,V«irr/i Amcriftin Jourmil <>t Fisheries Manttnt'tnenf 17:1144-1157. 1997American Fisheries Society 1997

Population Viability Assessment of Salmonids by UsingProbabilistic Networks

DANNY C. LEE' AND BRUCE E. RIEMANU.S. Forest Service. Rocky Mountain Research Station

3/6 East Myrtle Street. Boise. Idaho 83702. USA

Abstract.—Public agencies are being asked to quantitatively assess the impact of land manage-ment activities on sensitive populations of salmonids. To aid in these assessments, we developeda Bayesian v iab i l i t y assessment procedure (BayVAM) to help characterize land use risks to salmo-nids in the Pacific Northwest. This procedure incorporates a hybrid approach to viability analysisthat blends qualitative, professional judgment wi th a quant i ta t ive model to provide a generalizedassessment of risk and uncertainty. The BayVAM procedure relies on three main components: ( 1 )an assessment survey in which users judge the relative condition of the habitat and estimate survivaland reproductive rates for the population in question: (2) a stochastic simulation model that providesa mathematical representation of important demographic and environmental processes; and ( 3 ) aprobabilistic network that uses the results of the survey to define l ikely parameter ranges, mimicsthe stochastic behavior of the simulation model, and produces probability histograms for averagepopulation si/e. min imum population size, and t ime to extinction. The structure of the probabilisticnetworks allows partit ioning of uncertainty due to ignorance of population parameters from thatdue lo unavoidable environmental variation. Although based on frequency distributions of a formalstochastic model, the probability histograms also can be interpreted as Bayesian probabilities (i.e..the degree of belief about a future event). We argue that the Bayesian interpretation provides arational framework for approaching viability assessment from a management perspective. TheBayVAM procedure offers a promising step toward tools that can be used to generate quant i ta t iverisk estimates in a consistent fashion from a mix ture of information.

Public agencies such as the U.S. Forest Service(FS) and the U.S. Bureau of Land Management(BLM) must consider the potential effects of landmanagement activities on aquatic species occur-ring on lands under their administration. Frequent-ly, biological evaluations are based largely on thejudgment and experience of local biologists be-cause requisite information on local populations isunavailable and unlikely to be obtained within areasonable lime. Recent federal listings of severalspecies under the U.S. Endangered Species Act(e.g.. Snake River chinook salmon Oncorhynchustshawytscha and sockeye salmon 0. nerka) haveincreased the importance and visibility of theseanalyses. In turn, the higher analytical profile hashighlighted the need for more rigorous, consistent,and defensible analyses that recognize the uncer-tainty inherent in decisions made in information-poor environments. Watershed analysis has beenproposed as the formal framework for assessingpotential threats from land use decisions and forproposing mitigation measures that might be takento ensure population viability (FEMAT 1993). Toensure the success of watershed-scale analysis, de-cision support tools are being developed by FS

1 Corresponding author: dlee/inuboised'fs.fed.us

and BLM to guide evaluation of physical and bi-ological systems.

We have developed a Bayesian viability assess-ment procedure (BayVAM) as a tool to character-ize land use risks to salmonids. Two versions ofthe procedure exist, one for anadromous salmonidsand one for potamodromous, or resident, trouts andchars. The BayVAM procedure incorporates a hy-brid approach to viability analysis that blendsqualitative judgment with quantitative models toprovide a generalized assessment of risk and un-certainty. Herein we highlight distinguishing fea-tures of the BayVAM approach, using examplesfrom the potamodromous version. A more specificapplication is discussed in the companion paperby Shepard et al. (1997, this issue).

OverviewThe purpose of the BayVAM procedure is to

provide a rigorous and repeatable method for ex-amining watershed conditions in light of species'life history requirements in order to determinewhether a population faces a serious risk of extir-pation or significant decline. The BayVAM ap-proach is an extension of the coarse filter approachpresented by Rieman et al. (1993), which providesa concise method for distinguishing healthy pop-ulations from those at risk. In contrast to coarse

1144

PROBABILISTIC NHTWORKS FOR RISK ASSESSMENT 1145

TABI.I-: 1.—Model parameters used in the Bay VAM simulations.

Parameter

fecundincsuvalpha ( u )parrcap•»'i\2mv0_CVimmigrantmtadtcal

Description

Number of female eggs per adult femaleSpawning and incubation successFry survival at low densityAsymptotic parr capacityJuvenile survivalAdult survival rateAge at maturityCoefficient of variation in fry survival41

Mean immigrants per generationInit ial number of adult femalesExpected time between catastrophes

Range

50-1.250 eggs10-70'*10-40'*-I.OOO-KMXX) parr15-6031 5-90'*3-6 years15-90'*0-20 adult equivalents50- 1 .250 females20- 1 70 years

•' U K ) SD/mean.

filters, the BayVAM approach not only estimatesrisk, it clearly identities presumed links betweenwatershed conditions and life history parametersand distinguishes uncertainty due to natural ran-dom processes from uncertainty due to lack ofknowledge or information (both of which contrib-ute to overall assessed risk). The BayVAM pro-cedure consists of three to rive steps, the numberdepending on the scale of analysis, as follows.

Step I.—Define the scale of analysis in terms ofthe populations under consideration and the land-scape involved. Watershed boundaries form logi-cal boundaries for many salmonid populations. Be-cause the scales of administrative interest and pop-ulation processes may not coincide (see Ruggieroet al. 1994), it is important to define individualpopulations or regional aggregations of popula-tions clearly and in a biologically meaningful way.That is, the population boundaries should be con-sistent with the biology of the organism, not re-stricted by administrative boundaries. Boundariesplaced on individual populations may vary widelyby species and life history type.

Step 2.—Survey watershed conditions in termsof how they affect vital population parameters (seeShepard et al. 1997, this issue, for a survey ex-ample). Judge the relative condition of the habitatand estimate survival and reproductive rates forthe population in question.

