ2(2) +(β(3)
2)2
σ2(3) !
+2x2
2[MIS] −β(2)
0−β(2)
1x1
1
σ2(2) +−β(3)
2(x3
3−β(3)
0)
σ2(3) !+C#!
∝ N x2
2[MIS]
µ∗=σ2∗· β(2)
0+β(2)
1x1
1
σ2(2) +β(3)
2(x3
3−β(3)
0)
σ2(3) !,
σ2∗= 1
σ2(2) +(β(3)
2)2
σ2(3) !−1

3
We hen conside he DBN depic ed in Figu e A.1b, and ocus on a missing
alue x
3[MIS] o a iable X3a ime > 0. The Ma ko blanke o a iable
X
3is cons i u ed o wo pa en s (X −1
1,X −1
2), wo child en (X +1
4,X +1
5), and
one a iable (X
4) sha ing a common child. De i ing he FCD o X
3in his
se ing illus a es he p ocedu e in case o mo e complex Ma ko blanke s, in
he di ec ion o demons a ing he ac abili y o he FCD in he b oade case.
P(x
3[MIS]| · )∝P(X
3|X −1
1, X −1
2)·P(X +1
4|X
3)·P(X +1
5|X
3, X
4)
∝ N(x
3[MIS]|µ3=β(3)
0+β(3)
1x −1
1+β(3)
2x −1
2, σ2(3))
· N(x +1
4|µ4=β(4)
0+β(4)
3x
3[MIS], σ2(4))
· N(x +1
5|µ5=β(5)
0+β(5)
3x
3[MIS]+β(5)
4x
4, σ2(5))
∝exp −(x
3[MIS]−(β(3)
0+β(3)
1x −1
1+β(3)
2x −1
2))2
2σ2(3) !
·exp −(x +1
4−(β(4)
0+β(4)
3x
3[MIS]))2
2σ2(4) !
·exp −(x +1
5−(β(5)
0+β(5)
3x
3[MIS]+β(5)
4x
4))2
2σ2(5) !
We de ine µ(4)
{−3}:= µ(4) −β(4)
3x
3[MIS]and µ(5)
{−3}:= µ(5) −β(5)
3x
3[MIS]. Then
(x +1
4−(β(4)
0+β(4)
3x
3[MIS]))2= (x +1
4−β(4)
3x
3[MIS]−µ(4)
{−3})2and (x +1
5−(β(5)
0+
β(5)
3x
3[MIS]+β(5)
4x
4))2= (x +1
5−β(5)
3x
3[MIS]−µ(5)
{−3})2. Fo j∈ {4,5}, each e m
(x +1
j−β(j)
3x
3[MIS]−µ(j)
{−3})2is a squa ed inomial. Since we a e compu ing he
FCD o x
3[MIS], we can ocus on he e ms ha in ol e his quan i y. Namely:
(x +1
j−β(j)
3x
3[MIS]−µ(j)
{−3})2∝(x
3[MIS]β(j)
3)2−2β(j)
3x
3[MIS](x +1
j−µ(j)
{−3})
Indeed, we could ew i e (x
3[MIS]−(β(3)
0+β(3)
1x −1
1+β(3)
2x −1
2))2= (x
3[MIS]−
µ3)2. Again, we a e in e es ed in he e ms o he squa ed binomial in ol ing
x
3[MIS], hus:
(x
3[MIS]−µ3)2∝(x
3[MIS])2−2x
3[MIS]µ3
Then:
4
P(x
3[MIS]| · )∝exp −1
2(x
3[MIS])2·1
σ2(3) −(2x
3[MIS])µ3
σ2(3) 
·exp 
−1
2
(x
3[MIS])2· (β(4)
3)2
σ2(4) !−(2x
3[MIS])

β(4)
3(x +1
4−µ(4)
{−3})
σ2(4) 



·exp 
−1
2
(x
3[MIS])2· (β(5)
3)2
σ2(5) !−(2x
3[MIS])

β(5)
3(x +1
5−µ(5)
{−3})
σ2(5) 



∝exp −1
2 (x
3[MIS])2· 1
σ2(3) +(β(4)
3)2
σ2(4) +(β(5)
3)2
σ2(5) !+
−(2x
3[MIS])

µ3
σ2(3) +β(4)
3(x +1
4−µ(4)
{−3})
σ2(4) +β(5)
3(x +1
5−µ(5)
{−3})
σ2(5) 
+C


∝ N 
µ∗=σ2∗·

µ3
σ2(3) +β(4)
3(x +1
4−µ(4)
{−3})
σ2(4) +β(5)
3(x +1
5−µ(5)
{−3})
σ2(5) 
,
σ2∗= 1
σ2(3) +(β(4)
3)2
σ2(4) +(β(5)
3)2
σ2(5) !−1

