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Appendixes

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Appendix A
Lexicon
INTRODUCTION ally called “analysis” (indeed, employees assigned to
information gathering are called intelligence analysts),
The lexicon in this appendix, prepared by the Commit-
and the second step is called an “assessment” of the
tee on Methodological Improvements to the Department of
situation.
Homeland Security’s Biological Agent Risk Analysis, is in-
• The risk and decision community reverses these
tended to be an exemplar of what might be used in any public
definitions: the first step of gathering information (in
presentation and discussion of a probabilistic risk analysis
particular, obtaining information about the uncertainty
and presented as a supplement to this report. Without a clear
of events and their possible consequences) is usu-
and consistent use of language in this technical arena, there
ally called “assessment,” while the second step—the
will be a tendency for conclusions to be misinterpreted and
process of using this information and combining it in
for policy recommendations based on these conclusions to
such a way that a decision maker can make better deci-
be misguided.
sions—is usually called “analysis.”
Because many of the terms in this lexicon (Table A.1) are
found in everyday usage, often with implications or mean-
For this reason, in the lexicon the committee has taken pains
ings different from those presented here, it was suggested
to break out the various components of “risk analysis” as
that the committee also include “lay definitions” in order
used in its report.
to provide a comparison and to help in interpreting various
loosely written documents and statements made available
ALTERNATIVE DEFINITIONS FOR “RELATIVE RISK”
to the committee (and the public). However, the commit-
tee has intentionally not done this, in order to avoid giving
The term “relative risk” has a well-accepted definition
credence to analyses that might be flawed by improper use
in the biomedical community: “The risk of harm among a
or interpretation of various technical terms. The committee
population exposed to a potentially damaging substance,
recommends that any governmental agency issuing a report
compared to the risk amongst an unexposed population.”1
on or engaging in a discussion of risk analysis consider using
The term may also be used to describe the ratio: {cumula-
terms as defined in this lexicon, or establish from the begin-
tive incidence rate in the exposed population}/{cumulative
ning reasons for using alternative definitions.
incidence rate in the unexposed population}. However, the
Department of Homeland Security (DHS) has chosen to
ANALYSIS AND ASSESSMENT use the term for a completely different concept. In particu-
lar, “relative risk” for a particular agent is determined as
There is an unfortunate (but readily dealt with) incon-
follows:2
sistency in usage between two communities importantly
involved in understanding the risk of terrorist events: intel-
• For each agent i an expected consequence E(Ci) is
ligence analysts and risk analysts.
calculated (by Monte Carlo simulation),
• In the intelligence community it is customary first to 1 R.M. Anderson and R.M. May. 1991. Infectious Diseases of Humans.
gather information about an opponent’s intentions and Oxford, United Kingdom: Oxford University Press.
capabilities and then to use this information to present a 2 Department of Homeland Security. 2006. Bioterrorism Risk Assess-
statement of the current situation. The first step is usu- ment. Biological Threat Characterization Center of the National Biodefense
Analysis and Countermeasures Center, Fort Detrick, Md., p. C-95.

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DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT
available on the World Wide Web but developed for
• Probability pi is assigned to the event {agent i will be
promoting commercial software packages, consulting
used},
services, and such. These sites are, for the most part,
• An overall “total” expected consequence (or “risk”) is
computed R = ∑ piE(Ci), poorly conceived and, more problematic, have not been
• The relative risk for agent i is Ri = piE(Ci)/R. vetted by any professional independent set of experts,
academics, practitioners, or professional societies.
• The main portion of the lexicon (Part A.1.A), although
That is, “relative risk” is the proportion of the total expected
developed for biological risks, can also be appropri-
risk contributed by a particular agent. Since this definition
ately applied to nonbiological (chemical, radioactive,
is quite different from that used by the biomedical com-
agricultural, and other) threats. The second part of the
munity, it presents a major source of potential confusion
lexicon (Part A.1.B), specifically, the terms used in
and misinterpretation, particularly among readers who are
susceptible, exposed, infected, and recovered (SEIR)
knowledgeable in epidemiology.
modeling, applies only to biological risk analysis.
• Although the committee recognizes the long philo-
COMMENTS ON THE CONSTRUCTION
sophical history of the controversy surrounding the
AND USE OF THE LEXICON
nature of uncertainty, it takes the position that, for the
purposes of policy development and decision making
• Since the committee’s primary objective is to provide
(the eventual goal of DHS’s risk analysis), all uncer-
consistency among the various terms, the terms are
tainty (subjective, frequency-derived, and so on) must
cross-referenced as needed. Column 1 provides syn-
eventually be encoded into probabilities.
onyms and cross-references for the terms defined. It
• The entry “[None]” in second column, “Committee’s
also gives quoted definitions from the DHS document
Recommended Definition,” indicates a conclusion by
entitled “A Lexicon of Risk Terminology and Meth-
the committee that it is not necessary (or it is poten-
odological Description of the DHS Bioterrorism Risk
tially confusing) to provide a definition. Indeed, the
Assessment” (DHS, 2007).
committee recommends that such terms not be used in
• References are given in footnotes to the table. Rather
any formal discussion of methods, results, and so on,
than using highly theoretical sources, the commit-
unless they are used as exemplars of what not to say.
tee chose to rely on widely accepted introductory or
basic texts3 or more contemporary but focused refer- • Due to the (committee) process by which the lexicon
ences (e.g., Meyer and Booker4). Where appropriate, was developed, it may not include terms that some
readers might find important; further, choices among
selected Web sites from well-regarded sources have
alternative accepted definitions were made where
also been used. However, the committee has intention-
necessary.
ally avoided the use of glossaries and lexicons readily
3 For example: W. Feller, 1968, An Introduction to Probability Theory
REFERENCE
and Its Applications, New York: Wiley; D.V. Lindley, 1965, Introduction to
Probability and Statistics from a Bayesian Viewpoint; Part : Probability, DHS. 2007. “A Lexicon of Risk Terminology and Methodological De-
Cambridge, U.K.: Cambridge University Press; and B. deFinetti, 1974, scription of the DHS Bioterrorism Risk Assessment.” Written com-
Theory of Probability, Hoboken, N.J.:Wiley. munication to the Committee on Methodological Improvements to the
4 For example: M.S. Meyer and J.M. Booker, 2001, Eliciting and Analyz-
Department of Homeland Security’s Biological Agent Risk Analysis.
ing Expert Judgment: A Practical Guide, Philadelphia, Pa.: American Statis- April 14, 2007.
tical Association and the Society for Industrial and Applied Mathematics.