Guidelines normally are provided to help withdrawing inferences from existing information andwith designing data collection. For example, if anarea has abundant silt-free gravel of the appropri-ate size for a salmon species' redd building andreceives adequate instream flow during the egg andalevin incubation period, one might presume thatspawning and incubation success are relativelyhigh. Users of BayVAM do not choose preciseparameter values; rather, they assign probabilitiesor likelihood values to specific ranges for each

parameter (Table 1). The available parameter rang-es span values consistent with salmonid popula-tions in western North America. Parameter uncer-tainty due to lack of information or experience isincorporated in the probability distributions. Asbetter information becomes available, responsescan be updated to reflect the improved accuracyor precision of the information.

Step 3.—Complete a quantitative viability anal-ysis for each population of interest using surveyresults from step 2. Central to this analysis arestochastic life cycle models, recast as probabilisticnetworks (details are provided below). The lifecycle models integrate information on critical pop-ulation processes and rates (e.g., fecundity, sur-vival) and habitat capacity to simulate the dynam-ics of a population in a variable environment.These models track population numbers throughannual time steps, introducing random fluctuationsat each step. The probabilistic networks summa-rize and mimic the behavior of the life cycle modelin a way that simplifies calculation and provideseasily interprctable outputs. The probabilistic net-work is provided in the form of a simple spread-sheet. The BayVAM users enter the probabilityvalues generated in step 2, and the spreadsheetcalculates corresponding probability values for aseries of intermediate variables (algebraic com-binations of the input parameters) and for threeoutput variables: minimum population size, av-erage population size, and time to extinction.

The output variables are based on simulated100-year periods, which are roughly 10-20 timesthe generation time of stream-dwelling salmonids.Although longer time frames may be appropriatefor some species (Marcot and Murphy 1996), 100years were sufficient for characterizing the dynam-ics of model populations and provide useful in-dices of risk for the species we examined. Changesin environmental conditions and management that

1146 LKK AND R1HMAN

would make any projection moot are certain tooccur within the next 100 years. Because ecosys-tems are dynamic, the chances of population in-creases or decreases will be different at any pointin the future. Thus, time frame is only a standardof reference for an assessment based on conditionsat the time of analysis.

The models consider only demographic featuresof populations; they incorporate no explicit rep-resentation of population genetics, including hy-bridization. Although small populations may bethreatened by inbreeding depression and lost ge-netic diversity, it is difficult to incorporate geneticrisks other than in qualitative terms (Hedrick andMiller 1992). An implicit assumption of theBayVAM approach is that most populations are atrisk more from demographic processes than fromgenetic effects of small population size (see Shaf-fer 1987 and Lande 1988). Management actionsaimed at natural reproduction (i.e., no hatchery fishare involved), which minimizes risks associatedwith population demographics, also should mini-mize most genetic risks, hybridization being a no-table exception.

Step 4.—Identify, if desired, alternative hypoth-eses concerning population parameters and rerunthe analysis. Multiple iterations can be used toevaluate the sensitivity of predictions to alterna-tive hypotheses, determine the potential value ofnew information, or identify possible changes thatmight follow alternative management scenarios.For example, values for juvenile and adult survivalmight be adjusted upward to reflect a proposedmanagement scheme to improve the number andqual i ty of pool habitats through increased recruit-ment of large woody debris, if such action canlegitimately improve habitat.

Step 5.—Complete analyses for all individualpopulations in the regional population, if a re-gional analysis is required, or for a representativesubsample of populations. The examples below arelimited to single populations. Shepard el al. (1997)provide an example of a regional analysis.

Quantitative Model StructureThe BayVAM procedure relics on stochastic

population simulation models as the quantitativeagents supporting the analysis. These models havebeen recast as probabilistic networks in order toenhance their utility. The following sections de-scribe the structure of the underlying model usedin the potamodromous version and the correspond-ing probabilistic network. The anadromous version

is similar but is based on the life cycle model ofLee and Hyman (1992).

Underlying Model Structure

The underlying population viability model usedin the potamodromous version is a stochastic, age-structured demographic model that tracks popu-lation numbers in annual time steps. To simplifymatters, modeled populations consist of femalesonly: any reference below to a specific life stageimplies females only. This makes it unnecessaryto track maturity and survival rates for both sexesor to account for time-varying sex ratios. The pop-ulation is divided into three main stages: subyear-lings (age 0), juveniles, and adults. Subyearlingsprogress from eggs to parr during the course oftheir first year, whereas juveniles and adults arefurther divided into discrete year-class cohorts,numbered from 1 to m - 1, and m to m + 9,respectively, where m is the age of maturity (Fig-ure 1).

The model requires eight demographic param-eters, an estimate of the variation in subyearlingsurvival, the expected frequency of major habitatdisruptions (catastrophes), and an initial numberof spawning adults (Table 1). The length of thesimulation period is variable. As mentioned above,a 100-year simulation period was used in devel-oping the probabilistic network, but no such re-striction is inherent in the simulation model itself.

Spawning initiates each simulated year. Thenumber of eggs produced is the product of averagefecundity and the number of adults remaining atthe end of the prior time step plus any immigrants.Immigrants are expressed as adult equivalents perspawning brood; that is, any differences in repro-ductive success between immigrants and residentadults are incorporated by conversion to adultequivalents. For example, if immigrants are pre-sumed to have a relative fitness of 0.8, then fouradult equivalents would be added for every fiveimmigrants. Immigrants contribute only to thenumber of eggs produced; they are not counted inadult numbers. Each year, the number of immi-grant adult equivalents is generated as a Poissonrandom variate with expected value equal to im-migrant.

