Alan R. Washburn, Ph.D.
Distinguished Professor Emeritus of Operations Research
Naval Postgraduate School, Monterey, California
July 10, 2007
MEMORANDUM FOR THE NATIONAL ACADEMY OF SCIENCES (NAS)
Review of the Department of Homeland Security (2006) work on bioterrorism.
Background. The Department of Homeland Security (DHS) has produced a 2006 bioterrorism study, and is working on subsequent versions. DHS has asked NAS to assess the 2006 work, which I will refer to hereafter as “the 2006 work.” I have become acquainted with the work through contacts with the NAS committee, and have been invited to provide a review. This is the review. It is intended for a scientific audience, so I will not hesitate to use the language of probability in describing what I think was done in 2006, or in how things might be handled differently in the future. Random variables are uppercase symbols, P() and E() are the probability and expected value functions, respectively.
My Qualifications. After working five years for the Boeing Company, I joined the Operations Research faculty at the Naval Postgraduate School in 1970, where I did the usual academic things until retiring in 2006. My teaching includes probability and decision theory, which are relevant here. See my resume at http://www.nps.navy.mil/orfacpag/resumePages/washbu.htm for details. I have no biological or medical qualifications. My acquaintance with the work is mainly through the references listed at the end of this review.
Event Trees. The fundamental idea behind the 2006 work is an event tree. As I will use the term in this review, an event tree is a branching structure whose root corresponds to the assertion that some event has occurred, the event in this case being what I will call an “incident.” The tree branches repeatedly until a “scenario” is encountered, at which point one will find a probability distribution that determines the consequence of the incident, a random variable that I will call Y. I think of consequences as being “lives lost,” but any other scalar measure would do. Each node of the tree has a set of successor arcs, and there is a given probability distribution over these arcs. One can imagine starting at the root and randomly selecting an arc at each node encountered until finally the consequence is determined. In addition to Y, the event tree involved in the 2006 work is such that every path from root to consequence also defines two other random variables:
A, the biological agent, one of 28 possibilities, and
S, the scenario.
The scenario might be null in the sense that Y is 0 because the incident is terminated prematurely, but is nonetheless always defined.
DHS determines the consequence distributions through Monte Carlo simulation based on expert input. The results are collected into decade-width histograms. I will not comment further on the methodology for producing the consequence distributions, since I have not examined it in detail.
DHS has modified the above definition of an event tree in three senses. One is that the initial branches from the root are rates, rather than probabilities. Call the rate on branch i λ_{i}, and let the sum of all of these rates be λ. If one interprets these rates as independent Poisson rates of the various kinds of incident, then it is equivalent to think of incidents as occurring in a Poisson process with rate λ, with each incident being of type i with probability λ_{i}/λ. These ratios can be the first set of branch probabilities, so this is all equivalent to the standard event tree definition, except that we must remember that incidents occur at the given rate λ. This first modification is thus of little import.
The second modification is that an incident might involve multiple attacks, each with separate consequences. This is a more significant modification, and will be discussed separately below.
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Appendix I
Review of BTRA Modeling
Alan R. Washburn, Ph.D.
Distinguished Professor Emeritus of Operations Research
Naval Postgraduate School, Monterey, California
July 10, 2007 consequence of the incident, a random variable that I will
call Y. I think of consequences as being “lives lost,” but
MEMORANDUM FOR THE NATIONAL ACADEMY OF any other scalar measure would do. Each node of the tree
SCIENCES (NAS) has a set of successor arcs, and there is a given probability
distribution over these arcs. One can imagine starting at the
Review of the Department of Homeland Security (2006) root and randomly selecting an arc at each node encountered
work on bioterrorism. until finally the consequence is determined. In addition to Y,
the event tree involved in the 2006 work is such that every
Background. The Department of Homeland Security (DHS) path from root to consequence also defines two other random
has produced a 2006 bioterrorism study, and is working on variables:
subsequent versions. DHS has asked NAS to assess the 2006
work, which I will refer to hereafter as “the 2006 work.” I • A, the biological agent, one of 28 possibilities, and
have become acquainted with the work through contacts • S, the scenario.