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APPENDIX A
TABLE A.1 Lexicon of Probabilistic Risk Assessment Terms
PART A.1.A TERMS APPLICABLE TO BIOLOGICAL RISKS AND TO OTHER, NONBIOLOGICAL THREATS
Term, with Synonyms, Cross-
References, and DHS Lexicona
Definitions Committee’s Recommended Definition Notes, Comments, and References
accuracy See note under precision
A measure of agreement between the estimated
value of some quantity and its true value.
See also precision. (Adapted from Society for Risk Analysis [SRA]
Glossary.b)
agent-conditional expected risk The conditional expected risk computed using
probabilities conditional upon the use of a
See also conditional expected risk. particular agent.
agent-conditional relative risk The conditional relative risk using probabilities
conditional upon the use of a particular agent.
See also agent-conditional
expected risk.
aleatory probability “A measure of the uncertainty of an unknown
event whose occurrence is governed by some
Synonym: aleatory uncertainty random physical phenomena that are either
(1) predictable, in principle, with sufficient
See also probability, epistemic information (e.g., tossing a die) or (2) essentially
unpredictable (radioactive decay).”c
probability.
approximation “The result of a computation or assessment that
may not be exactly correct, but that is adequate
for a particular purpose.”d
See also estimation.
arc (directed arc) In an event tree: an outcome from a preceding
event to a subsequent event; in a decision tree:
Synonym: branch either an action or an outcome from a preceding
event to a subsequent event.
See also split fraction.
The sum of n numbers divided by n.e,f,g
arithmetic average The average is a simple arithmetic operation, requiring a set
of n numbers. It is often confused with the mean (or expected
Synonyms: arithmetic mean, sample value), which is a property of a probability distribution.
mean One reason for this confusion is that the average of a set of
realizations of a random variable is often a good estimator of
See also mean. the mean of the random variable’s distribution.
conditional expected risk Expected risk computed using conditional The conditioning event is typically the choice of agent;
probabilities. however, it could be other events such as good weather,
See also agent-conditional successful manufacture, or ineffective countermeasures.
expected risk.
conditional probability The probability of an event supposing (i.e., It is important to note that subjectively assessed probabilities
“conditioned upon”) the occurrence of other are based on the state of knowledge that holds at the time of
See also probability. specified events. In the aleatory theory of the probability assessment.
probability, the conditional probability of event
A given event B is equal to the probability of
the joint occurrence of events A and B divided
by the probability of event B, if the probability
of eent B is not zero. (After Feller [1968],g
DeFinetti [1974],h and Lindley [1965].i)
If pi = P{conditioning event i} and Ci = expected consequence
conditional relative risk The proportion of the total expected risk
associated with event i, then total expected risk is R = Σ piCi
contributed by a particular conditioning event.
and the total conditional relative risk associated with event i
is piCi/R.
continued

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DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT
TABLE A.1 Continued
Term, with Synonyms, Cross-
References, and DHS Lexicona
Definitions Committee’s Recommended Definition Notes, Comments, and References
conditional risk Risk computed using conditional probabilities The expected risk associated with a particular agent, as
(follows from the definition of conditional measured by the expected number of deaths, may be
See also risk, conditional probability). conditioned upon (for example) the direction of the wind.
probability.
confidence interval A range of values [a,b] determined from a Confidence intervals should not be interpreted as implying
sample, using a predetermined rule chosen that the parameter itself has a range of values; it has only one
See also uncertainty range. such that, in repeated random samples from the value. The confidence limits a and b, being computed from
same population, the fraction α of computed a sample, are random variables, the values of which (for a
ranges will include the true value of an unknown particular sample) either do or do not include the true value
parameter. The values a and b are called a of the parameter. However, in repeated samples, a certain
confidence limits; α is called the confidence fraction of these intervals will include the parameter, provided
coefficient (commonly chosen to be .95 or .99); that the actual population satisfies the initial hypothesis.
and 1 – α is called the confidence level.
(Adapted from SRA.b)
consequence A description of a scenario, in terms of For DHS risk analyses, typical and important consequence
measurable factors, that decision makers may measures are lives lost, morbidities, direct and indirect dollar
Synonym: outcome consider in assessing preferences over different losses, and others.
scenarios; these factors are often random
variables. (Adapted from McCormick [1981],j
with “damage” replaced by consequences.”)
continuous random variable A random variable that has an absolutely
continuous cumulative distribution function.i,e
See also cumulative distribution
function, probability density
function.
cost-benefit analysis “A formal quantitative procedure comparing SRA also includes in its definition: “To determine a rank
costs and benefits of a proposed act or policy.”b ordering of projects to maximize rate of return when available
funds are unlimited, the quotient of benefits divided by costs
is the appropriate form; to maximize absolute return given
limited resources, benefits minus costs is the appropriate
form.” This method of rank-ordering is inappropriate for risk
analysis in that it implies specific (and presumably known)
trade-offs between noncommensurable benefits and costs. A
better procedure is to plot the costs and benefits associated
with each possible decision and then to present the results to
decision makers to assess the trade-offs, which may (or may
not) result in the linear or multiplicative functions inherent in
the cost-benefit computations.
The function ƒ(x) whose value is the probability
cumulative distribution function The cumulative distribution function always exists for any
(CDF) that a random variable, X, will be less than or random variable; it is monotonic and nondecreasing in x,
equal to a value x; written as P{X ≤ x}.e,g,k and (being a probability) 0 ≤ P{X ≤ x} ≤ 1. If P{X ≤ x} is
Synonyms: cumulative distribution, absolutely continuous in x, then X is called a “continuous”
distribution function random variable; if it is discontinuous at a finite or countably
infinite number of values of x, and constant otherwise, X is
See also probability distribution, called a “discrete” random variable.
probability density function,
probability mass function.

