Modeling decisions: influence diagrams and probabilistic networks

Modeling decisions: influence diagramsModeling decisions: influence diagramsand probabilistic networksand probabilistic networks

James Peterson, OR CFWRUJames Peterson, OR CFWRU

You You are hereare here

The The

Identify the decision Identify the decision situation and objectivessituation and objectives

Identify the management alternativesIdentify the management alternatives

Decompose and model the problemDecompose and model the problem

Identify the best alternativeIdentify the best alternative

Perform sensitivity analysisPerform sensitivity analysis

Is further Is further analysis needed?analysis needed?

Implement the best alternativeImplement the best alternative

NONO

YESYES

From From ClemenClemen and Reilly 2001and Reilly 2001

Influence DiagramInfluence Diagram

The decision

Uncertain events

Consequence(outcome, utility)

Improve habitat?

FutureFuturePopulation Population

sizesize

Conservation value

CurrentCurrentPopulation Population

sizesize


Arcs represent causality

Represent flow of information(no feedbacks)

NOT a flowchartNOT a flowchartImprove habitat?


sizesize

Conservation value


sizesize


Improve habitat?


sizesize

Conservation value


sizesize

Root node


ONLY type of link to representONLY type of link to representtiming timing and flow and flow of of informationinformation(current population size is (current population size is known when decision is made)known when decision is made)

Improve habitat?


sizesize

Conservation value


sizesize

No link: Current population size is No link: Current population size is Not known with certainty when Not known with certainty when decision is madedecision is made

Influence Influence Diagram: imperfect informationDiagram: imperfect information

Improve habitat?Improve habitat?


sizesize

Conservation Conservation valuevalue


Size?Size?

Field Field SamplingSampling

resultsresults



sizesize



Size?Size?

Field Field ssamplingamplingresultsresults

Influence Influence Diagram: linked/sequential decisionsDiagram: linked/sequential decisions

SampleSamplepopulation?population?

Reintroduce species?

Species Species persistencepersistence


Future Future habitathabitat

Influence Influence Diagram: linked/sequential decisionsDiagram: linked/sequential decisionsRestoringhabitat?

Influence Influence Diagram: Diagram: Dynamic (Dynamic (MarkovianMarkovian) ) decisionsdecisions

PopulationPopulationsizesizet = t = 22

Harvest Harvest DecisionDecisionTime t = 1Time t = 1

Harvest Harvest DecisionDecisionTime t = Time t = 22

Cumulative Cumulative harvestharvest

TotalTotalHarvest Harvest

t=1t=1


t=2t=2

PopulationPopulationsizesizett = 1= 1

PopulationPopulationsizesizet = t = 33

Harvest Harvest DecisionDecisionTime t = Time t = 33


t=3t=3

Common problemsCommon problemsFailing to account for direct effects of decisions on consequencFailing to account for direct effects of decisions on consequenceses

Cost of action = freeCost of action = freeImprove habitat?


sizesize

Conservation value


sizesize

Common problemsCommon problemsFailing to account for direct effects of decisions on consequencFailing to account for direct effects of decisions on consequenceses

Improve habitat?


sizesize

Conservation value


sizesize

Cost of action NOT freeCost of action NOT free

Common problemsCommon problemsMissing important uncertaintiesMissing important uncertainties



sizesize



sizesize

FutureFuturehabitathabitat

Common problemsCommon problemsFailing to consider the current state of the systemFailing to consider the current state of the system



sizesize



sizesize

FutureFuturehabitathabitat

CurrentCurrenthabitathabitat

Common problemsCommon problemsMissMiss--specifying the relationships specifying the relationships

BB

AA

CC

BB AA

CC

OROR

Common problemsCommon problems

BB

AA

CC

Is B really necessary?Is B really necessary?