We i s de i ed he FCD o a missing alue a a speci ic ime in a sim-
ple DBN wi h h ee nodes. We hen illus a ed i s o m o a a iable wi h
wo pa en s and wo child en a a gene ic ime . Since all FCDs a e Gaus-
sian dis ibu ions wi h closed- o m pa ame e s, we now p esen he gene al case
o a missing alue on a a iable Xia ime , cha ac e ized by npa en s
π(i)={X −1
p1, . . . , X −1
pn}and mchild en {X +1
c1, . . . , X +1
cn}..
The FCD’s o he missing alue x
i[MIS]depends on he condi ional likelihood
o Xiand he condi ional likelihoods o i s child en. As be o e, we isola e he
con ibu ion o x
i[MIS]in i s child en likelihoods. Speci ically, o each child
j∈ {X +1
c1, . . . , X +1
cn}, gi en he mean e m µ +1
jand he linea coe icien β(j)
i
associa ed wi h he Gaussian likelihood o Xj, we de ine
µ(j)
{−i}
( +1) =µ +1
j−β(j)
ix
i[MIS]
We can hen de i e he missing alue FCD as ollows:
5
P(x
i[MIS]| · )∝P(X
i|π(i))·Y
j∈{c1,...,cm}
P(X +1
j|π(j))
∝ N(x
i[MIS]|µ
i=β(i)
0+β(i)
1x −1
p1+· · · +β(i)
nx −1
pn, σ2(i))
·Y
j∈{c1,...,cm}
N(x +1
j|µ +1
j=µ(j)
{−i}
( +1) +β(j)
ix
i[MIS], σ2(j))
∝exp −1
2(x
i[MIS])2·1
σ2(i)−(2x
i[MIS])µ
i
σ2(i)
·Y
j∈{c1,...,cm}
exp −1
2 (x
i[MIS])2· (β(j)
i)2
σ2(j)!+
−(2x
i[MIS])

β(j)
i(x +1
j−µ(j)
{−i}
( +1))
σ2(j)



∝exp 
−1
2
(x
i[MIS])2·

1
σ2(i)+X
j∈{c1,...,cm}
(β(j)
i)2
σ2(j)
+
−(2x
i[MIS])

µ
i
σ2(i)+X
j∈{c1,...,cm}
β(j)
i(x( +1)
j−µ(j)
{−i}
( +1))
σ2(j)
+C


∝ N 
µ∗=σ2∗·

µ
i
σ2(i)+X
j∈{c1,...,cm}
β(j)
i
(x +1
j−µ(j)
{−i}
( +1))
σ2(j)
,
σ2∗=

1
σ2(i)+X
j∈{c1,...,cm}
(β(j)
i)2
σ2(j)

−1


We can now exp ess he FCD o a gene ic missing alue x
i[MIS] o a a iable
Xia ime in he con ex o a DBN, accoun ing o an a bi a y numbe o
pa en s and child en:
P(x
i[MIS]| ·) = Nµ∗, σ2∗
whe e: σ2∗=

1
σ2(i)+X
j: (X
i∈π(j))
(β(j)
i)2
σ2(j)

−1
,
µ∗=σ2∗·

µ
i
σ2(i)+X
j: (X
i∈π(j))
β(j)
ix +1
j−µ(j)
{−i}
( +1)
σ2(j)
.

6
B Lea ning DBNs om Incomple e Da a
Algo i hm B.1 Pa ame e Se Upda e ia Collapsed Gibbs Sampling
1: Inpu : Da a {Y, X}, p io s {ασ, βσ, a, b, µ}, cu en {σ2, δ2, π}
2: Sample σ2(Collapsed Gibbs s ep)
σ2∼In −GAMασ+NT
2, βσ+1
2(Y−X[π]µ[π])⊤(I+σ2X[π]X⊤
[π])−1(Y−X[π]µ[π])
3: Sample β
β∼ N(σ−2I+X⊤
[π]X[π])−1(σ−2µ[π]+X⊤
[π]Y), σ2(σ−2I+X⊤
[π]X[π])−1
4: Sample δ2
δ2∼In −GAMa+|π|+ 1
2, b +1
2σ−2(β−µ[π])⊤(β−µ[π])
5: Ou pu : Pos e io sample {β, σ2, δ2}
Algo i hm B.2 Co a ia e Se Upda e ia Me opolis–Has ings
1: Inpu : Da a {Y, X}, cu en co a ia e se π, pa ame e s {δ2, σ2}
2: Choose a mo e ype (Dele ion, Addi ion, o Exchange) uni o mly a andom
3: Gene a e candida e co a ia e se π⋆based on he chosen mo e
4: Compu e accep ance p obabili y:
A(π→π⋆) = min 1,p(Y|π⋆, δ2)
p(Y|π, δ2)·p(π⋆)
p(π)·HR
wi h HR =|π|
n−|π⋆|(Dele ion), n−|π|
|π⋆|(Addi ion), o 1(Exchange);
p(Y|δ2, π) = Γ(A)
Γ(ασ)·π−T
2(2βσ)ασde (C)−1/22βσ+M⊤C−1M−A.a
5: D aw u∼ U(0,1)
6: i u<A hen
7: Accep : π←π⋆
8: Sample β om FCD:
β∼ N(σ−2I+X⊤
[π]X[π])−1(σ−2µ[π]+X⊤
[π]Y), σ2(σ−2I+X⊤
[π]X[π])−1
9: end i
10: Ou pu : Pos e io sample {π, β}
aA=T
2+ασ,C=I+δ2XXT,M=Y−Xµ
7
Algo i hm B.3 Missing Da a Upda e ia Gibbs Sampling
1: Inpu : Missing Va iable Index i,
2: D= [x
j]j=1:k, =1:3,B= [β(j)
i]i=0:k,j=1:k,Σ={σ2(j)}j=1:k
3: Compu e p io mean:
µi=X
p:(X1
p∈π(i))
x1
pβ(i)
p
4: Fo each j, compu e:
µ(j)
{−i}=X
m=i:(X2
m∈π(j))
x2
m·β(j)
m
5: Compu e pos e io a iance:
σ2∗=