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APPENDIX A
TABLE A.1 Continued
Term, with Synonyms, Cross-
References, and DHS Lexicona
Definitions Committee’s Recommended Definition Notes, Comments, and References
decision tree A tree with event nodes that are random The operations used in a decision tree are elementary:
variables or decision nodes that represent expectation over consequences at event nodes and
See also event tree, fault-tree decisions of an active agent. Each branch (path maximization (or minimization) at decision nodes.
analysis. of event and decision nodes leading to a terminal
node) may have consequences (e.g., in dollars, Decision trees can be infinite (with no terminal nodes,
lives, utility) associated with its terminal node. as in recursive game trees) and/or can have intermediate
consequences at nonterminal nodes.
directed arc In an event tree: an ordered pair of nodes,
representing a preceding event, followed by a
Synonym: branch subsequent event. It is usual to interpret an arc
as the outcome of an event.
See also split fraction.
In a decision tree or game tree: an ordered
pair of nodes representing either an action or
a preceding event, followed by a subsequent
action or event or terminal (“payoff”) node.
discrete random variable “A random variable that has a non-zero A probability mass function is used to represent the set of
probability for only a finite, or countably infinite, probabilities for all values of a discrete random variable.
set of values.”c
See also cumulative distribution
function, probability mass
function.
epistemic probability “A representation of uncertainty about Some examples of epistemic probability are (1) the assigning
propositions due to incomplete knowledge. Such of a probability to the proposition that a proposed law of
Synonym: epistemic uncertainty propositions may be about either past or future physics is true; (2) determination of the probability that
events.”c a terrorist will use a particular agent, based on evidence
See also aleatory probability, presented.
uncertainty.
DHS Lexicon: “arising from limited
state of knowledge”a
estimation (of parameters in “A procedure by which sample data are used to Estimation procedures are usually based on statistical analyses
assess the value of an unknown quantity.”f
probability models) that address their efficiency, effectiveness, limiting behaviors,
degrees of bias, etc. The most common methods of parameter
Also see approximation. estimation are maximum likelihood and the method of
moments. Bayesian methods tend to avoid producing estimates
and instead treat parameters as unknown quantities, with
associated probability distributions.
A subset of the sample space.f,g In a decision or
event Events are the basic building blocks of a probabilistic risk
event tree, a random variable whose values are assessment; they are the entities for which probabilities
See also random variable, event possible outcomes. are assessed and/or computed. Event descriptions must be
space. carefully and unambiguously articulated. The terminal event
“100 people die”—without making explicit the time frame
within which they die, their geographical distribution, their
demographics, etc.—is quite different from “100 people die”
within the first 48 hours of the attack, all of whom are within
5 km of the city center, 60% of whom are age 65 and older,
and so on. The important thing to consider here is that the
granularity of probability risk assessment events should be
only as fine as needed to capture the consequences of the
scenarios that include the events.
continued

