Can Hidden Process Models be represented as Dynamic Bayes Nets?

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Can Hidden Process Models be represented as
Dynamic Bayes Nets?

• (Yes, but its not pretty!)

2
Outline

• Review HPM formalism
• Some failed attempts
• How to turn an HPM into a DBN
• The next questions to ask

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HPM Formalism

• HPM ltH,F,C,lts1,,sVgtgt
• H lth1,,hHgt, a set of processes
• h ltW,Q,W,dgt, a process
• W response signature
• W allowable offsets
• Q multinomial parameters over values in W
• d process duration
• C ltc1,, cCgt, a set of configurations
• c ltp1,,pLgt, a set of process instances
• lth,l,Ogt, a process instance
• h process ID
• associated stimulus landmark
• O offset (takes values in W(h))

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Hidden Process Models
Name Read sentence Process ID 1 Response
Name View Picture Process ID 2 Response
Name Decide whether consistent Process ID
3 Response
Processes
Process ID 1
Process ID 1
Process Instances
Process ID 2
View picture
Process ID 3
Decide whether consistent
Observed fMRI cortical region 1 cortical
region 2
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Synthetic Data Example
Process 1
Process 2
Process 3
Process responses
ProcessID1, S1
Process instances
ProcessID2, S17
ProcessID3, S21
Predicted data
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Synthetic Data Example
ProcessID1, S1
Configuration 1
ProcessID2, S17
ProcessID3, S21
ProcessID2, S1
Configuration 2
ProcessID1, S17
ProcessID3, S23
Observed data
Prediction 1
Prediction 2
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HPM Example
Name Decide whether consistent Process ID
3 Offsets W 1,2,3 Duration d5 Response W
Name Read sentence Process ID 1 Offsets W
0,1 Duration d6 Response W
Name View picture Process ID 2 Offsets W
0,1 Duration d6 Response W
Processes (H3)
c1
c2
c3
p1 lth1,l1,O0gt p2 lth2,l9,O0gt p3
lth3,l9,O2gt (L3)
p1 lth2,l1,O0gt p2 lth1,l9,O0gt p3
lth3,l9,O2gt (L3)
p1 lth1,l1,O1gt p2 lth2,l9,O1gt (L2)
Configurations (C3)
c1
c2
c3
t 1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16
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HPM Formalism

• HPM ltH,F,C,lts1,,sVgtgt
• H lth1,,hHgt, a set of processes
• h ltW,Q,W,dgt, a process
• W response signature
• Q multinomial parameters over values in W
• W allowable offsets
• d process duration
• C ltc1,, cCgt, a set of configurations
• c ltp1,,pLgt, a set of process instances
• lth,l,Ogt, a process instance
• h process ID
• l associated stimulus landmark
• O offset (takes values in W(h))

MIGHT WANT TO LEARN
KNOWN IN ADVANCE
TYPICALLY DO INFERENCE TO CHOOSE A C
KNOWN GIVEN C
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First Attempt
Process A (d5)
4
2
1
0
5
3
0
0
Process B (d5)
3
2
1
0
0
0
5
4
2 instances of process C (d5)
5
4
7
6
2
1
0
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PROBLEM!
Did this come from instances whose counts are at
3 and 4 or at 2 and 5? Dont know which parts of
the response signature to use.
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Chains must be instances

• We know in advance the maximum possible number of
  process instances maxcÎC(L(c)).
• If a configuration has a fewer number of
  instances, all timepoints for the excess
  instances will be zero (never active).