The number of fry produced is a random bi-nomial variate with parameters W (number of eggs)and p (spawning and incubation success). First-year survival of fry is modeled as the asymptotic,density-dependent relationship

PROBABILISTIC NETWORKS FOR R I S K ASSESSMENT 1147

Eggs —— *• Fry

Juveniles(agel) —— *» Juveniles

(age 2)

Adults(age m) —— »» Adults

(agem+1)

Spawning

Fioi'Ki I.—Structure of the underlying population v iab i l i t y s imula t ion model.

s() = otf 1 - exp-par reap

a - f r y (1)

where so is the expected survival, a is the maxi-mum fry survival rate at a low density, parrcap isthe maximum number of fry expected to survivethe first year (i.e.. the parr carrying capacity of thesystem), and fry is the number of fry or alevinshatching.

The actual number of juveniles produced is arandom binomial-beta variate, whereby the prob-ability of survival is randomly drawn from a betadistribution with mean SQ and coefficient of vari-ation (CV = 100-SD/mean) specified within themodel parameters. This random draw of the first-year survival probability introduces an element ofvariability consistent with annual environmentalfluctuations in rearing conditions. Successive tran-sitions within juvenile age-classes, from juvenilesto adult class m. and within adult age-classes aredensity-independent, random binomial processesconsistent with the stochastic, discrete nature ofthe model. Transitions within juvenile age-classesuse the juvenile survival parameter X[: juvenile-to-adult and adult-to-adult transitions use the adultsurvival parameter^, which includes prespawningmortality (which affects first-time and repeat

spawners). These transition probabilities are con-stant across years for any realization of the model.

Modeling Catastrophic DisturbanceThe simulation model also includes catastrophic

disturbances. A catastrophic disturbance refers toa random event that substantially diminishes sur-vival of all age-classes in the year of its occurrenceand reduces spawning and incubation success andfirst-year survival for some years into the future.In the model, the expected time period (in years)between catastrophes (cat. Table 1) determines theprobability of occurrence in a given year; the prob-ability of catastrophe equals the reciprocal of theexpected time between catastrophes. Occurrenceof a catastrophe is modeled simply as a Bernoullitrial each year: a catastrophe either occurs or not.

When a catastrophe occurs, two things happen.First, the number of juveniles and adults is im-mediately reduced, but their survival rates in sub-sequent years remain the same as before. Second,the habitat carrying capacity for parr (reflected inparrcap) and the spawning and incubation success(incsuv) are also reduced but return to their pre-calastrophe levels more slowly according to theequation

1148 LKE AND RIEMAN

10 20 30 40 50 60

Year70 80 90 100

FICHIRK 2.—Simulated habitat qual i ty in an environment experiencing three catastrophic disturbances wi th in a100-year period (years 10. 19. and 93).

H = (1 - D) + Dt-

4oo (2)

where H is a scalar between 0 and 1 that is mul-tiplied by parrcap and incsuv, D is the disturbancecoefficient, and / is the time steps since distur-bance. In model applications thus far, D is a ran-dom variate from a beta distribution with a meanof 0.5 and a standard deviation of 0.13. Thus, onaverage, each catastrophe reduces the populationby one-half. Because multiple catastrophes can oc-cur in sequence, the habitat quality index H isadjusted to reflect cumulative impacts (Figure 2).

Model BehaviorEach realization of the model produces a sim-

ulated time series of population abundance thatexhibits random fluctuations. Conceptually, eachtime series reflects a combination of an underlyingdeterministic population trajectory with the annualvariation introduced by the stochastic processesincluded in the model (Figure 3). This variationin a single time series arises from three parts: (1 )demographic variability that arises because pop-ulations are composed of discrete individuals, andeach individual independently contributes to birthand death processes through the binomial pro-cesses; (2) environmental variability that arisesfrom the annual variation in fry survival, inde-pendent of density-dependent variation; and (3)catastrophic disturbances. Each time series is ame-nable to the same calculations that one can perform

on empirical data. For example, summary statistics(e.g., mean and variance) or statistics designed fortime series analysis can be used to summarize re-sults. Because of random fluctuations within themodel, simulated extinction can occur even whenthe overall trend in population numbers seems sta-ble or increasing. Normally, multiple iterations ofthe model would be run for any given parameterset in order to identify central tendencies and therelative frequency of various outcomes such asextinction.

The relatively simple structure of the simulationmodel permits calculation of intermediate vari-ables that can aid interpretation of model behavior.Two straightforward intermediate variables are re-production and replacement. Reproduction is sim-ply the product of fecundity and spawning andincubation success. Replacement is the hypothet-ical number of parr required to produce one adultspawner. Replacement (R) can be calculated fromjuvenile survival (.vj), maturation age (w), andadult survival (^2) as

R = /V"' I

(3)

Replacement, reproduction, and the parametersof the fry survival equation combine to determinetwo key intermediate variables, equilibrium sizeand resiliency. Equilibrium size is the number ofadults that would be expected to exactly replace

PROBABILISTIC NETWORKS FOR R I S K ASSbSSMHNT 1149

1000-

1<*1z

900-

800

700

600

500

400

300

200

100

/

D

10 20 30 40 50

Year60 70 90 100

FuiURh 3.—Simulated time trace of population abundance with various levels of random variation included:deterministic model with no random variation (A), model A plus demographic variation (B). model B plus envi-ronmental variation in first-year survival (C), and model C plus catastrophic disruptions from Figure 2 (D) .

themselves if there were no stochastic variationwithin the model (Figure 4). It is calculated as

Neq =-par reap

,og(.(, - JL\n\ /

(4)

•F

where Neq is the equilibrium size, R is replace-ment, F is reproduction, and a and parrcap arc asdefined in equation ( 1 ) . Equilibrium size is mean-ingful only if OL-F is greater than R. If R exceedsOL-F, then there is no equilibrium point other thanzero in the deterministic analog of the model. Inthese situations, rapid population extinction is ex-pected unless the population is supported by im-migration.