with the NAS committee, and have been invited to provide a
review. This is the review. It is intended for a scientific audi- The scenario might be null in the sense that Y is 0 because
ence, so I will not hesitate to use the language of probability the incident is terminated prematurely, but is nonetheless
in describing what I think was done in 2006, or in how things always defined.
might be handled differently in the future. Random variables DHS determines the consequence distributions through
are uppercase symbols, P() and E() are the probability and Monte Carlo simulation based on expert input. The results
expected value functions, respectively. are collected into decade-width histograms. I will not com-
ment further on the methodology for producing the conse-
My Qualifications. After working five years for the Boeing quence distributions, since I have not examined it in detail.
Company, I joined the Operations Research faculty at the DHS has modified the above definition of an event tree
Naval Postgraduate School in 1970, where I did the usual aca- in three senses. One is that the initial branches from the root
demic things until retiring in 2006. My teaching includes prob- are rates, rather than probabilities. Call the rate on branch i
λi, and let the sum of all of these rates be λ. If one interprets
ability and decision theory, which are relevant here. See my
resume at http://www.nps.navy.mil/orfacpag/resumePages these rates as independent Poisson rates of the various kinds
/washbu.htm for details. I have no biological or medical quali- of incident, then it is equivalent to think of incidents as oc-
curring in a Poisson process with rate λ, with each incident
fications. My acquaintance with the work is mainly through
being of type i with probability λi/λ. These ratios can be the
the references listed at the end of this review.
first set of branch probabilities, so this is all equivalent to the
Event Trees. The fundamental idea behind the 2006 work is standard event tree definition, except that we must remember
that incidents occur at the given rate λ. This first modification
an event tree. As I will use the term in this review, an event
tree is a branching structure whose root corresponds to the is thus of little import.
assertion that some event has occurred, the event in this The second modification is that an incident might involve
case being what I will call an “incident.” The tree branches multiple attacks, each with separate consequences. This is
repeatedly until a “scenario” is encountered, at which point a more significant modification, and will be discussed sepa-
one will find a probability distribution that determines the rately below.
22
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2
APPENDIX I
The third and most significant modification is that the However, summing to 1 is not sufficient for the SME
marginals to be meaningful. This is most obvious when N =
branching probabilities (DHS on occasion also calls them
“branch fractions”) are not fixed, but are instead themselves 2. If the first branch has probability A, then the second must
have probability 1 - A, and therefore the second probability
determined by sampling from beta distributions provided
indirectly by Subject Matter Experts (SMEs). Let θ be the distribution has no choice but to be the mirror image of the
first. If the experts feel that the first marginal has α = 1 and
collection of branching probabilities. In each incident we
therefore observe (θ, A, S, Y), with θ determining the event b = 1, while the second has α = 2 and b = 2, then we must
tree for the other three random variables. This modification explain to the experts that what they are saying is meaning-
will also be discussed separately below. less, even though both marginals have a mean of 0.5. The
second marginal has no choice but to be the mirror image
The Second Modification: Repeated Attacks per Incident. of the first, and must therefore be the first, by symmetry.
The vision is that a cell or group of terrorists will not plan Any other possibility is literally meaningless, since there is
a single attack, but will plan to continue to attack until no pair of random variables (A1, A2) such that Ai has the ith
marginal distribution and also A1 + A2 is always exactly 1.
interrupted, with the entire group of attacks constituting
I think DHS recognizes the difficulty when N = 2, and has
an incident. The effect of this is to change the distribution
of consequences of an incident, since a successful attack basically fixed it in that case by asking the SMEs for only
one marginal, but the same difficulty is present for N > 2,
will be accompanied by afterattacks, the number of which
I will call X. I believe that the formula used for calculating and has not been fixed. The sampling procedure offered on
E(X) is incorrect. Specifically, let λ′ be the probability that page C-81 of Department of Homeland Security (2006) will
any one of the afterattacks will succeed, assume that after- reliably produce probabilities A1, …, AN that sum to 1, and
attacks continue until one of them fails, and assume that the which are correct on the average, but they do not have the
failed afterattack terminates the process and itself has no marginal beta distributions given by the SMEs. This is most
consequences. Then the average value of X is E(X) = λ′/(1 - obvious in the case of the last branch, since the Nth marginal
λ′), the mean of a geometric-type random variable. This is is never used in the sampling process, but I believe that the
not the formula in use. Using the correct formula would be marginal distribution is correct only for the first branch.