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DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT
TABLE A.1 Continued
Term, with Synonyms, Cross-
References, and DHS Lexicona
Definitions Committee’s Recommended Definition Notes, Comments, and References
event space The set of all possible outcomes of an
experiment or of some (unknown) phenomenon.
(After Feller [1968]g and Statistical Education
Synonym: sample space
Through Problem-Solving [STEP] Consortium.f)
See also event.
event tree A tree formed of a sequence of random An event tree is essentially a decision tree with the decisions
variables, called events. The branching point at removed or replaced by nodes representing events that are
Synonyms: probability tree, chance which a new variable is introduced in the tree the result of probabilistic decisions (made either by the
tree is called a node. Each node is followed by the decision maker or some other agency). If a node in an event
possible random variable realizations, called tree represents a decision taken by an adversary, then the
See also tree, decision tree, fault- outcomes, and their probability distributions (conditional) probabilities of the resulting events must be
tree analysis. conditional on outcomes of previous random assessed or computed just as those for any other event nodes.
variables in the tree. The outcomes are Note that some computations (perhaps based on game-
DHS Lexicon: “a logic diagram represented as arcs leading from one event to the theoretic approaches) might produce event probabilities of 0
consisting of both decisions and next. The joint probability of the intersection of or 1, associated with “knowing” with certainty what action the
physical events in which the events that constitute a sequence (or scenario) adversary will take.
potential outcomes are represented is found by multiplication. A natural way to
by a finite, complete, discretized set construct an event tree is to place events in the There is no need to disallow infinite or continuous outcomes,
of outcomes (branches). The events chronological order in which they occur, if this as the DHS definition would imply.
order is known.l
are not necessarily consecutive
in time and are, in general, not
independent.”a
expected risk A summary measure of risk for an event, The committee strongly recommends that, wherever possible,
scenario, etc., as expressed by the expected the term “expected risk” be replaced by the specific measure
value of any one of the measurable
Synonym: expected consequences of consequences, such as “expected deaths,” “expected loss of
consequences associated with the risk. (Adapted income,” “expected illnesses.” If these measures are combined
from McCormick [1981]j with “damage”
Although “expected risk” is not in some functional way, for example via a utility function,
in the DHS Lexicon, DHS reports replaced by “consequence.”) then “expected risk” should be replaced by “expected utility.”
and presentations seem to imply One difficulty with defining “expected risk” is the historical
synonymy among the terms “risk” reality that the discipline of probabilistic risk assessment
(as related to a specific set of events arose from an understanding of the risks associated with
or scenarios), “expected risk,” and nuclear reactors, chemical plants, and such. In these situations,
“total risk.” expected risk is defined to be [expected frequency of
occurrence of an event] times [expected consequences of that
event].
expected value The first moment of the probability distribution The arithmetic average of random samples taken from the
of a random variable X; often denoted as E(X) distribution converges to the mean for all sufficiently large
and defined as ∑ xi p(xi) if X is a discrete random
Synonym: expectation sample sizes, under certain conditions.
variable and as ∫xƒ(x)dx if X is a continuous
random variable.g,e
See also mean. Ironically, in many cases the expected value of a random
variable is a numerical value that the random variable can
neer take on. For example, if a random variable X has P{X =
0} = .5 and P{X = 100} = .5, then E(X) = 50, even though X
can only take on values of 0 or 100. There is also a common
confusion between expected value and average, due to the
fact that, in the limit, as the sample size becomes very large,
the average of a set of observations of a random variable
will approach the mean of the random variable’s probability
distribution. (A curious linguistic note: in French the
expectation is called l’esperance, which in rough translation
means “hoped for.” Being simply the result of a mathematical
operation, it is neither hoped for nor truly “expected.”)

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APPENDIX A
TABLE A.1 Continued
Term, with Synonyms, Cross-
References, and DHS Lexicona
Definitions Committee’s Recommended Definition Notes, Comments, and References
fault-tree analysis “A technique by which events that interact to
produce other events can be related using simple
logical relationships permitting a methodical
building of a structure that represents the
system.”b
frequency “The fraction of events that satisfy some The two DHS definitions confound four different ideas:
prespecified criterion; a record of how often expected value, rate, fraction of past events that satisfy some
DHS Lexicon: “1. The number of each value (or set of values) of the variable in criterion, fraction of future events that satisfy some criterion.
question occurs.”f
events that would be expected to
occur in a time period.”a “2. A rate
(with units, #/time).”a
in-degree The number of arcs resulting in an event. In a
tree, the in-degree is one for all events, except
the initial event, which has an in-degree of 0.
initial event The first node in an event tree.
Synonym: initial node
initiating event An event with the potential to initiate a
sequence of other events leading to undesirable
Synonym: initial event consequences.
DHS Lexicon: “An action taken by
a terrorist organization to begin the
process that may culminate in an act
of terrorism.”a
The likelihood, L(A | D), of an event A, given
likelihood In informal usage, “likelihood” is often a qualitative
the data D and a specific model, is often taken description of probability or frequency. However, equally
to be proportional to P(D | A), the constant of
See also likelihood function, often these descriptions do not satisfy the axioms of
proportionality being arbitrary.m
probability, uncertainty. probability. For example, “likelihood” has been used by DHS
as a “weight” when informally assessing uncertainties, even
though the collection of these weights do not add to 1.
likelihood function A weighting function interpreted as a function of The maximum (with respect to the parameter value) of
parameters with the random variable(s) replaced the likelihood function often produces an estimator of the
by its (their) observed values.h,n
See also likelihood. parameter with desirable properties.
mean The first moment of a probability distribution, See note under expected value.
with the same mathematical definition as
See also expected value, expected value. The mean is a parameter
arithmetic average. that represents the central tendency of the
distribution. (After Glossary of Statistics Terms,e
STEP Consortium,f Ross [2000],o Devore
[2000].k)
measurement error “The unexplainable discrepancy between Measurement error is often decomposed into two components:
a measurement and the quality that the (1) random variation of measurements on objects of identical
measurement instrument is intended to quality; (2) a systematic error in measurement (e.g., a
measure.”p measurement device may be out of adjustment).
continued