Resiliency also plays an important role in pop-ulation persistence. Resiliency is defined as a nor-malized index of the average surplus productionof juveniles over the range of spawning adults be-tween zero and Neq, divided by the number ofrecruits produced at point INeq, Jeq]. This isequivalent to the shaded area in Figure 4 dividedby Neq-Jeq. Populations with high resiliency ex-hibit a high degree of density dependence that

tends to quickly return populations to near-equi-librium conditions following a perturbation. Re-siliency and equilibrium size are linked becausezero equilibrium size implies zero resiliency aswell. For any positive equilibrium value, a rangeof resiliency values is possible.

Probabilistic NetworksIn real-world applications, model parameters arc

never known with certainty. As described in step2 above, this uncertainty in parameters is quanti-fied in the BayVAM procedure during the surveyof watershed conditions. When ample data areavailable to estimate model parameters, uncertain-ty in the parameters can be estimated through con-ventional means such as confidence limits. Whendata are lacking, professional judgment is used toidentify the probability of various parameter rang-es. The use of professional judgment to estimateparameters and their uncertainty is much discussedin the literature (Morgan and Hcnrion 1990; Lud-wig 1996) and is not without its detractors (seealso Dennis 1996). There is no disagreement, how-ever, that parameter uncertainty works in concert

1150 LEE AND RIRMAN

16000

14000

120001

10000

8000

6000

4000

2000

Oi

100 200 300 400 500 600Adults

700 800 900 1000

Fuii 'Ri 4.—Graphical representation of the underlying relationship between juveniles and adults. Curve A rep-resents the expected number of juveniles produced for each level of spawning adults: curve B is the reciprocalnumber of adults that are expected from a given level of juvenile recruits to the population. The point |Neq. Jeqjrepresents the equi l ibr ium point of the deterministic analog of the model. The shaded area between the two curvesreflects the degree of population resiliency present.

with natural variability to reduce the certainty withwhich one can anticipate future stales of nature.

In the current context, uncertainty arises fromthe combination of parameter uncertainty and thethree sources of natural variability addressedabove (demographic variability, environmentalvariability, and catastrophes). To explore the im-plications of this uncertainty with the populationsimulation model, a series of Monte Carlo simu-lations can be used in which various combinationsof parameters arc drawn from the probability dis-tributions defined for each parameter. Each real-i/ation would represent a combination of randomlyselected parameters (one set drawn per realization)and a random sequence of annual fluctuations fromnatural processes (a different sequence for everyrealization). Combining large numbers of such re-alizations would identify the probability of variousoutcomes given all four sources of uncertainty.(There is additional uncertainty due to alternativemodel forms, but we ignore this for now.)

The Bay VAM procedure permits analysis of thecombination of parameter uncertainty and naturalvariability, not by requiring Monte Carlo simula-tion in each analysis, but by recasting the simu-lation model as a probabilistic network. A largenumber of simulations over all possible combi-nations of parameters are done once, and the re-sults of these simulations are used to build a prob-abilistic network. Probabilistic networks, alsocalled Bayesian belief networks or causal proba-bilistic networks, are relatively new developments

in the field of artificial intelligence and form aspecial class of graphical models that are becom-ing increasingly popular in statistics (see Whit-taker 1990). Examples of probabilistic networksused in natural resource management are rare;Haas (1991) is a notable exception. Pearl (1988),Olson et al. (1990), and Haas (1991) provide usefulintroductions to the topic.

Key to understanding the probabilistic networkis viewing the simulation model as a graph (Figure5), wherein each input parameter, intermediatevariable, and output variable defines a node orpoint on the graph. Because they are intricatelylinked, equilibrium size and resiliency are capturedin a single node. Nodes are connected by directedarcs (arrows), which indicate the causal relation-ships and the conditional dependencies amongnodes. For example, replacement is a function ofjuvenile survival, age at maturity, and adult sur-vival. This relationship, which was defined math-ematically in equation three, is depicted graphi-cally in the upper right portion of Figure 5. Re-production is a similar function of fecundity andspawning and incubation success. The relationshipbetween reproduction, a. parrcap, replacement,and equilibrium size and resiliency raises an in-teresting point that is made explicit in the graph:once the values for reproduction and replacementare known, their antecedent parameters are notneeded to estimate equilibrium size and resiliency.The graphical view of the model also suggests thatonce equilibrium size and resiliency arc known,

PROBABILISTIC NETWORKS FOR RISK ASSESSMKNT 1151

Fecundity Spawning/Incubation

5.—Schematic diagram of the BayVAM probabilistic network. Input nodes (highlighted in boldface i tal ic)and a diagnostic node (Adul l CV) correspond to elements assessed in a watershed and population survey. All othersare either model intermediates or outputs.

the only additional parameters needed to projectthe output variables are initial adults, catastrophicfrequency, immigration rate, and the coefficient ofvariation in first-year survival (Figure 5).

In order to depict the uncertainty associated witheach node, nodes are subdivided into 3-10 discretelevels. Generally, these levels refer to a specifiedrange of a continuous variable. At a given pointin time, each node has one true value that reflectsthe state of nature. This true value, however, willnot be known with absolute certainty. In the prob-abilistic network, this uncertainty is displayed byassigning a degree of belief (i.e., Bayesian prob-abil i ty) to each value within each node. Each nodethen has an array of probability values associatedwith it, creating a probability histogram that isknown as a belief vector (Figure 6). These vectorsare constrained such that the sum of all valueswithin each vector is 100%. The allocation of be-liefs within a node is used to quantify the strengthor quality of the available information. For ex-ample, if several years of data provide good es-timates of juvenile survival that vary around amean of 30%, the belief vector for juvenile sur-vival might be concentrated in the 27-34% range.Conversely, if there are no data for a particular

watershed, but survival values in similar systemshave been reported that uniformly range from 20to 40%, a belief vector spread over the lower fourclasses might be appropriate. If there is absolutelyno information available to judge juvenile surviv-al, the default uniform probability distribution(equal probability assigned to each class) wouldbe appropriate. Biologists with some knowledgeof the watershed, the species and life history, andthe relevant fish habitat relations should be ableto make classifications that fall somewhere be-tween the extremes of no information and a valueknown with certainty.