a simple enough change, but I believe the numerical effect There is a multivariable distribution (the Dirichlet distri-
might be significant. bution) whose marginals are all beta distributions, but the
Dirichlet distribution has only N + 1 parameters. The SME
Other changes may also be necessary to implement the
original vision. If the afterattacks all have independent con- marginals require 2N, in total, so the Dirichlet distribution is
sequences, then the distribution of total consequences is the not a satisfactory joint distribution for A1, …, AN.
(1 + X)-fold convolution of the consequence distribution, a
Estimation of the Spread in Agent-Damage Charts. I have
complicated operation that I see no evidence of. The docu-
mentation is mute on what is actually assumed about the defined Y to be the consequence and A to be the agent. Define
Ya to be the consequence if A = a, or otherwise 0, so that the
independence of after attacks, and on how the E(X) computa-
tion is actually used. Simply scaling up the consequences of 28 random variables Ya sum to Y. Most of the DHS output
one attack by the factor (1 + E(X)) is correct on the average, deals with the random variable E(Ya | θ), the expected conse-
regardless of independence assumptions, but will not give quence contribution from agent a, given the sampled branch
probabilities θ. This quantity is random only because of its
the correct distribution of total consequences.
dependence on θ, the natural variability of Ya having been
averaged out. A sample E(Ya | θj), j = 1,…, 500 is produced
The Third Modification: “Random Probabilities.” DHS
has accommodated SME uncertainty by allowing the branch by Latin Hypercube Sampling (LHS) of the branch prob-
probabilities themselves to be random quantities, with the abilities, each sample including the standard average risk
ˆ
SMEs merely agreeing to a distribution for each probability, computations for the event tree. A sample mean estimate Ya of
500
rather than a specific number. I will refer to each of these
∑
ˆ
E(Y ) is then made by Y = (1 / 500 ) E (Y | θ ) . The agents
a a a j
probability distributions as a “marginal” for its branch. If a j =1
are then sorted in order of decreasing sample mean, and
node has N branches, the experts contribute N marginals, one
displayed in what I will call “agent-damage” charts showing
for each branch. Except at the root, these marginals are all
the expected values and spreads as a function of agent. The
beta distributions on the interval [0 1], and each therefore
sample means are normalized before being displayed, prob-
has two parameters, alpha (α) and beta (b). Each of these
ably by forcing them to sum to 1. The normalization destroys
distributions has a mean, and since the probabilities them-
information that is relevant to the decisions being made. I do
selves must sum over the branches to 1, the same thing must
not know the motivation for doing so.
logically be true of the means. The same need not be true
The spreads display the epistemic variability due to SME
of the SME inputs, but DHS seems to have disciplined the
uncertainty about θ, but suppress all of the aleatoric vari-
elicitation process so that the SME marginal means actually
ability implied by the event tree. If there were no uncertainty
do sum to 1. That is true in all of the data that I have seen.
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2 DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT
about θ, all of the spreads would collapse to a single point long as i and j are not the same, which is all that is required
(the mean) for each agent. I am not sure how the variability for my conclusion to be true. While it is certainly true that
displayed in agent-damage charts is supposed to relate to the branches chosen at nodes i and j are in general dependent,
decision making, but I guess that the graphs are intended to the branch probabilities are not.)
support conclusions such as the following: “I know that the
Use of SMEs. It is inevitable in a project like this that
mean damage for agent 1 is larger then the mean damage for
agent 2, but I still think that we ought to spend our money probabilities will have to be obtained from Subject Matter
defending against agent 2 because of its high associated vari- Experts, rather than experimentation. The important thing is
ability. Even a small prospect of the high damages associated that the SMEs at least know what they are estimating, and
with agent 2 is not acceptable.” If that is the kind of logic that that estimates be used correctly once they are obtained. I
the agent-damage charts are intended to support, then they have already mentioned that SME estimates of the marginal
should include aleatoric variability. Without it, the spreads branch distributions are not reproduced by the sampling pro-
associated with each agent are too small. This issue affects cedure. Another concern is at the third stage of the event tree,
infectious agents more than the other kind, since infectious where SMEs are asked to deal with agent selection. At that
stage there are 4 × 8 = 32 nodes in the event tree where an
diseases will have especially high damage variances.