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I11
I13
I12
I14
I15
I16
I17
I18
I19
I111
I110
I112
I113
I115
I114
I116
I21
I23
I22
I24
I25
I26
I27
I28
I29
I211
I210
I212
I213
I215
I214
I216
I31
I33
I32
I34
I35
I36
I37
I38
I39
I311
I310
I312
I313
I315
I314
I316
Y3
Y2
Y4
Y5
Y6
Y7
Y8
Y9
Y11
Y10
Y12
Y13
Y15
Y14
Y16
Y1
13
C
H1
L1
I11
I13
I12
I14
I15
I16
I17
I18
I19
I111
I110
I112
I113
I115
I114
I116
H2
L2
I21
I23
I22
I24
I25
I26
I27
I28
I29
I211
I210
I212
I213
I215
I214
I216
H3
L3
I31
I33
I32
I34
I35
I36
I37
I38
I39
I311
I310
I312
I313
I315
I314
I316
Y3
Y2
Y4
Y5
Y6
Y7
Y8
Y9
Y11
Y10
Y12
Y13
Y15
Y14
Y16
Y1
14
C
H1
L1
I11
I13
I12
I14
I15
I16
I17
I18
I19
I111
I110
I112
I113
I115
I114
I116
H2
L2
I21
I23
I22
I24
I25
I26
I27
I28
I29
I211
I210
I212
I213
I215
I214
I216
H3
L3
I31
I33
I32
I34
I35
I36
I37
I38
I39
I311
I310
I312
I313
I315
I314
I316
Y3
Y2
Y4
Y5
Y6
Y7
Y8
Y9
Y11
Y10
Y12
Y13
Y15
Y14
Y16
Y1
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C
H1
L1
I11
I13
I12
I14
I15
I16
I17
I18
I19
I111
I110
I112
I113
I115
I114
I116
H2
L2
I21
I23
I22
I24
I25
I26
I27
I28
I29
I211
I210
I212
I213
I215
I214
I216
H3
L3
I31
I33
I32
I34
I35
I36
I37
I38
I39
I311
I310
I312
I313
I315
I314
I316
Y3
Y2
Y4
Y5
Y6
Y7
Y8
Y9
Y11
Y10
Y12
Y13
Y15
Y14
Y16
Y1
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C
Nnumber of process instances
Ln
In1
In3
In2
In4
In5
In6
In7
In8
In9
In11
In10
In12
In13
In15
In14
In16
Hn
Y3
Y2
Y4
Y5
Y6
Y7
Y8
Y9
Y11
Y10
Y12
Y13
Y15
Y14
Y16
Y1
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CPDs
P(H3C)
These priors can be generated using the F from
the HPM formalism.
1 2 3
0 0 0 1
1 0 0 0
2 0 0 0
3 1 1 0
C
H3
C 1 2 3
P(c) 0.4 0.4 0.3
C
P(L3C)
1 2 3
0 0 0 1
1 0 0 0
9 1 1 0
C
H3
L3
L3
P(I3tI3t-1,L3,H3)
p QH3(t-L3)
5 4 3 2 1 0
5 0 0 0 0 0 p
4 1 0 0 0 0 0
3 0 1 0 0 0 0
2 0 0 1 0 0 0
1 0 0 0 1 0 0
0 0 0 0 0 1 1-p
I3t-1
I3t
I31
I33
I32
I31 P(I31L3,H3)
0 1 - QH3(L3-1)
d5 QH3(L3-1)
Y3
Y2
Y1
mt Sn WHn(Int)
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Problem!

• Process instance specified by a configuration
  might not show up.
• Solution 1 in CPD for the last possible start
  time, if the process hasnt started yet, force it
  to start.
• This only works if the window of possible start
  times is shorter than the process duration!
• Solution 2 Add another Markov chain to keep
  track of whether the process has started. Then
  use solution 1.

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C
Nnumber of process instances
Ln
In1
In3
In2
In4
In5
In6
In7
In8
In9
In11
In10
In12
In13
In15
In14
In16
An3
An2
An4
An5
An6
An7
An8
An9
An11
An10
An1
Hn
Y3
Y2
Y4
Y5
Y6
Y7
Y8
Y9
Y11
Y10
Y12
Y13
Y15
Y14
Y16
Y1
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HPM to DBN Algorithm Sketch

• N max number of process instances in any
  configuration
• S set of possible start times given by C (all
  possible landmarks all possible offsets).
  (SLUh(W(h))

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HPM to DBN Algorithm Sketch

• create nodes C and Y1T
• for n1N
• create nodes Hn and Ln
• connect C-gtHn and C-gtLn
• create a Markov chain In1T
• for t1T connect Int-gtYt
• create a Markov chain Anmin(S)max(S)
• for tÎS
• connect Hn-gtInt
• connect Ln-gtInt
• connect Ant-1-gtInt
• connect Int-gtAnt

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HPM to DBN Algorithm Sketch

• Populate CPDs with HPM parameters as hinted at
  above
• NOTE This DBN structure does not require any new
  (not already in HPM) free parameters!

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Where we are

• Yes, we can represent HPMs as DBNs.
• An HPM with N possible process instances and T
  timepoints can be turned into a DBN with O(TN)
  nodes.
• The DBN representation has no more free
  parameters than the HPM representation.

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Where were going (but not today)

• The DBN representation is less elegant than the
  HPM representation. Is it worth this loss of
  simplicity to be able to use existing DBN
  machinery?
• Can we do inference better? Faster?
• Can we learn parameters better? Faster?
• Can we do the extensions we want in the DBN
  framework?
• Stimulus specific durations
• Process communication