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0 DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT
TABLE A.1 Continued
Term, with Synonyms, Cross-
References, and DHS Lexicona
Definitions Committee’s Recommended Definition Notes, Comments, and References
model A representation of some portion of the world Mathematical models are used to aid our understanding of
in a readily manipulable form. A mathematical some aspects of the real world and to aid in decision making.
See also simulation. model is an abstraction that uses mathematical They are also valuable rhetorical tools for presenting the
language to describe the behavior of system. rationale supporting various decisions, since they arguably
(Adapted from Wikipedia.q) allow for transparency and reproduction of results by others.
However, models are only as good as their (validated)
relationship to the real world and within the context for which
they are designed. It is wise to remember the advice of George
E.P. Box: “All models are wrong, but some may be useful.”
node A representation of an event or decision in a
decision tree. A representation of an event in an
See also event. event tree.
node-to-node branch An ordered pair of nodes; a course of action
leading from a preceding event to a subsequent
See also course of action. one.
normal distribution A symmetric “bell-shaped” probability density The normal distribution commonly used, since (1) it is (with
function, (1/(σ√2π))(-exp((x - μ)2/(2σ2))), certain conditions) the limiting distribution of the sum of
Synonym: Gaussian distribution completely characterized by two parameters: random variables, (2) it has a certain degree of mathematical
mean μ and standard deviation σ.g,k,o tractability, (3) there exist many well-known methods for
estimating its parameters, and (4) it represents a reasonable fit
to data obtained for a wide variety of situations.
normalized risk The proportion of the total expected risk
contributed by a particular agent.
See also conditional relative risk,
relative risk.
out-degree The number of directed arcs leaving a node.
path A sequence of arcs.
Synonym: scenario
Poisson distribution A commonly used probability mass function The Poisson distribution is often used to reflect “randomness”
associated with a random variable X = number of events over time—P{time between consecutive occurring
events ≥ t} = e–µt, which does not depend on the time of any
of events that occur in a given period of time.
The formula is P{X = x} = μxe-μ / x!, for previous event.
x = 0, 1, . . ., where the parameter μ = E(X) is
the mean of the distribution.k
Consider two statements assessing “W = Bill Gates’s
precision The implied degree of certainty with which a
value is stated, as reflected in the number of net worth”: A precise but inaccurate assessment is “W is
See also accuracy. significant digits used to express the value—the $123,472.89”; an imprecise but accurate assessment is: “W is
more digits, the more precision. (Adapted from more than $8 billion.”
SRA.b)
probabilistic risk assessment An analytical tool that (1) identifies and
delineates logical combinations of basic (not
Synonym: risk assessment analyzed further) events that, if they occur,
could lead to an accident (or other undesired
event, called the top event); (2) assesses or
approximates the probability of the top event
from the probabilities of logical combinations
of basic events; and (3) assesses the probable
consequences associated with occurrence of the
top event.

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APPENDIX A
TABLE A.1 Continued
Term, with Synonyms, Cross-
References, and DHS Lexicona
Definitions Committee’s Recommended Definition Notes, Comments, and References
probability One of a set of numerical values between 0 and The definition holds for all quantification of uncertainty:
1 assigned to a collection of random events subjective or frequentist.
See also likelihood, conditional (which are subsets of a sample space) in such a
probability. way that the assigned numbers obey two axioms:
1. 0 ≤ P{A} ≤ 1 for any A, and
DHS Lexicon: “1. A probability
2. P{A} + P{B} = P{A ∪ B} for two mutually
assignment is a numerical encoding
exclusive events A and B.o
of the relative state of knowledge
(Society for Risk Analysis).
2. The subjectivist viewpoint of
probability: the analyst’s state of
knowledge or degree of belief.”a
probability density function (PDF) The derivative of an absolutely continuous The PDF is the common way to represent the probability
cumulative distribution function.p distribution of a continuous random variable, because its shape
often displays the central tendency (mean) and variability
(standard deviation). From its definition, P{a ≤ X ≤ b} is the
For a scalar random variable X, a function f such
that, for any two numbers, a and b, with a ≤ b, integral of the PDF between a and b.
P{a ≤ X ≤ b} = ∫ ab f(x)dx.
probability distribution See cumulative distribution function.
probability elicitation “A process of gathering, structuring, and coding There are many approaches for probability elicitation,
expert judgment (about uncertain events or the most common of which are those used for obtaining a
quantities).”r
Synonym: probability assessment priori subjective probabilities. However, in some sense all
probabilities, even those that result from statistical analysis of
large data sets, are subjective and therefore require elicitation.
This is because the conditions under which the data have
been collected, and the relevance of these conditions to
future events for which probabilities are desired, are a matter
of expert and subjective judgment. Note that the results of
probability elicitations are sometimes called probability
“assessments” or “assignments.”
probability mass function (PMF) A function that gives the probability that a
discrete random variable takes on a particular
value.k,o
See also discrete random variable.
random error [None] This term is meaningful only in the context of analyzing the
results of a particular experiment and therefore should not be
See also measurement error. used.
random variable “A real valued function defined on a sample (or The random variables of interest to a PRA are those that
event) space.”g describe the consequences of a particular event. For example,
See also event, probability suppose that the event space consists of only three events:
A = “100 deaths, 500 illnesses”; B = “0 deaths, 0 illnesses”;
distribution, continuous random
C = “75 deaths, 375 illnesses”; and their respective probabilities
variable, discrete random
are P{A} = .3, P{B} = .2, P{C} = .5. Then, if the random
variable.
variables are defined to be X = “the number of deaths
associated with the event space,” and Y = “the number of
illnesses associated with the event space,” this implies P{X =
100} = P{A} = .3; P{X = 12} = 0 (there are no events with
X = 0); P{Y = 375} = .5; P{Y/X = .5} = P{A} + P{C} = .8, etc.
A probability distribution, constructed on the range of the
random variable, can then be used to assign probabilities to
events in the event space.
continued