1
σ2(i)+X
j:(X2
i∈π(j))
(β(j)
i)2
σ2(j)

−1
6: Compu e pos e io mean:
µ∗=σ2∗·

µi
σ2(i)+X
j:(X2
i∈π(j))
β(j)
i·x3
j−µ(j)
{−i}
σ2(j)

7: Sample xi[MIS]∼ N (µ∗, σ2∗)
8: Ou pu : Pos e io Sample xi[MIS]
8
Algo i hm B.4 LUME-DBN
1: Inpu : Incomple e Da ase DM= [x
i,j]i=1:N;j=1:k; =1:T, numbe o epochs
E, Missing Impu a ion eq EM, p io pa ame e s {ασ, βσ, a, b, µ =
{µ(j)
i}i=0:k,j=1:k}
2: Ini ialize: Ini ial models M(0) = [π(j)
i(0)]i,j=1:k,
3: Ini ial linea coe icien s B(0) = [β(j)
i(0)]i=0:k;j=1:k,
4: Ini ial noise pa ame e s Σ(0) ={σ2(j)
(0)}j=1:k,
5: Ini ial unce ain y pa ame e s ∆(0) ={δ2(j)
(0)}j=1:k,
6: Ini ial comple ed da ase D(0) = [x
i,j
(0)
[MIS]]i=1:N;j=1:k; =1:T
7: e←0
8: while e<Edo
9: e←e+ 1
10: X(e)←Lag(D(e−1))
11: o emin {1,...,EM}do
12: e←e+ 1
13: o jin {1, . . . , k}do
14: Y←X
j;X← X( −1)
{−j};µ←µ(j);
15: π←π(j)(e−1),β←β(j)
(e−1);σ2←σ2(j)
(e−1);δ2←δ2(j)
(e−1)
16: Pa ame e s Mo e:
17: Upda e ia Algo i hm B.1:{β(j)(e), σ2(j)
(e), δ2(j)
(e)}←{β, σ2, δ2}
18: Model Mo e:
19: Upda e ia Algo i hm B.2:{π(j)(e), β(j)
(e)}←{π, β}
20: end o
21: end o
22: M(e)←[π(j)
i(e)]i,j=1:k,B(e)←[β(j)
i(e)]i=0:k;j=1:k
23: Σ(e)← {σ2(j)
(e)}j=1:k,∆(e)← {δ2(j)
(e)}j=1:k
24: Missing Da a Impu a ion Mo e:
25: o { , i}:∃x
i,j =NA (wi h j∈ {1, . . . , k}do
26: D ← [x
i,j
(e−1)
[MIS]]j=1:k; =( −1: +1)
27: M←M(e),B ← B(e),Σ←Σ(e)
28: o j:x
i,j is missing do
29: Upda e Missing ia Algo i hm B.3:x
i,j
(e)
[MIS]←xj[MIS]
30: end o
31: end o
32: D(e)←[x
i,j
(e)
[MIS]]i=1:N;j=1:k; =1:T
33: end while
34: Ou pu : Pos e io samples {M(e),B(e), Σ(e), ∆(e),D(e)}e=0:E
9
C Con e gence Diagnos ics
C.1 Con e gence Diagnos ics on Simula ed Da a
Fig. C.2. Con e gence Diagnos ic o Ne wo k econs uc ion a e aged o e 5 simula-
ions o simula ed da ase s wi h di e en missingness a es.
Fig. C.3. Con e gence Diagnos ic o Missing Value impu a ion a e aged o e 5 sim-
ula ions o simula ed da ase s wi h di e en missingness a es.