Bayesian Networks II:

Dynamic Networks and

Markov Chains

By Peter Woolf (pwoolf@umich.edu)

University of Michigan

Michigan Chemical Process

Dynamics and Controls

Open Textbook

version 1.0

Creative commons

Existing plant

measurements

Physics, chemistry, and chemical

engineering knowledge & intuition

Bayesian network models to

establish connections

Patterns of likely

causes & influences

Efficient experimental design to

test combinations of causes

ANOVA & probabilistic models to eliminate

irrelevant or uninteresting relationships

Process optimization (e.g. controllers,

architecture, unit optimization,

sequencing, and utilization)

Dynamical

process modeling

From http://www.norsys.com/netlib/car_diagnosis_2.htm

Static Bayesian Network Example 1:

Car failure diagnosis network

Static Bayesian Network Example

2:

ALARM network: A Logical Alarm Reduction Mechanism

A medical diagnostic system for patient monitoring with 8

diagnoses, 16 findings, and 13 intermediate values

From Beinlich, Ingo, H. J. Suermondt, R. M. Chavez, and G. F. Cooper (1989) "The ALARM monitoring

system: A case study with two probabilistic inference techniques for belief networks" in Proc. of the

Second European Conf. on Artificial Intelligence in Medicine (London, Aug.), 38, 247-256. Also Tech.

Report KSL-88-84, Knowledge Systems Laboratory, Medical Computer Science, Stanford Univ., CA.

weight

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ALT

survival

RBC

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Yesterday

(t

i-1

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Today

(t

i

)

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Unrolled Network

Dynamic Bayesian Networks

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These are both examples of

Dynamic Bayesian Networks

(DBNs)

OR

Collapsed Network

Predicts

future

responses

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t

i-1

t

i

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i-2

t

i-3

t

i+1

t

i+2

Model derived from past data

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(t

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Tomorrow

(t

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)

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Dynamic Bayesian Networks:

Predict to explore alternatives

DBNs provide a suitable environment for MPC!

DBN: Thermostat example

From http://www.norsys.com/networklibrary.html#

N: fluctuations

Time(t)

ti

ti+1

H: Heater

T: Temperature

G: Temp Set Pt.

S: Switch

V: Value/Cost

N: fluctuations

H: Heater

T: Temperature

G: Temp Set Pt.

S: Switch

V: Value/Cost

Unrolled network

Collapsed network

N: fluctuations

Time(t)

ti

ti+1

H: Heater

T: Temperature

N: fluctuations

H: Heater

T: Temperature

Simplified DBN

A Dynamic Bayesian Network can

be recast as a

Markov Network

Assume each variable is

binary (has states 1 or

0), thus any

configuration could be

written as {010} meaning

N=0, H=1, T=0

A Markov network describes

how a system will transition

from system state to state

{000}

{001}

{010}

{110}

{011}

{111}

{101}

{100}

A Dynamic Bayesian Network can

be recast as a

Markov Network

A Markov network describes

how a system will transition

from system state to state

{000}

{001}

{010}

{110}

{011}

{111}

{101}

{100}

Each edge has a probability associated with it.

!

to

{000} {001} {010} {100} {101} {110} {011} {111}

from

{000}

{001}

{010}

{100}

{101}

{110}

{011}

{111}

0 p1 0 0 0 p2 0 0

p3 p4 0 0 0 0 0 0

0 0 0 p5 0 0 0 0

0 0 0 0 0 p6 p7 0

0 0 p8 p9 p10 0 0 0

0 0 0 0 0 p11 0 0

0 0 0 0 p12 0 0 p13

0 0 0 0 0 0 p14 p15

"

#

$

$

$

$

$

$

$

$

$

$

%

&

'

'

'

'

'

'

'

'

'

'

Note: All rows must sum to 1

P1+P2=1

P5=1 e

tc.

Case Study

: Synthetic Study

Situation:

Imagine that we are exploring the effect of a DNA

damaging drug and UV light on the expression of 4 genes.

GFP

Gene A

Gene B

Gene C

Case Study 1

: Synthetic Study

GFP

Gene A

Gene B

Gene C

Idealized

Data

Case Study 1

: Synthetic Study

GFP

Gene A

Gene B

Gene C

Noisy

data

Case Study 1

: Synthetic Study

Noisy data

Idealized

Data

Given idealized or noisy data, can we find any relationships

between the drug, UV exposure, GFP, and the gene

expression profiles?

See miniTUBA.demodata.xls

Case Study 1

: Synthetic Study

Google

“

miniTUBA

”

or go to

http://ncibi.minituba.org

Case study 1: synthetic data

•

Observations:

–

Stronger relationships require fewer

observations to identify

–

Noise in measurements are okay

–

Moderate binning errors are forgivable

–

Uncontrolled experiments can be your

friend in model learning

Take Home Messages

•

Noisy, time varying processes can be

modeled as a Dynamic Bayesian

Network (DBN)

•

A DBN can be recast as a Markov

model of a stochastic system

•

DBNs can be learned directly from data

using tools such as miniTUBA

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