Value of B depends on A, B = AxValue of B depends on A, B = Ax

Value of C depends of B, C = Value of C depends of B, C = BxBx

IF B known with certainty, the value ofIF B known with certainty, the value ofA no longer affects CA no longer affects C

C is conditionally independent of AC is conditionally independent of A

Common problemsCommon problemsJoint effects of A and B on CJoint effects of A and B on C

BB AA

CC

C = A + BC = A + B

CC

AA

B B ““highhigh””

B B ““lowlow””

C = A + B + A*BC = A + B + A*B

CC

AA

B B ““highhigh””

B B ““lowlow””

Common problemsCommon problemsJoint effects of A and B on CJoint effects of A and B on C

BB

AA

CC

The value of B depends on AThe value of B depends on Aand the value of C depends on and the value of C depends on the value of A and Bthe value of A and B

Fairly rare, make sure this is Fairly rare, make sure this is what you meantwhat you meant

Interpret the relationships Interpret the relationships forfortimber harvest timber harvest Influence DiagramInfluence Diagram

Timber harvestdecision

Stream Habitat

CurrentPopulation

Size

SocioeconomicSocioeconomicvaluevalue

Future Population

Size

Interpret the relationships Interpret the relationships forfortimber harvest timber harvest Influence DiagramInfluence Diagram


Stream Habitat

CurrentPopulation

Size


Future Population

Size

Defining NodesDefining Nodes

Stream Habitat


CurrentPopulation

Size

Potential StatesPotential States

No, YesNo, Yes Low, Medium, HighLow, Medium, High 0, 10ha, 100ha0, 10ha, 100ha

Good, BadGood, Bad Excellent, Good, Poor, BadExcellent, Good, Poor, Bad

Low, Medium, HighLow, Medium, High 00, 1, 1--10, 1010, 10--2020

States are mutually exclusive and collectively exhaustiveStates are mutually exclusive and collectively exhaustive

Influence diagram with node statesInfluence diagram with node states

Improve habitat?Yes, No

FutureFuturePopulation SizePopulation SizeLow, Moderate, HighLow, Moderate, High

Conservation ValueLow, High

CurrentCurrentPopulation SizePopulation SizeLow, Moderate, HighLow, Moderate, High

Defining Node StatesDefining Node States

States must be explicitly definedStates must be explicitly definedTransparency avoids confusionTransparency avoids confusionValues more explicit than narrativesValues more explicit than narratives

States can be discretized continuous valuesStates can be discretized continuous values

States often based on ecological/management States often based on ecological/management considerationsconsiderations

Timber Timber HHarvest arvest Influence DiagramInfluence Diagram

Timber harvestTimber harvestddecisionecision

yes, noyes, no

Stream Stream hhabitatabitatggood, poorood, poor

Current Current ppopulation sizeopulation size

ssmall, largemall, large


Uncertain Uncertain eventsevents

FutureFutureppopulation sizeopulation size


Remove the decision and utilityRemove the decision and utility……....

where where future fish population size future fish population size is influenced by is influenced by stream habitat and stream habitat and current fish population sizecurrent fish population size






Bayesian Belief NetworkBayesian Belief Network(DAG, probabilistic network, causal network) (DAG, probabilistic network, causal network)

Conditional ProbabilitiesConditional Probabilities






FutureFutureCurrentCurrent population sizepopulation size

Stream habitatStream habitat population sizepopulation size smallsmall largelargegoodgood smallsmall 0.30.3 0.70.7goodgood largelarge 0.10.1 0.90.9poorpoor smallsmall 0.60.6 0.40.4poor poor largelarge 0.50.5 0.50.5