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2 DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT
TABLE A.1 Continued
Term, with Synonyms, Cross-
References, and DHS Lexicona
Definitions Committee’s Recommended Definition Notes, Comments, and References
relative risk (in an epidemiological See health terms in Part A.1.B of this table.
context
Synonyms: risk ratio; odds ratio
reload capacity A measure of the ability to introduce a pathogen (Formulated by former Navy Secretary Danzig, according to
into more than one country and/or on more than Marc Lipsitch of the Harvard School of Public Health.)
one occasion.
risk It is important to distinguish between the term risk, which
“The potential for unwanted, adverse
consequences.”b involves uncertainties, consequences and conditioning
See also expected risk. statements, and expected risk, which combines these factors
using the linear additive expectation operation. It is essential
DHS Lexicon: “when used in a to be absolutely clear when using these two these terms.
Unfortunately, even SRA’s Glossaryb intermixes them, since
general sense: The potential for
realization of unwanted, adverse after giving the definition in Column 2, it goes on to say,
consequences to human life, health, “estimation of risk is usually based on the expected alue of
property or the environment” the conditional probability of the event occurring times the
consequence of the event given that it has occurred”b—which
[American Heritage Dictionary];
“(‘technical meaning’): The set of is technically incorrect as well as misleading.
triplets of frequency, scenario and
consequences, for all scenarios

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APPENDIX A
TABLE A.1 Continued
Term, with Synonyms, Cross-
References, and DHS Lexicona
Definitions Committee’s Recommended Definition Notes, Comments, and References
risk communication The process used by risk analysts, decision
makers, policy makers, and intelligent
adversaries to provide data, information, and
knowledge to change the risk perceptions of
individuals and organizations and enable them
to assess the risk more accurately than they
otherwise might.
risk curve A graph describing frequency of events as
a function of consequences. Alternatively,
a curve describing frequency of events with
consequences greater than or equal to some level
as a function of that level.
risk estimation “The determination of the characteristics of risks Although SRA provides a definition, the committee sees no
such as the magnitude, spatial scale, duration, need to include this term in a formal lexicon, since the term
See also risk analysis. and intensity of adverse consequences and their “risk” by itself has many connotations and in any event is a
associated probabilities of the cause-and-effect random variable which, by definition, cannot be “estimated.”
links.”b
DHS Lexicon: “The scientific There are also many overlaps with “risk assessment.”
determination of the characteristics
of risks, usually in as quantitative
a way as possible. These include
the magnitude, spatial scale,
duration and intensity of associated
probabilities as well as adverse
consequences and their description
of the cause and effect links. (from
SRA)”a
risk management The process of constructing, evaluating, Taken from the definition in the committee’s interim report:
implementing, monitoring, and revising “In the case of an individual, private sector or public sector
See also risk analysis. strategies for reducing (or distributing) losses organization, these strategies enable them to transfer, mitigate,
from future hazards and dealing with the or accept their perceived risks. Risk management strategies
DHS Lexicon: “The process recovery process should a hazard occur. Risk can be evaluated by undertaking cost-benefit analyses to
of constructing and evaluating management strategies include a combination determine the tradeoff between the reduction of risk and
strategies for reducing losses from of options such as providing information (i.e., the costs of undertaking such measures. In evaluating a risk
future hazards and dealing with the risk communication), economic incentives (e.g., management strategy one needs to be concerned with the way
recovery process should a disaster subsidies, fines), insurance, compensation, resources are allocated (i.e. efficiency considerations) as well
occur.”a regulations, and standards. (Adapted and as the impact of these measures on different stakeholders (i.e.
expanded from SRA.b) distribution or equity considerations).”t
risk perception Beliefs, attitudes, judgments, and perceptions
held by individuals, communities, societies,
See also risk analysis. groups, or organizations about the risks of
a hazard. Risk perception is concerned with
DHS Lexicon: “Beliefs held the psychological and emotional factors. Risk
by individuals or organizations perceptions can be influenced by personal
about the risks of a hazard. Risk knowledge, experience, and beliefs; they can be
perception is concerned with affected by changing perceptions of the threat,
the psychological and emotional the vulnerabilities, and/or the consequences;
factors, which have been shown they may be influenced by information about
to have an enormous impact on hazards, risk assessments, risk policies, and risk
behavior.”a management decisions. (Adapted and expanded
from SRA.b)
continued

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DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT
TABLE A.1 Continued
Term, with Synonyms, Cross-
References, and DHS Lexicona
Definitions Committee’s Recommended Definition Notes, Comments, and References
scenario A complete enumeration of one path on a tree,
from the initial event to the terminal node (if
DHS Lexicon: “One of a possible any).
combination of approaches leading
to the execution of an act of
terrorism. An end of an event tree.”a
simulation “A model constructed so that the input of a large By its inherent nature, each set of “runs” of a simulation
number of random variables drawn from defined represents the outcomes of a series of experiments. Analysis
Synonym: Monte Carlo simulation probability distributions will generate outputs of simulation output data therefore requires a proper
that are representative of the random behavior of experimental design, followed by the use of statistical
See also model. a particular system, phenomenon, consequences, techniques to estimate parameters, test hypotheses, etc.
etc., of a series of events.”u
split fraction [None] Presumably this term has been used by DHS to be
synonymous with “conditional probability.” However, the
See also conditional probability. DHS definition is not consistent with the DHS definition of
“frequency,” and “relative frequency” is not defined by DHS.
DHS Lexicon: “For an event, the
relative frequency of a branch.”a
standard deviation “The square root of the variance of a
distribution.”o
See also variance.
terminal event An event in an event tree or a decision tree with
out-degree 0.
Synonym: terminal node
total expected risk The probability-weighted sum of expected risks It is preferable that “total risk” should depend on the specific
associated with all agents. (Implied by DHS context: the consequence (deaths, utility, etc.) and the events
Synonym: total risk usage). over which the sum is taken (e.g., agents, other conditioning
events, etc.).
For example, if pi = P{conditioning event i}, and Ci is the
expected consequence associated with event i, then total
expected risk is R = Σ piCi.
tree A connected acyclic directed graph with exactly
one distinguished (root) node with in-degree 0,
See also event tree, decision tree. and every other node with in-degree 1.
uncertainty The condition of being unsure about something; The formal definition of “uncertainty” is really not important
a lack of assurance or conviction.d to the understanding of any PRA method. However, having
See also probability. a clear and agreed on definition of the uses to which any
quantification of “uncertainty” is put is crucial. The DHS
DHS Lexicon: “Two types of Lexicon definitions are neither clear nor agreed on, and in
uncertainty are considered and fact they confuse the notion of “uncertainty” with the various
treated differently: aleatory methods used to quantify it in a useful way.
uncertainty—arising from
variability (e.g., weather
variability); epistemic uncertainty—
arising from limited state of
knowledge.”a

