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Appendix B
Mathematical Characterization of the Biological Threat Risk Assessment Event Tree and Risk Assessment

Gerald G. Brown, Ph.D.

Distinguished Professor of Operations Research

Naval Postgraduate School, Monterey, California


An event tree can be defined as a directed-out-tree (i.e., a connected di-graph that contains no cycle with exactly one, distinguished, root node with in-degree 0, and every other node with in-degree 1).1 Each node represents some event, and each directed out-arc represents a randomly-chosen outcome that selects a successor event node. Every directed path in this tree starts with the root node, and ends at a node with out-degree zero (a leaf node). Each directed path from the root node to a leaf node in the event tree represents a possible sequence of alternating events and outcomes (i.e., a scenario).

Figure B.1 defines the Biological Threat Risk Assessment (BTRA) event tree mathematically and shows how to solve for all path probabilities. This event tree is a restriction of a completely general one: This tree consists of successive stages, or echelons of events, with each stage restricted to offer the same branch opportunities.

Figure B.2 defines the BTRA risk analysis mathematically.

If we attach a set of mutually-exclusive, exhaustive probabilities to the arcs branching out of each node, we can trace each directed path in the event tree and reckon its joint probability of selection by multiplying the successive arc selection probabilities on the path. Note that we need not assume independence among successive probabilities, and can in fact condition each arc probability on all prior outcomes in its path.

If we associate a consequence (i.e., a measured outcome) with each end state node, we can assess the total expected consequence of each path by multiplying this consequence by its path probability. We can also generalize to a distribution of consequences for each end state node, and accumulate an expected distribution of consequences.

Many of the scenario paths terminate early (e.g., due to interdiction), so the actual number of paths terminating with non-zero consequences is in the thousands, rather than billions.

The distributions of consequences for all scenarios (paths) share the same “bin structure” (discrete intervals), and random sampling of paths can be used to induce a random sampling of consequence distribution. From this expected consequence distribution, we can estimate, for instance, the 5th and 95th percentiles.

1

See, for example, R. Ahuja, T. Magnanti, and J. Orlin, 1993, Network Flows: Theory, Algorithms, and Applications, Upper Saddle River, N.J.: Prentice Hall, Chapter 2.



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Appendix B Mathematical Characterization of the Biological Threat Risk Assessment Event Tree and Risk Assessment Gerald G. Brown, Ph.D. Distinguished Professor of Operations Research Naval Postgraduate School, Monterey, California An event tree can be defined as a directed-out-tree (i.e., a joint probability of selection by multiplying the successive connected di-graph that contains no cycle with exactly one, arc selection probabilities on the path. Note that we need distinguished, root node with in-degree 0, and every other not assume independence among successive probabilities, node with in-degree 1).1 Each node represents some event, and can in fact condition each arc probability on all prior and each directed out-arc represents a randomly-chosen outcomes in its path. outcome that selects a successor event node. Every directed If we associate a consequence (i.e., a measured outcome) path in this tree starts with the root node, and ends at a node with each end state node, we can assess the total expected with out-degree zero (a leaf node). Each directed path from consequence of each path by multiplying this consequence the root node to a leaf node in the event tree represents a by its path probability. We can also generalize to a distribu- possible sequence of alternating events and outcomes (i.e., tion of consequences for each end state node, and accumulate a scenario). an expected distribution of consequences. Figure B.1 defines the Biological Threat Risk Assessment Many of the scenario paths terminate early (e.g., due (BTRA) event tree mathematically and shows how to solve to interdiction), so the actual number of paths terminating for all path probabilities. This event tree is a restriction of with non-zero consequences is in the thousands, rather than a completely general one: This tree consists of successive billions. stages, or echelons of events, with each stage restricted to The distributions of consequences for all scenarios (paths) offer the same branch opportunities. share the same “bin structure” (discrete intervals), and Figure B.2 defines the BTRA risk analysis mathematically. random sampling of paths can be used to induce a random If we attach a set of mutually-exclusive, exhaustive sampling of consequence distribution. From this expected probabilities to the arcs branching out of each node, we consequence distribution, we can estimate, for instance, the can trace each directed path in the event tree and reckon its 5th and 95th percentiles. 1 See, for example, R. Ahuja, T. Magnanti, and J. Orlin, 1993, Network Flows: Theory, Algorithms, and Applications, Upper Saddle River, N.J.: Prentice Hall, Chapter 2. 

OCR for page 78
 APPENDIX B Index Use [cardinality] g = {1,2,…,G} ordinal set of successive stages of events leading from initiation of attack planning to final attack consequence. (alias g′) [18] ag ∈ Ag outcome at stage g < G [2-28] pg = {a1,…, ag} ∈ Pg = {a1 × … × ag} sequence of outcomes chosen through stage g < G ∏g′1 FIGURE B.1 Mathematical definition of BTRA event tree and solution for tree probabilities. This defines a BTRA event tree and shows how to completely evaluate all probabilities for every path. This definition applies whether or not the tree includes all agents, or just one of them. Additional Index Use [cardinality] c ∈ AG-1 ≡ C set of final consequences, outcomes in penultimate stage G - 1 [10] Additional Data [units] cost of consequence c [cost] costc Computed Parameters [units] cost_ pr (c) probability of consequence c with costc [cost] total risk (i.e., expected cost) [cost] R Computation ∑ cost _ pr (c) = path _ pr ( pg ) × cost _ pr (c), ∀c ∈C Pg ∈PG -  R = ∑ costc × cost _ pr(c) = ∑ costc × path _ pr ( pg ) × cost _ pr (c) c∈C c∈C , Pg ∈PG -1 FIGURE B.2 Mathematical definition of BTRA risk analysis. This shows how to completely evaluate all cost consequences and risk (expected cost). The paths here have one extra, final stage that BTRA does not: This stage eliminates the necessity for separate notation for consequence distributions, with each of its outcomes resulting in a scalar cost consequence. A Monte Carlo sampling to estimate these computed parameters would proceed by randomly selecting a path p G-1={a 1,a 2, … ,a G-1} (the probability of this path could be computed by ∏ branch _ prob pg (ag ), but this is not essential) and collecting this result as a sample statistic. g