The agent-damage charts are intended for a high level of agent might be selected, each of which has 28 branches. I can
decision-making audience, and devote considerable space certainly understand DHS’s reluctance to conduct 896 inter-
(one of the two available dimensions) to showing the spread views with SMEs, each to determine one of the needed beta
associated with each agent. Without the need to show spread, distributions. Some kind of a shortcut is needed, but I wonder
they could be replaced by bar charts or simple tables. If whether the one adopted is a good one. The SMEs are first
spread is important enough to be displayed, then it ought to asked to determine an “input regarding known preferences of
be displayed in a manner that facilitates good decisions. I terrorists” for each agent. If I were an SME and somebody
doubt that that is currently the case. asked me to determine the quoted expression for agent a, I
would announce my estimate of P(A = a), the probability
Even without the aleatoric issue, I still have concerns
about the spread that is displayed. The object ought to be that agent a is actually selected in an incident. Given all of
to display the mean and fractiles (the spread) of the random these SME inputs, DHS then goes over the 896 branches,
variable E(Ya | θ) for each value of a. The mean of E(Ya | θ) some of which have a logical 0 for the agent, and assigns
is simply E(Ya) by the conditional expectation theorem, and probabilities using the rule that the probability is either 0 or
ˆ
is estimated by Ya. DHS claims graphically that the LHS else proportional to the SME’s agent input, the proportional-
sample fractiles are also the fractiles of the random variable ity constant being selected in each of the 32 cases so that the
E(Ya | θ). I suspect that this claim is false. LHS is basically probabilities sum to 1. My objections are that
a variance reduction technique that makes the variance of
ˆ
Ya smaller than it would be with ordinary sampling. While • The quoted expression above does not make it clear that
the SME input is supposed to be P(A = a). There is a
this effect is welcome, LHS also has an unpredictable effect
on variability. The spread that is shown for each agent may danger of every SME making a different interpretation
not be a good estimate of the spread of the random variable of what is being asked for.
E(Ya | θ). • If the SME does input the probabilities P(A = a), and if
One final point on estimation. As long as there is no de- DHS applies the shortcut procedure to fill out the third
pendence between the branch probabilities at different nodes, stage of the event tree, and if the probabilities of the 28
as there is not in the 2006 work, it is characteristic of an event agents are then computed from the tree, they will not
tree that P(Ya ≤ y) = E(P(Ya ≤ y | θ)) = P(Ya ≤ y | E(θ)). The first necessarily agree with the SME’s inputs. This would
equality is due to the conditional expectation theorem, and be true even without my next objection.
the second is because no event tree probability enters more • The SME’s inputs are subsequently modified by vari-
than once into calculating the probability of any scenario. ous formulas involving agent lethality, etc. What is an
In other words, all information pertinent to the distribution SME who is already acquainted with agent lethality
of Ya could be obtained without sampling error by simply to think of this? Should he adjust his input so that the
replacing the marginal branch distributions by their means. net result of all this computation is the number that he
This information includes E(Ya), which is currently being wanted in the first place? If one is going to elicit SME
ˆ
estimated (with sampling error) by Ya. (Note added in June inputs on probabilities, then it seems to me that one
2007. Let me expand the notation to clarify this final point, ought to use them as they are intended.
since it has caused some confusion. Let θ = (Q1, …, Qn),
where n is the number of nodes and Qi is the collection of Given that the agent probabilities strongly influence the
branch probabilities at node i. Also let Qij be the jth branch agent-damage charts, the procedure for eliciting and using
probability at node i. In the sampling procedure used by DHS them should be an object of concern in future work.