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APPENDIX A
TABLE A.1 Continued
Term, with Synonyms, Cross-
References, and DHS Lexicona
Definitions Committee’s Recommended Definition Notes, Comments, and References
uncertainty range [None] Depending on the context, DHS apparently means either
(1) a range of probabilities associated with a particular event,
DHS Lexicon: “Typically, a scenario, etc.—possibly due to disagreements among subject-
confidence interval. For the matter experts, outputs of a simulation or analytical model,
common definition of risk given or results of an experiment, etc.; or (2) the range of uncertain
above [presumably ‘expected outcomes associated with a particular event, scenario, etc.
consequences per unit time or
within a time interval,’ but not
shown in this table since the
committee does not display ‘lay
definitions’], ‘the confidence
interval associated with the
epistemic uncertainty’.”a
utility A real valued function of a consequence. In economics, “utility” captures “relative happiness” or
satisfaction gained by goods and services.
Synonym: utility function
In decision analysis, “utility” captures returns to scale and risk
DHS Lexicon: “function preference.
that transforms measures of
consequences into a number.”a In both cases, the assessment of utility values (and hence
utility functions) for consequences is an inherently subjective
exercise and so depends on the individual (or organization)
confronting the possible consequences.
Formally, let A be the most-preferred possible outcome of a
risky prospect, B be the least-preferred, and C be any other
outcome. If a decision maker is indifferent between C and
a prospect having probability u of getting A and probability
(1 - u) of getting B, then u is defined as the (von Neumann-
Morgenstern) utility of C.
variance The second moment of a probability distribution, The variance is a common measure of variability around the
defined as E(X - μ)2, where μ is the first moment mean of a distribution. Its square root, the standard deviation,
See also standard deviation. of the random variable X. having dimensional units of the random variable, is a more
intuitively meaningful measure of dispersion from the mean.
The logarithm of K = P{x | A} / P {x | B}, where
weight of evidence A nonstandard and nonstatistical definition, used by some
x is a realization of a random variable, and A and analysts but not recommended, is as follows: “An elicitation
See also probability risk B are alternative hypotheses. (K is also called the of uncertainty that results in a non-normalized set of numbers
likelihood ratio.)q
assessment. which can be normalized (by dividing by the sum over all
possible events) to produce probabilities.”
In some statistical usage, the “weight of evidence” is defined
to be 10 times the log-likelihood ratio.
More generally, a loosely defined or undefined term indicating
the extent to which studies are judged to support a conclusion.
continued

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DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT
TABLE A.1 Continued
PART A.1.B TERMS USED IN SUSCEPTIBLE, EXPOSED, INFECTED, AND RECOVERED (SEIR) MODELING AND APPLICABLE ONLY TO
BIOLOGICAL RISK ANALYSIS
Term, with Synonyms, Cross-
References, and DHS Lexicona
Definitions Committee’s Recommended Definition Notes, Comments, and References
Bioshield A federal program authorized in 2004 to For more information see http://www.whitehouse.gov/infocus/
improve medical countermeasures protecting bioshield/.
Americans against a chemical, biological,
radiological, or nuclear (CBRN) attack.
contagious A person who is infected and capable of This can be used as an adjective or noun, but most often, in
transmitting an infectious agent to another host. the modeling context, as a noun.
(Adapted from Thomas and Weber [2001].)
DHS Lexicon: “infected and
capable of spreading disease.”a
dose The amount (or concentration) of desired
matter or energy deposited at the site of effect.
(Adapted from SRA.b)
exposed A person or population that came in contact with For SEIR modeling, but generally not other usage, “exposed”
the infectious agent or toxin. includes only those who received an infectious dose. This
See also infected. can be used as an adjective or noun, but most often, in the
modeling context, as a noun.
DHS Lexicon: “population who
came in contact with the infectious
agent or toxin and received an
infectious dose.”a
ill Infected or intoxicated population showing This can be used as an adjective or noun, but most often, in
clinical signs and symptoms. the modeling context, as a noun.
DHS Lexicon: “infected or
intoxicated population showing
symptoms.”a
infected An individual or population that has an This can be used as an adjective or noun, but most often, in
infectious agent enter and multiply in its tissues. the modeling context, as a noun.
DHS Lexicon: “population that
has been exposed and received an
infectious dose.”a
Typically X = 50, but it is sometimes set to 10, 90, or other
infectious dose X (IDX) A dose that is expected to lead to the infection of
X percent of individuals exposed. values, depending on the intent of the analysis.
intoxicated Population that has been exposed to a threshold
amount of toxin and will become ill in the
DHS Lexicon: “population that has absence of intervention.
been exposed and received a toxic
dose of a toxin.”a
lethal concentration X (LCX) A concentration that is calculated to kill X
percent of a population. (Adapted from SRA.b)
Typically X = 50, but it is sometimes set to 10, 90, or other
lethal dose X (LDX) A dose that is expected to kill X percent
of a population in the absence of medical values, depending on the intent of the analysis.
intervention(s).b
R0 The mean number of secondary cases of R0 is a property of the pathogen. R0 is a theoretical number
infection to which one primary case gives rise and does not hold if the population is not entirely susceptible,
Synonym: basic reproduction throughout its infectious period, if introduced or even in the case where there is more than 1 contagious
number into a population consisting solely of susceptible person (since the entire population is not susceptible).
individuals. (Adapted from Anderson and May
[1991].w)
See also R.

