Maximum a posteriori (MAP) inference is a particularly complex type of probabilistic inference in Bayesian networks. It consists of finding the most probable configuration of a set of variables of interest given observations on a collection of other variables. In this paper we study scalable solutions to the MAP problem in hybrid Bayesian networks parameterized using conditional linear Gaussian distributions. We propose scalable solutions based on hill climbing and simulated anneal- ing, built on the Apache Flink framework for big data processing. We analyze the scalability of the solution through a series of experiments on large synthetic networks. Full text paper: http://www.jmlr.org/proceedings/papers/v52/ramos-lopez16.pdf

1. Scalable MAP inference in Bayesian
networks based on a Map-Reduce
approach
Darío Ramos-López1, Antonio Salmerón1, Rafael Rumí1,
Ana M. Martínez2, Thomas D. Nielsen2, Andrés R. Masegosa3,
Helge Langseth3, Anders L. Madsen2,4
1Department of Mathematics, University of Almería, Spain
2Department of Computer Science, Aalborg University, Denmark
3 Department of Computer and Information Science,
The Norwegian University of Science and Technology, Norway
4 Hugin Expert A/S, Aalborg, Denmark
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 1

2. Outline
1 Motivation
2 MAP in CLG networks
3 Scalable MAP
4 Experimental results
5 Conclusions
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 2

3. Outline
1 Motivation
2 MAP in CLG networks
3 Scalable MAP
4 Experimental results
5 Conclusions
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 3

4. Motivation
Aim: Provide scalable solutions to the MAP problem.
Challenges:
Data coming in streams at high speed, and
a quick response is required.
For each observation in the stream, the
most likely configuration of a set of
variables of interest is sought.
MAP inference is highly complex.
Hybrid models come along with specific
difficulties.
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 4

5. Context
The AMiDST project: Analysis of MassIve Data STreams
http://www.amidst.eu
Large number of variables
Queries to be answered in real time
Hybrid Bayesian networks (involving discrete and continuous
variables)
Conditional linear Gaussian networks
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 5

6. Outline
1 Motivation
2 MAP in CLG networks
3 Scalable MAP
4 Experimental results
5 Conclusions
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 6

7. Conditional Linear Gaussian networks
Y
W
TU
S
P(Y ) = (0.5, 0.5)
P(S) = (0.1, 0.9)
f (w|Y = 0) = N(w; −1, 1)
f (w|Y = 1) = N(w; 2, 1)
f (t|w, S = 0) = N(t; −w, 1)
f (t|w, S = 1) = N(t; w, 1)
f (u|w) = N(u; w, 1)
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 7

8. Querying a Bayesian network
Belief update: Computing the posterior distribution of a variable:
p(xi |xE ) =
xD
xC
p(x, xE )dxC
xDi
xCi
p(x, xE )dxCi
Maximum a posteriori (MAP): For a set of target variables XI , seek
x∗
I = arg max
xI
p(xI |XE = xE )
where p(xI |XE = xE ) is obtained by first marginalizing out from
p(x) the variables not in XI and not in XE
Most probable explanation (MPE): A particular case of MAP where
XI includes all the unobserved variables
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 8

9. Querying a Bayesian network
Belief update: Computing the posterior distribution of a variable:
p(xi |xE ) =
xD
xC
p(x, xE )dxC
xDi
xCi
p(x, xE )dxCi
Maximum a posteriori (MAP): For a set of target variables XI , seek
x∗
I = arg max
xI
p(xI |XE = xE )
where p(xI |XE = xE ) is obtained by first marginalizing out from
p(x) the variables not in XI and not in XE
Most probable explanation (MPE): A particular case of MAP where
XI includes all the unobserved variables
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 8

10. Outline
1 Motivation
2 MAP in CLG networks
3 Scalable MAP
4 Experimental results
5 Conclusions
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 9

11. MAP by Hill Climbing
HC_MAP(B,xI ,xE ,r)
while stopping criterion not satisfied do
x∗
I ← GenerateConfiguration(B, xI , xE , r)
if p(x∗
I , xE ) ≥ p(xI , xE ) then
xI ← x∗
I
end
end
return xI
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 10