Sums to 1Sums to 1

Unconditional Unconditional ProbabilitiesProbabilities






CurrentCurrentpopulation sizepopulation size ProbabilityProbability

smallsmall 0.80.8largelarge 0.20.2

StreamStreamhabitathabitat ProbabilityProbabilitygoodgood 0.50.5poorpoor 0.50.5

The top two nodes do not depend on anything else inthe model

Sum to 1Sum to 1


StreamStreamhabitathabitat

goodgood

smallsmall

smallsmall

smallsmall

smallsmallpoorpoor

largelarge

largelarge

largelarge

largelarge

largelarge

largelarge

0.300.30

0.700.70

0.100.10

0.900.90

0.600.60

0.400.40

0.500.50

0.500.50

0.800.80

0.200.20

0.800.80

0.200.20

0.500.50

0.500.50

Decision TreeDecision Tree

smallsmall

smallsmall



StreamStreamhabitathabitat

goodgood

smallsmall

smallsmall

smallsmall

smallsmallpoorpoor

largelarge

largelarge

largelarge

largelarge

largelarge

largelarge

0.300.30

0.700.70

0.100.10

0.900.90

0.600.60

0.400.40

0.500.50

0.500.50

0.800.80

0.200.20

0.800.80

0.200.20

0.500.50

0.500.50

Estimating probability future population is smallEstimating probability future population is small

smallsmall

smallsmall


0.300.300.800.80

0.500.50

0.50*0.80*0.300.50*0.80*0.30

CalculationCalculation

+ 0.50*0.20*0.10+ 0.50*0.20*0.10 + 0.50*0.80*0.60+ 0.50*0.80*0.60 + 0.50*0.20*0.50+ 0.50*0.20*0.50

0.100.100.200.20

0.600.600.800.80

0.500.50

0.500.500.200.20

= 0.42= 0.42

Bayesian Bayesian Belief NetworkBelief Network

StreamStreamhabitathabitat ProbabilityProbabilitygoodgood 0.50.5poorpoor 0.50.5

CurrentCurrentpopulation sizepopulation size ProbabilityProbability

smallsmall 0.80.8largelarge 0.20.2

Future population sizesmalllarge

42.058.0

Current population sizesmalllarge

80.020.0

Stream habitatgoodpoor

50.050.0


42.058.0


80.020.0


50.050.0

Interpreting the graphical model in Interpreting the graphical model in NeticaNetica

Model created with Model created with NeticaNetica softwaresoftware

Name of nodeName of node

StatesStates

StateState--specific probabilitiesspecific probabilities(expressed as %)(expressed as %)

Bar length representsBar length representsssize of probabilityize of probability

Bayesian Bayesian Belief Networks, conditional independenceBelief Networks, conditional independence

Stream habitat now depends on the width of the riparian zoneStream habitat now depends on the width of the riparian zone


42.857.2


80.020.0


47.552.5

Riparian widthnarrowwide

50.050.0


Riparian Riparian Stream Stream habitathabitat

widthwidth goodgood poorpoornarrownarrow 0.200.20 0.800.80

widewide 0.750.75 0.250.25

Predicted probabilitiesPredicted probabilitieswwhen riparian is narrowhen riparian is narrow


51.648.4


80.020.0


20.080.0


100 0


Riparian Riparian Stream Stream habitathabitat

widthwidth goodgood poorpoornarrownarrow 0.200.20 0.800.80widewide 0.750.75 0.250.25

Predicted probabilitiesPredicted probabilitieswwhen riparian is widehen riparian is wide


34.066.0


80.020.0


75.025.0


0 100


26.074.0


80.020.0


100 0


0 100


Assume that Assume that kknew habitatnew habitatwwas goodas good

Predicted Predicted probabilitiesprobabilities

Riparian widthRiparian widthiis wides wide


26.074.0


80.020.0


100 0


100 0

Bayesian Belief Networks, conditional independence

Assume that Assume that kknew habitatnew habitatwwas goodas good

Predicted Predicted ProbabilitiesProbabilitiesddo not changeo not change

Riparian widthRiparian widthiis narrows narrow

Once the condition of Stream habitat is known Riparian width no Once the condition of Stream habitat is known Riparian width no longerlongeraffects Future population status.affects Future population status.