to obtain θ, Qij and Qkl are independent random variables as
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2
APPENDIX I
Tree Flipping? The process described earlier for generat- software is sometimes used diagnostically, which might be
ing agent-damage charts may not be a correct statement of of use in bioterrorism. One might observe that the agent is
what DHS actually did in 2006. The DHS documentation in known to be anthrax, for example, and instantly recompute
several places, after describing a single event tree with 17 the target probabilities based on that known condition.
ranks, states that a separate analysis was actually done for Another suggestion is to examine the potential for op-
each agent (paragraph C.3.4.2 of Department of Homeland timization. Given that the basic problem is how to spend
Security [2006], for example). Now, it is possible to end money to reduce risk, it is too bad that a problem that simple
up with the single-tree analysis described earlier by doing in structure cannot be posed formally. It is possible that
that. The essential step is to first calculate P(A = a) for each some actions that we might take would be effective for all
agent, and then make a new tree where the agent is selected contagious diseases. This should make them attractive, but
at the root, with the agent selection probabilities on the 28 the low rank of most contagious diseases individually in the
branches from the root. The second and third ranks of the agent-damage charts tends to suppress their attractiveness.
tree would then be what were originally the first and second, My last suggestion is to report future results in a scientific
with new probabilities as computed by Bayes’ theorem, and fashion that can be reviewed by scientists. English is a notori-
the rest of the tree would be unchanged. Since the agent is ously imprecise language for describing operations involving
at the root of the resulting “flipped” tree, using the flipped chance, so I have repeatedly struggled to understand what
tree is in effect doing a separate analysis for each agent. The was actually done in making my way through the references.
flipped tree would lead to the same earlier described agent- As a result, I may well have misinterpreted something above
damage charts—the two trees are stochastically equivalent. that I hope DHS will correct. If I were reviewing the 2006
But I don’t see the motivation for doing all this extra work work for a journal, my first act would be to send the material
in flipping the tree, and I have some concerns about whether back to the authors with a request that it be written up using
the flipping operation was actually done correctly, or done mathematics embedded in English, instead of just English. I
at all. know that DHS has to communicate complicated ideas about
One concern is that the thing being manipulated is not an risk to laypeople. That task should be in addition to reporting
ordinary event tree, and there is no reason to expect that beta the results scientifically, not a replacement for it.
distributions will remain beta distributions in the flipping In summary, my opinion is that the 2006 DHS methodol-
process. Of course, the flipping could occur after the tree is ogy is not yet the “rigorous and technically sound methodol-
instantiated in each of the 500 replications, but that would get ogy” demanded by the 2004 Homeland Security Presidential
to be a lot of work. I doubt if that has been the case. Directive 10: Biodefense for the 2st Century. Let me also
The documentation is mute about the tree flipping pro- add that I consider the report as a whole to be a remarkable
cess. I can only hope that the method actually used for pro- accomplishment, given the magnitude of the task and the
ducing agent-damage charts is equivalent to analyzing the time available to do it.
single event tree as described above.
References. Materials that I have examined before writing
Suggestions. My main suggestion for future work is that this review include the following:
distributions for branch probabilities be abandoned in favor
of direct branch probabilities, as in a standard event tree. In Department of Homeland Security. 2006. Bioterrorism Risk
other words, keep it simple. SMEs will not be comfortable Assessment. Biological Threat Characterization Center of the
expressing definite values for the probabilities, but then they National Biodefense Analysis and Countermeasures Center.
are probably not comfortable with expressing definite values Fort Detrick, Md.
for α and b, either. Most people are simply not comfortable
quantifying uncertainty. There is very little to be gained by I have also examined various drafts of the following:
including epistemic uncertainty about the branch probabili-
ties in an analysis like this, and much to be lost in terms of Department of Homeland Security. 2007. “A Lexicon of
complication. Epistemic uncertainty is not even discussed in Risk Terminology and Methodological Description of the
most decision theory textbooks. Standard software for han- DHS Bioterrorism Risk Assessment.” April 16.
dling decision trees would become applicable (event trees are
just a special case where there are no decisions) if epistemic Of all the documents, this last one comes closest to the tech-
uncertainty were not present. There is also standard software nical appendix that I recommend. It has been of considerable
for handling influence diagrams, which ought to be consid- use to me, but even it does not address tree flipping.
ered as an alternative to decision trees. Influence diagram