The purpose of a probabilistic network is to rig-orously track these belief vectors and to system-atically update them as information is added to thesystem, ft does this using the calculus of condi-tional probability first articulated by Bayes (1763).Information is added to the system by changingthe belief vectors associated with the input param-eters. This information is then propagated throughthe probabilistic network to update beliefs aboutthe intermediate and output variables. Essential topropagation are link matrices of conditional prob-abilities that define the relationships betweennodes. Wherever there is a directed arc, there is a

1152 LEE AND R I E M A N

50-200 13 Bl200-350 13B1350-500 13 BB500-650 13 •660-800 13 •800-960 13 BB960-1100 13 Bl1100-1260 13 Bl

10-20%20-30%30-40%40 50%50-60%60-70%

7mm7fln7mt7mm7mt7mm

Juvenffe Survival

15-21%21 -27%27-34%34-41%41-48%48-54%54-60%

14 BB14 BB14 BB14 BB14 Ml14 BB14 BB

^-^^—

3yrs4yrsSyrs6VT5

Mature Age

25BBBI25BBBI25 BIB2SBBB

Adult Survival

15-30%30-45%45-60%60-75%75-90%

20BB120BB120 BB120BBB20BBB

Rcoroduction

<7070-120120-170170-240240-350350-500>500

15BB13BB13B114 BB17BBI15 BB13flB

Man

10-16%16-22%22 28%28-34%34-40%

num Fry Survival

20mm20mm20mm20 mm20mm

————— Pan Capacity —————

1000-4000 33BBBBI4000- 7000 33 BUM7000-10000 33BBBBB1

2 - 55-1010-2020-50>50

low. low 8Bmed.low 6mhigh.low 7atow.med 8*

9BNgh.med 6mlow.Wph 9*mtd.hlgh 8B

<30 23 a30 250 26 B250 750 20 •750-1500 12 •>1500 19B

zero 24 a1-30 18B30-130 20 B130 300 16 B>300 22 B

33 yt 3134 66yr 3167 99 yr 1 1>99yr 92 •

<40% 25 fl40 - 60% 28 B60-90% 14 a>90% 32 •

FiGt 'Rh 6.—Graphical representation of the BayVAM probabilistic network in the ini t ial i /ed state, wherein noprior information regarding the condition of the model input parameters is given. Histograms at each node reflectthe marginal probabilities (i.e.. belief vectors) associated with each node, expressed as percentages.

l ink matrix that explicitly defines the conditionalrelationship between the two nodes. For example,if a hypothetical node A is connected to node Bthrough a directed arc from A to B, then there isan explicit statement of the form, "given that Ahas a value a, then the probability that B has thevalue h is /V* for all combinations of possiblevalues of A and B. Generally, the values withinl ink matrices come from empirical evidence, or ifthat is lacking, from expert opinion. The BayVAMprobabilistic networks are unconventional in thatthe l ink matrices are derived from simulation re-sults (see below).

Viability IndicesThree variables are used within the BayVAM

procedure to reflect viability: average populationsize, minimum population size, and time to ex-tinction. These variables are generated from eachrealization of the simulation model. In the prob-

abilistic network, the belief vectors associatedwith each viability index are comparable to thefrequency distributions produced in a Monte Carloanalysis as described above, in that they representthe observed frequency of outputs given a largenumber of simulations involving random samplingof parameters from the parameter space defined bythe parameter belief vectors.

The viability indices can be used to assess risk,where "risk" is used in the colloquial sense thatimplies a probabilistic loss (i.e., there is somethingof value which could be lost, though the loss isuncertain). On these terms, risk is expressed in twodimensions, one dimension expressing the mag-nitude of potential loss and the second expressingthe probability of loss. Because the magnitude ofloss can be described in an infinite number of waysdepending on the circumstances, direct compari-sons of risks measured on different scales are notpossible. There is no such ambiguity with proba-

P R O B A B I L I S T I C NETWORKS FOR R I S K ASSESSMENT 1153

bility, which can be defined rigorously on a scaleof zero to one, where one implies certainty. AsRamsey (1926). de Finctti (1974), and others haveshown, the rules of probability apply even whenthe probability merely reflects one's subjective de-gree of belief (Press 1989; Howson and Urbach1991).

In the case of fish populations, any declinewould represent loss, be it loss of recreational fish-ing opportunities, aesthetic values, or unique ge-netic resources. The extent of the potential loss isreflected in average adult numbers and minimumpopulation size; the probability of the loss is re-flected in the belief vectors. Where populationsmay go extinct, the urgency of the situation isreflected in the belief vector of time to extinction.

Diagnostic NodesMost of the inferences within the probabilistic

network follow causal pathways; that is, infor-mation about parameters is used to infer beliefabout intermediate variables and viability indices.One exception is the adult CV node, which is afourth output of the simulation model that tracksthe coefficient of variation in adult numbers forextant populations. This node can be used option-ally to provide additional information about thevariation in first-year survival when time seriesinformation on adult counts is available. If thisoption is invoked, the probabilistic network uses( 1 ) the relationships defined in the link matricesamong equilibrium size and resiliency, immigra-tion, first-year variation in survival, and adult CV,(2) the calculated belief vectors for equilibriumsize and resiliency and immigration given the otherinput parameters, and (3) adult CV values fromempirical data to update belief vectors for first-year survival.