12. MAP by Hill Climbing
HC_MAP(B,xI ,xE ,r)
while stopping criterion not satisfied do
x∗
I ← GenerateConfiguration(B, xI , xE , r)
if p(x∗
I , xE ) ≥ p(xI , xE ) then
xI ← x∗
I
end
end
return xI
Max. number of non-improving iterations
Target prob. threshold
Max. number of iterations
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 10

13. MAP by Hill Climbing
HC_MAP(B,xI ,xE ,r)
while stopping criterion not satisfied do
x∗
I ← GenerateConfiguration(B, xI , xE , r)
if p(x∗
I , xE ) ≥ p(xI , xE ) then
xI ← x∗
I
end
end
return xI
Never move to a worse configuration
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 10

14. MAP by Hill Climbing
HC_MAP(B,xI ,xE ,r)
while stopping criterion not satisfied do
x∗
I ← GenerateConfiguration(B, xI , xE , r)
if p(x∗
I , xE ) ≥ p(xI , xE ) then
xI ← x∗
I
end
end
return xI
Estimated using importance sampling:
p(xI , xE ) =
x∗∈ΩX∗
p(xI , xE , x∗
) =
x∗∈ΩX∗
p(xI , xE , x∗
)
f ∗(x∗)
f ∗
(x∗
)
= Ef ∗
p(xI , xE , x∗
)
f ∗(x∗)
≈
1
n
n
i=1
p xI , xE , x∗(i)
f ∗ x∗(i)
,
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 10

15. MAP by Hill Climbing
HC_MAP(B,xI ,xE ,r)
while stopping criterion not satisfied do
x∗
I ← GenerateConfiguration(B, xI , xE , r)
if p(x∗
I , xE ) ≥ p(xI , xE ) then
xI ← x∗
I
end
end
return xI
We’ll see later
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 10

16. MAP by Simulated Annealing
SA_MAP(B,xI ,xE ,r)
T ← 1 000; α ← 0.90; ε > 0
while T ≥ ε do
x∗
I ← GenerateConfiguration(B, xI , xE , r)
Simulate a random number τ ∼ U(0, 1)
if p(x∗
I , xE ) > p(xI , xE )/(T · ln(1/τ)) then
xI ← x∗
I
end
T ← α · T
end
return xI
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 11

17. MAP by Simulated Annealing
SA_MAP(B,xI ,xE ,r)
T ← 1 000; α ← 0.90; ε > 0
while T ≥ ε do
x∗
I ← GenerateConfiguration(B, xI , xE , r)
Simulate a random number τ ∼ U(0, 1)
if p(x∗
I , xE ) > p(xI , xE )/(T · ln(1/τ)) then
xI ← x∗
I
end
T ← α · T
end
return xI
Default values of the temperature parameters
T 1: almost completely random
T 1: almost completely greedy
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 11

18. MAP by Simulated Annealing
SA_MAP(B,xI ,xE ,r)
T ← 1 000; α ← 0.90; ε > 0
while T ≥ ε do
x∗
I ← GenerateConfiguration(B, xI , xE , r)
Simulate a random number τ ∼ U(0, 1)
if p(x∗
I , xE ) > p(xI , xE )/(T · ln(1/τ)) then
xI ← x∗
I
end
T ← α · T
end
return xI
Accept x∗
I if its prob. increases
or decreases < T · ln(1/τ)
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 11

19. MAP by Simulated Annealing
SA_MAP(B,xI ,xE ,r)
T ← 1 000; α ← 0.90; ε > 0
while T ≥ ε do
x∗
I ← GenerateConfiguration(B, xI , xE , r)
Simulate a random number τ ∼ U(0, 1)
if p(x∗
I , xE ) > p(xI , xE )/(T · ln(1/τ)) then
xI ← x∗
I
end
T ← α · T
end
return xI
Cool down the temperature
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 11

20. Generating a new configuration of variables
Only discrete variables
The new values are chosen at random
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 12

21. Generating a new configuration of variables
Hybrid models
We take advantage of the properties of the CLG distribution
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 12

