The third and most significant modification is that the branching probabilities (DHS on occasion also calls them “branch fractions”) are not fixed, but are instead themselves determined by sampling from beta distributions provided indirectly by Subject Matter Experts (SMEs). Let θ be the collection of branching probabilities. In each incident we therefore observe (θ, *A, S, Y*), with θ determining the event tree for the other three random variables. This modification will also be discussed separately below.

* The Second Modification: Repeated Attacks per Incident.* The vision is that a cell or group of terrorists will not plan a single attack, but will plan to continue to attack until interrupted, with the entire group of attacks constituting an incident. The effect of this is to change the distribution of consequences of an incident, since a successful attack will be accompanied by afterattacks, the number of which I will call

Other changes may also be necessary to implement the original vision. If the afterattacks all have independent consequences, then the distribution of total consequences is the (1 + *X*)-fold convolution of the consequence distribution, a complicated operation that I see no evidence of. The documentation is mute on what is actually assumed about the independence of after attacks, and on how the E(*X*) computation is actually used. Simply scaling up the consequences of one attack by the factor (1 + E(*X*)) is correct on the average, regardless of independence assumptions, but will not give the correct distribution of total consequences.

* The Third Modification: “Random Probabilities.”* DHS has accommodated SME uncertainty by allowing the branch probabilities themselves to be random quantities, with the SMEs merely agreeing to a distribution for each probability, rather than a specific number. I will refer to each of these probability distributions as a “marginal” for its branch. If a node has

However, summing to 1 is not sufficient for the SME marginals to be meaningful. This is most obvious when *N* = 2. If the first branch has probability *A*, then the second must have probability 1 − *A*, and therefore the second probability 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 β = 1, while the second has α = 2 and β = 2, then we must explain to the experts that what they are saying is meaningless, even though both marginals have a mean of 0.5. The second marginal has no choice but to be the mirror image of the first, and must therefore *be* the first, by symmetry. Any other possibility is literally meaningless, since there is no pair of random variables (*A*_{1}, *A*_{2}) such that *A*_{i} has the *i*th marginal distribution and also *A*_{1} + *A*_{2} is always exactly 1.

I think DHS recognizes the difficulty when *N* = 2, and has basically fixed it in that case by asking the SMEs for only one marginal, but the same difficulty is present for *N* > 2, and has not been fixed. The sampling procedure offered on page C-81 of Department of Homeland Security (2006) will reliably produce probabilities *A*_{1}, …, *A*_{N} that sum to 1, and which are correct on the average, but they do not have the marginal beta distributions given by the SMEs. This is most obvious in the case of the last branch, since the *N*th marginal is never used in the sampling process, but I believe that the marginal distribution is correct only for the first branch.

There is a multivariable distribution (the Dirichlet distribution) whose marginals are all beta distributions, but the Dirichlet distribution has only *N* + 1 parameters. The SME marginals require 2*N*, in total, so the Dirichlet distribution is not a satisfactory joint distribution for *A*_{1}, …, *A*_{N}.

* Estimation of the Spread in Agent-Damage Charts.* I have defined

The spreads display the epistemic variability due to SME uncertainty about θ, but suppress all of the aleatoric variability implied by the event tree. If there were no uncertainty