Developing Link MatricesA key step in developing the probabilistic net-

work was specifying the link matrices that definethe relationships between a target or receptor nodeand the set of donor nodes that conditionally affectthe belief vector of the receptor node. (Within thegraphical model, directed arcs connect donornodes to receptor nodes.) Because the probabilisticnetwork was designed to mimic the behavior ofthe simulation model, developing the link matricesinvolved analyzing the results of Monte Carlo sim-ulation. This process involved ( 1 ) randomly sam-pling model parameters from defined ranges andcalculating intermediate attributes, (2) running themodel once for each random set of parameters, (3)

mapping the parameter values into discrete levelsconsistent with the ranges used in the assessmentsurvey and mapping intermediate attributes andmodel outputs into similar variables that were usedas nodes within the probabilistic network, and (4)computing contingency tables based on the jointdistribution of parameters, intermediate variables,and viability indices. These contingency tablesthen were translated into the link matrices neededin the probabilistic network. The matrices that linkeach receptor node to donor nodes are based onthe relative frequency with which each level of thereceptor node was observed during the simulationexercise, given each combination of donor values.

The simulation exercise involved 600,000 ran-dom combinations of parameters chosen from se-lected ranges (Table 1). Uniform distributions wereused to describe the distribution of parameterswithin each range. Each simulation consisted of100 years of dynamics. Several summary statisticswere calculated for each simulation, including themean and variance of the population size over theentire period and for only those years when adultswere present. Extinction time was defined as thefirst year in which there were neither adults norjuveniles; runs that did not result in extinctionwere assigned an extinction time of more than 100years.

Parameter ranges generally were chosen to spanthe range of expected values for trout and charpopulations in a variety of conditions. One excep-tion is the parr capacity parameter, parrcap. Parrcapacity was used as a scaling parameter to set therange of possible equilibrium sizes. Our intent wasto develop a model that would be applicable toself-sustaining populations in small to moderatelysized watersheds (<20,000 ha). Thus, the upperl imit on parrcap was set to correspond to an Neqvalue of roughly several thousand, given medianvalues of the other parameters. The lower limitwas set to correspond to Ncq values of a few hun-dred under good conditions. Obviously, applica-tion of the model to populations outside theseranges would be inappropriate.

The link matrix for the diagnostic node, adultCV, was derived from the subset of simulations inwhich the simulated population did not go extinctand there were no catastrophes. In such cases, theobserved adult CV is sensitive to population re-siliency and variation in juvenile survival; adultCV declines as resiliency increases and variationin juvenile survival decreases.

1154 LEE AND RIEMAN

Interpreting Belief VectorsThe principal outputs of the probabilistic net-

works are the belief vectors associated with theviability indices. As stated above, these vectorsrepresent the expected frequency of results givena large number of simulations involving randomsampling of parameters from the parameter spacedefined by the parameter belief vectors. The in-terpretation of these vectors thus must be guidedby an understanding of the model. Interpretationalso must be tempered by an understanding of un-certainty, how it is propagated through a proba-bilistic network, and what the belief vectors implyin terms of ecological risk.

For starters, remember that any model is, at best,nothing more than a caricature of reality; it willexaggerate some features of the system while un-derstating or ignoring others. Even models param-eterized with the best of data and ecological un-derstanding should not be confused with real eco-systems. Similarly, model outputs such as esti-mated probability of extinction should not beconfused with any true system parameter. The sur-vival of a particular species or stock is not a con-trolled experiment that will be repeated infinitetimes, leading to an empirical probability of per-sistence. Time will proceed only once and the spe-cies will either persist or not. Thus, the only de-fensible interpretation of probability of extinctionis in the Bayesian sense of a measure of our beliefabout a future event (see Howson and Urbach1991; Ludwig 1996). Even staunch frequentiststatisticians cannot claim a non-Baycsian inter-pretation of future species extinction or quasiex-t inction. Frequcntists can define probability dis-tributions only in terms of the long-term behaviorof known statistical models or the result of an in-f inite series of identical trials (Dennis 1996; Lud-wig 1996). Translating the behavior of a statisticalmodel to a degree of belief about a future eventin the real world is uniquely Bayesian.

For those who value species persistence, itseems prudent to manage natural resources cau-tiously and to proceed with changes in a naturalsystem only when the weight of available evidenceconvincingly suggests that future actions will notjeopardize a species (i.e., the species has a highBayesian probability of persistence). Any uncer-tainty, whether it arises from unpredictable eventsin nature, or from an inabili ty to identify or mea-sure ecosystem parameters, lessens the certainlywith which one can anticipate future states of na-ture. In the probabilistic network, increased un-

certainty translates into more even probability dis-tributions at each node. For example, when no in-formation is available, the belief vectors for pa-rameter nodes are uniform distributions (Figure 6).In this initial state, the belief vectors for averagepopulation size and minimum population size arcfairly uniform and the belief vector for extinctiontime suggests a low probability of extinction,mainly due to the high probability of the popu-lation being supported by immigration. Closer in-spection reveals that the probability of zero equi-librium and resiliency is 26%, which would leadone to expect that at least 26% of the populationsare ultimately doomed, while simultaneously 75%of the simulated populations are being augmentedby immigration (Figure 6). Multiplying 26% times25% (the fraction unsupported by immigration)gives 7%, which is essentially the fraction thatwent extinct.

In this example, the probability of extinctionmeans little. The larger lesson is that uncertaintybegets uncertainty. Without information, nothingcan be said with certainty concerning the futurestatus of the stock. When all combinations of pa-rameters (over the ranges used here) are equallylikely, then the odds favor unforeseen combina-tions of parameters that can either enhance or in-hibit populations in an unpredictable fashion.

From a management perspective, knowing howmuch uncertainty is due to ignorance (i.e., param-eter uncertainty) and how much is due to randomenvironmental fluctuations can be important be-cause of the different management strategies need-ed to address each. While parameter uncertaintycan be decreased simply through increased infor-mation, a certain level of uncertainty is due toprocesses that may be beyond the control of man-agement. The role of management in this case isto plan for that uncertainly and mitigalc negalivceffecls lo ihc exlenl ihis is practical.