22. Generating a new configuration of variables
Hybrid models
We take advantage of the properties of the CLG distribution
A variable whose parents are discrete or observed
Y
W
TU
S
P(Y ) = (0.5, 0.5)
P(S) = (0.1, 0.9)
f (w|Y = 0) = N(w; −1, 1)
f (w|Y = 1) = N(w; 2, 1)
f (t|w, S = 0) = N(t; −w, 1)
f (t|w, S = 1) = N(t; w, 1)
f (u|w) = N(u; w, 1)
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 12

23. Generating a new configuration of variables
Hybrid models
We take advantage of the properties of the CLG distribution
A variable whose parents are discrete or observed
Y
W
TU
S
Return its conditional mean:
f (w|Y = 0) = N(w; −1, 1)
f (w|Y = 1) = N(w; 2, 1)
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 13

24. Generating a new configuration of variables
Hybrid models
We take advantage of the properties of the CLG distribution
A variable with unobserved continuous parents
Y
W
TU
S
Simulate a value using
its conditional distribution:
f (t|w, S = 0) = N(t; −w, 1)
f (t|w, S = 1) = N(t; w, 1)
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 13

25. Scalable implementation
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 14

26. Outline
1 Motivation
2 MAP in CLG networks
3 Scalable MAP
4 Experimental results
5 Conclusions
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 15

27. Experimental analysis
Purpose
Analyze the scalability in terms of
Speed
Accuracy
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 16

28. Experimental analysis
Purpose
Analyze the scalability in terms of
Speed
Accuracy
Experimental setup
Synthetic networks with 200 variables (50% discrete)
70% of the variables observed at random
10% of the variables selected as target ⇒ 20% to be
marginalized out
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 16

29. Experimental analysis
Computing environment
AMIDST Toolbox with Apache Flink
Multi-core environment based on a dual-processor AMD
Opteron 2.8 GHz server with 32 cores and 64 GB of RAM,
running Ubuntu Linux 14.04.1 LTS
Multi-node environment based on Amazon Web Services
(AWS)
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 17

30. Scalability: run times
100
200
300
400
500
600
1 2 4 8 16 24 32
Number of cores
Executiontime(s)
method HC Global HC Local SA Global SA Local
Scalability of MAP in a multi−core node
5
10
15
Speed-upfactor
5
10
15
20
1 2 4 8 16 24 32
Number of nodes (4 vCPUs per node)Speed−upfactor
method HC Global HC Local SA Global SA Local
Scalability of MAP in a multi−node cluster
25
50
75
100
125
1 2 4 8 16 24 32
Number of nodes (4 vCPUs per node)Executiontime(s)
method HC Global HC Local SA Global SA Local
Scalability of MAP in a multi−node cluster
5
10
15
20
Speed-upfactor
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 18

31. Scalability: accuracy (Simulated Annealing)
−230
−220
−210
−200
−190
1 2 4 8 16 32
Number of cores
logProbability
cores
1
2
4
8
16
32
Simulated Annealing global
−200
−195
−190
−185
1 2 4 8 16 32
Number of cores
logProbability
cores
1
2
4
8
16
32
Simulated Annealing local
Estimated log-probabilities of the MAP configurations found by each algorithm
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 19

32. Scalability: accuracy (Hill Climbing)
−240
−220
−200
1 2 4 8 16 32
Number of cores
logProbability
cores
1
2
4
8
16
32
Hill Climbing global
−240
−220
−200
1 2 4 8 16 32
Number of cores
logProbability
cores
1
2
4
8
16
32
Hill Climbing local
Estimated log-probabilities of the MAP configurations found by each algorithm
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 20

33. Outline
1 Motivation
2 MAP in CLG networks
3 Scalable MAP
4 Experimental results
5 Conclusions
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 21

34. Conclusions
Scalable MAP for CLG models in terms of accuracy and run
time
Available in the AMIDST Toolbox
Valid for multi-cores and cluster systems
MapReduce-based design on top of Apache Flink
8th Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 22

35. Thank you for your attention
You can download our open source Java toolbox:
http://www.amidsttoolbox.com
Acknowledgments: This project has received funding from the European Union’s
Seventh Framework Programme for research, technological development and
demonstration under grant agreement no 619209
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