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APPENDIX A
TABLE A.1 Continued
Term, with Synonyms, Cross-
References, and DHS Lexicona
Definitions Committee’s Recommended Definition Notes, Comments, and References
R The number of secondary cases of infection R is a property of both the pathogen and the population’s
to which a single contagious case gives rise relative susceptibility. Under conditions of stable endemic
infection, R = 1. Note that the R value can and does
Synonym: effective reproduction throughout its infectious period, in a host
number population where not all persons are susceptible. change as the outbreak progresses. The change in R may
be due to reduction in the susceptible population, through
See also R0. natural infections, changes in social behavior, or medical
interventions.
relative risk (Biomedical context) The ratio of the risk of
disease or death among the exposed to the risk
Synonyms: risk ratio; odds ratio among the unexposed.
removed Population that has recovered, has been “Removed” may also include vaccinated individuals in some
successfully immunized, or has died. models.
DHS Lexicon: “population that has
recovered or died.”a
susceptible Individual or population who, if exposed to an This can be used as an adjective or noun, but most often, in
infectious agent, could become infected. the modeling context, as a noun.
DHS Lexicon: “population who
[sic] is at risk of becoming infected
if exposed to an infectious agent.”a
aDepartment of Homeland Security. 2007. “A Lexicon of Risk Terminology and Methodological Description of DHS Bioterrorism Risk Assessment.”
April 14.
bSociety for Risk Analysis (SRA), Glossary of Risk Analysis Terms. Available at sra.org/resources_glossary.php. Accessed Feb. 22, 2008.
cCornell LCS Statistics Laboratory. See http://instruct1.cit.cornell.edu:8000/courses/statslab/Stuff/index.php. Accessed Feb. 22, 2008.
dAmerican Heritage Dictionary. 2000. Boston: Houghton, Mifflin.
eGlossary of Statistics Terms. Available at www.stat.berkeley.edu/users/stark/SticiGui/Text/gloss.htm. Accessed Feb. 22, 2008.
fStatistical Education Through Problem Solving [STEP] Consortium. Available at www.stats.gla.ac.uk/steps/index.html. Accessed Feb. 22, 2008.
gW. Feller. 1968. An Introduction to Probability Theory and Its Applications. New York, N.Y.: Wiley.
hB. DeFinetti. 1974. Theory of Probability. Hoboken, N.J.: Wiley.
iD.V. Lindley. 1965. Introduction to Probability and Statistics from a Bayesian Viewpoint; Part : Probability. Cambridge. U.K.: Cambridge University
Press.
jN.J. McCormick. 1981. Reliability and Risk Analysis. San Diego, Calif.: Academic Press.
kJ.L. Devore. 2000. Probability and Statistics for Engineering and the Sciences. Pacific Grove, Calif.: Duxbury Press.
lE. Paté-Cornell. 1983. “Fault Trees vs. Event Trees in Reliability Analysis.” Risk Analysis 4(3):177-186.
mA.F.W. Edwards. 1992. Likelihood. Baltimore, Md.: Johns Hopkins University Press.
nThe White House. 2004. Available at www.whitehouse.gov/infocus/bioshield. Accessed Feb. 22, 2008.
oS.M. Ross. 2000. Introduction to Probability Models. New York, N.Y.: Academic Press.
pDuke University. 1998. Statistical and Data Analysis for Biological Sciences. Available at isds.duke.edu/courses/Fall98/sta210b/terms.html. Accessed
Feb. 22, 2008.
qWikipedia: “Statistics.” Available at en.wikipedia.org/wiki/Statisics. Accessed Feb. 22, 2008.
rM.S. Meyer and J.M. Booker. 1991. Eliciting and Analyzing Expert Judgment. Los Alamos, N.M.: Los Alamos National Laboratory.
sDHS (Department of Homeland Security). 2006. Bioterrorism Risk Assessment. Biological Threat Characterization Center of the National Biodefense
Analysis and Countermeasures Center. Fort Detrick, Md.
tNational Research Council. 2006. Interim Report on Methodological Improements to the Department of homeland Security’s Biological Agent Risk
Analysis. Washington, D.C.: The National Academies Press.
uE.J. Henley and H. Kumamoto. 1981. Reliability Engineering and Risk Assessment. Upper Saddle River, N.J.: Prentice-Hall.
J.C. Thomas and D.J. Weber. 2001. Epidemiologic Methods for the Study of Infectious Diseases. Oxford, U.K.: Oxford University Press.
wR.M. Anderson and R.M. May. 1991. Infectious Diseases of Humans. Oxford, U.K.: Oxford University Press.