The probabilistic network is structured such thatthe uncertainly in viability indices lhal arises fromparameier uncertainty can be partitioned partiallyfrom thai due lo random environmenlal fluciua-lions. For example, if ihe parameier nodes lhaldeiermine equilibrium and resiliency are set lo sin-gle-value histograms, then the probability vectorfor ihe equilibrium-resiliency node is concenlraledover a narrow range (Figure 7). In ihis example,ihe uncerlainly in ihe viabilily indices (Figure 7)arises primarily from random processes. Il isslraighiforward lo ilerilavely adjusl ihe parameieruncerlainty and the level of random fluctuations

PROBABILISTIC NETWORKS FOR R I S K ASSESSMENT 1155

FeairK»y50-200200-350350-500500-650

800-950850-11001100-1250 -

10-20%20-30%30-40%40-50% 100C50-60%60-70%

3

V

Juvenile Survival

15-21%21 - 27%27-34%

41 -48%48-54%54-60%

3yrs4yrs5yns6yrt

Mature Age

20BBB51

15-30%30-45%45-60%60-75%75-90%

Adut Survival

<7070-120120-170170 - 240240-350350 500>500

Maximum Fry Survival

10-16%16-22%

28-34%34-40%

Parr Capacity

1000-4000 -4000- 7000

™^̂ "* BmiJmrmmMjtlnepucenwfn

<22 - 55 1010-2020-50>50

AM67aaaaaaaaaai15 •§9*21

FiGURH 7.—Graphical representation of the BayVAM probabilistic network i l lus t ra t ing updated belief vectorsgiven information on population parameters. Information input to the network is shown as unshaded bars wi th abelief value of 100. A probability distr ibution for age at matur i ty was added also.

in order to discern the relative contribution of eachto the final uncertainty in viability indices.

The probabilistic network displays sensitivitiesto changes in parameter inputs that are consistentwith the behavior of the analytical model, with oneimportant difference: the magnitude of those sen-sitivities may be reduced. In general, the proba-bilistic network exhibits less sensitivity to changesin parameters than would be exhibited in a moreconventional simulation approach. The reason forthis lies in the way that a signal (i.e., the changein an output due to a change in input) is attenuatedas it passes through a probabilistic network. In theprobabilistic network, parameter nodes are con-nected to intermediate nodes in probabilistic fash-ion, as opposed to the precise algebraic connectionwithin the analytical model. This results in a damp-ening in signal as it passes from one node to theother. Changing the mode of the belief vector forincubation success slightly, for example, does not

change the belief vector for extinction time to thesame extent that an equivalent change in a pointestimate for incubation success does in the sim-ulation model.

This relative inscnsitivity lends a certain inertiato the probabilistic network, in that substantivechanges in parameter belief vectors are requiredto shift the belief vectors or reduce the uncertaintyexhibited in the viability indices. Relative to initialconditions of no information (Figure 6), the prob-abilistic network requires considerable informa-tion about population parameters to achieve ac-ceptably high probabilities of persistence or lowlevels of uncertainty (e.g., Figure 7). Thus, theprobabilistic network implicitly places the burdenof proof on the network user to demonstrate thata watershed can support a given population. Inworkshops where we have presented the BayVAMprocedures and asked participants to apply them,we've noticed a tendency on the part of partici-

1156 LKH A N D R I H M A N

pants to skew the belief vectors toward the lessoptimistic parameter ranges. Whether this repre-sents either an unintentional or strategic bias isunknown, but clearly it contributes further to theconservative nature of the procedure.

ApplicationWe evaluated the BayVAM procedure in a series

of test applications in watersheds throughout thePacific Northwest in a cooperative effort with FSand BLM biologists. Participating biologists re-ceived rudimentary training and then were askedto apply the procedure to populations familiar tothem. The most extensive application of theBayVAM procedure to date is the application de-scribed by Shepard et al. (1997).

Results from the early testing phase are instruc-tive. Although the procedure seemed well receivedby those in training, there was an apparent reluc-tance by some to apply it. Much of this reluctanceundoubtedly is due to the novel nature of this pro-cedure and the natural hesitancy to use an approachthat one does not fully understand, yet is expectedto explain to others. As the procedure is morewidely applied and the results become more fa-miliar, we expect wider acceptance.

One of the more encouraging results of our ex-perience thus far with the BayVAM procedure isthe extent and focus of discussions that are gen-erated by each application. The transparent natureof the probabilistic network makes explicit all ofthe assumptions made by a user. If such assump-tions seem in conflict with the evidence presentedto justify those assumptions, or inconsistent withother assumptions, others are quick to notice. Thisoften leads to a focused discussion of a particularassumption. The overall structure of the simulationmodel has not been seriously questioned, althoughit is a legitimate concern. For example, our un-derlying assumption is a population driven by age-structured dynamics, a conventional approach thatcan be traced to the age-structured models of Les-lie (1945, 1948) and the stock recruitment modelsof Ricker (1954). Alternative, habitat-structuredmodels have been proposed (Bartholow et al.1993; Bowers 1993) that might lead to differentresults. As our understanding of population dy-namics improves, corresponding changes and en-hancements in the BayVAM procedure are antic-ipated.

We have identified the BayVAM procedure as atool for population viability assessment (PVA). Assuch, many of the questions and concerns that havebeen raised about PVA may apply (e.g., Simberloff

1988; Boyce 1992; Caughley 1994). In his critiqueof conservation biology, Caughley (1994) char-acterizes many PVAs completed on threatenedspecies as, "essentially games played with guess-es" (see also Hedrick et al. 1996). Caughley(1994) correctly notes that most PVAs are basedon computer simulations with simplistic models,and the basic model parameters are rarely knownwith precision. In the BayVAM approach, we havetried to mitigate this limitation by making theguesswork an explicit source of uncertainty in re-sults and by incorporating sufficient detail into thepopulation model to force users to think system-atically about the factors affecting multiple lifestages. When used within an integrated assessmentof land use activities (e.g., Shepard et al. 1997).the procedure helps to more fully identify popu-lation threats as well as quantify risks.