Consequences of independenceConsequences of independence

StreambedStreambedsedimentsediment

ModularityModularity

PrecipitationPrecipitationSoil Soil

disturbancedisturbance

Timber harvestTimber harvest

StreambedStreambedsedimentsediment

FutureFuturePop sizePop size

CurrentCurrentfish pop sizefish pop size

Physical habitat modelPhysical habitat model FishFish--habitat modelhabitat model

Combine 2 different models Combine 2 different models

Representing time in a BBN

Time is difficult to represent in an BBN because they cannot Time is difficult to represent in an BBN because they cannot contain feedback loopscontain feedback loops

There are, however, 2 basic way to represent timeThere are, however, 2 basic way to represent time

To examine the effects of a single decision at various points inTo examine the effects of a single decision at various points in time:time:

Represent time as a node, usually a constant or decision nodeRepresent time as a node, usually a constant or decision node

To examine the effects of sequential decisions at various pointsTo examine the effects of sequential decisions at various points in time:in time:

Design influence diagram as a sequence of decisionsDesign influence diagram as a sequence of decisions

At 100 yearsAt 100 years

At 10 yearsAt 10 years

Representing time with a Representing time with a constant node constant node

BBN of riparian corridor condition at 10 and 100 years from cessBBN of riparian corridor condition at 10 and 100 years from cessation of cattle ation of cattle grazinggrazing(from (from ReimanReiman et al. 2001) et al. 2001)

Riparian conditionIntactMod DegradedHi Degraded

42.237.220.7

MitigationHi MitigationMod MitigationLow Mitigation

33.333.333.3

Prior Riparian ConditionIntact low damagMod High

50.050.0

Time = t+100

Riparian conditionIntactMod DegradedHi Degraded

33.540.825.7

MitigationHi MitigationMod MitigationLow Mitigation

33.333.333.3

Prior Riparian ConditionIntact low damagMod High

50.050.0

Time = t+10

Modeling sequential processesModeling sequential processesRepeating the same process through Repeating the same process through timetimeMallard population at time t modeled as a function of the previoMallard population at time t modeled as a function of the previous us population size (tpopulation size (t--1) and the number of ponds at time t. 1) and the number of ponds at time t.

Mallard population size t= 2smalllarge

66.034.0


60.040.0


68.831.2

Number of ponds t = 3fewmany

20.080.0

Number of ponds t= 2fewmany

60.040.0

Identical conditional probability tableIdentical conditional probability table

Model relationships directly from dataModel relationships directly from data

ParameterizingParameterizing Bayesian Bayesian Belief NetworksBelief Networks

Hardwood density (no/ha)LowModerateHigh

33.333.333.3

10 ± 8.2

Snag density (no/ha)LowMediumHigh

33.333.333.3

4.33 ± 2.7

Woodpecker abundanceAbsentLowModerate

33.333.333.3

2.33 ± 1.8

Woodpecker data fileWoodpecker data fileWoodpecker modelWoodpecker modelNo data (uniform probabilities)No data (uniform probabilities)

Yx5x6x7x8x9x10x11x12x13x14x15

0.340.343.4510.355.919.07.931.720.340.340.34

X1x0x1x2x3x4x5x6x7x8x9x10

0 0 0 0 0

100 0 0 0 0 0

X2x0x1x2x3x4x5x6x7x8x9x10

0 0 0

100 0 0 0 0 0 0 0

For X1 = 5 and X2 = 3, predicted Y = 9.75

Network representation of linear regression Y = 3 + X1 + 0.5*X2 + e

Recall linear regression assumptions:

Confidence limits

Parameterizing Bayesian Belief NetworksUsing existing models

““ExpertExpert”” JudgmentJudgment

Combine subjective probabilities across expertsCombine subjective probabilities across experts

and when information is completely lackingand when information is completely lacking…………

Insect DiversityInsect DiversityLowLow ModerateModerate HighHigh

HabitatHabitatstabilitystability

RefoundingRefoundingand supportand support

LowLowMediumMedium

HighHigh

?? ?? ??LowLow

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

LowLowLowLow

On to the case studyOn to the case study……..

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Modeling decisions: influence diagrams and probabilistic networks

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