Whatever its shortcomings, the BayVAM pro-cedure offers a promising step towards tools thatare widely applicable and can be used to assessthe population viability of salmonid populationsin a rigorous and consistent fashion with a mixtureof information. For this reason, we believe thatcombining qualitative judgments with quantitativemodels using probabilistic networks is an approachdeserving of a second look by the wider resourcemanagement community.

AcknowledgmentsWe thank Tim Haas, Bruce Marcol, and three

anonymous reviewers for their helpful commentson the manuscript.

ReferencesBartholow, J. M.. J. L. Laake. C. B. Stalnakcr. and S.

C. Williamson. 1993. A salmonid population modelwith emphasis on habitat l imitat ions. Rivers 4:265-279.

Bayes. T. 1763. An essay towards solving a problem inthe doctrines of chances. Philosophical Transactionsof the Royal Society of London 53:370-418. (Re-printed 1958, Biometrika 45:293-315.)

Bowers. M. A. 1993. Dynamics of age- and habitat-structured populations. Oikos 69:327-333.

Boyce. M. S. 1992. Population v iab i l i t y analysis. AnnualReview of Ecology and Systematies 23:481-506.

Caughley. G. 1994. Directions in conservation biology.Journal of Animal Ecology 63:215-244.

de Finetti. B. 1974. Theory of probability, volumes Iand 2. Wiley, New York.

Dennis. B. 1996. Discussion: should ecologists becomeBaycsians? Ecological Applications 6:1095-1 103.

FEMAT (Forest Ecosystem Management AssessmentTeam). 1993. Forest ecosystem management: anecological, economic, and social assessment. U.S.Forest Service, Portland. Oregon.

PROBABILISTIC NETWORKS FOR RISK ASSESSMENT 1157

Haas. T. C. 1991. A Baycsian belief network advisorysystem for aspen regeneration. Forest Science 37:627-654.

Hedrick. P. W.. R. C. Lacy, F. W. Allendorf. and M. E.Soule. 1996. Directions in conservation biology:comments on Caughlcy. Conservation Biology 10:1312-1320.

Hedrick. P. W.. and P. S. Miller. 1992. Conservationgenetics: techniques and fundamentals. EcologicalApplications 2:30-46.

Howson. C.. and P. Urbach. 1991. Bayesian reasoningin science. Nature (London) 350:371-374.

Lande. R. 1988. Genetics and demography in biologicalconservation. Science 241:1455-1460.

Lee. D. C. and J. B. Hyman. 1992. The stochastic life-cycle model (SLCM): s imula t ing the population dy-namics of anadromous salmonids. U.S. Forest Ser-vice Research Paper INT-459.

Leslie. P. H. 1945. On the use of matrices in certainpopulation mathematics. Biometrika 33:183-212.

Leslie. P. H. 1948. Some further notes on the use ofmatrices in population mathematics. Biometrika 35:213-245.

Ludwig. D. 1996. Uncertainty and the assessment ofextinction probabilities. Ecological Applications 6:1067-1076.

Marcot. B. G.. and D. D. Murphy. 1996. On populationv i a b i l i t y analysis and management. Pages 58-76 inR. S/aro, editor. Biodiversity in managed land-scapes: theory and practice. Oxford UniversityPress. New York.

Morgan. M. G.. and M. Henrion. 1990. Uncertainty: aguide to dealing w i th uncertainty in quant i ta t ive riskand policy analysis. Cambridge University Press,New York.

Olson. R. L.. J. L. Willers. and T. L. Wagner. 1990. Aframework for modeling uncertain reasoning in eco-

system management II. Bayesian belief networks.AI Applications in Natural Resource Management4(4): 11-24.

Pearl. J. 1988. Probabilistic reasoning in intel l igent sys-tems: networks of plausible inference. MorganKaufmann. San Mateo, California.

Press. J. S. 1989. Bayesian statistics. Wiley. New York.Ramsey. F. R 1926. Truth and probability. In R. B.

Braithwaite, editor. The foundations of mathematicsand other logical essays (1931) . Humanit ies PressInternational. New York.

Ricker, W. E. 1954. Slock and recruitment. Journal ofthe Fisheries Research Board of Canada 11:559-623.

Rieman. B.. D. Lee. J. Mclntyre, K. Ovenon. and R.Thurow. 1993. Consideration of extinction risks forsalmonids. U.S. Forest Service. Fish Habitat Re-lationships Technical Bullet in 14. Eureka. Califor-nia.

Ruggiero, L. F., G. D. Hay ward, and J. R. Squires. 1994.Viab i l i ty analysis in biological evaluations: con-cepts of population viabi l i ty analysis, biologicalpopulation, and ecological scale. Conservation Bi-ology 5:18-32.

Shaffer. M. L. 1987. Min imum viable populations: cop-ing with uncertainty. Pages 68-86 in M. E. Soule.editor. Viable populations for conservation. Cam-bridge University Press. Cambridge. UK.

Shepard. B. B.. B. Sanborn. L. Ulmer. and D. C. Lee.1997. Status and risk of extinction for we.stslopecutthroat trout in the upper Missouri River basin.Montana. North American Journal of FisheriesManagement 17:1158-1172.

Simberloff, D. 1988. The contribution of population andcommunity biology to conservation science. AnnualReview of Ecology and Systematics 19:473-51 1.

Whittaker. J. 1990. Graphical models in applied mul t i -variate statistics. Wiley. West Sussex. UK.