How Do You Compute with P-Boxes

Probability bounds analysis combines p-boxes together in mathematical operations such as addition, subtraction, multiplication, and division. This is an alternative to what is usually done with Monte Carlo simulations, which usually evaluate a risk expression in one fell swoop in each iteration. In probability bounds analysis, a complex calculation is decomposed into its constituent arithmetic operations, which are computed separately to build up the final answer. The actual calculations needed to effect these operations with p-boxes are straightforward and elementary. This is not to say, however, that they are the kinds of calculations one would want to do by hand. In aggregate, they will often be cumbersome and should generally be done on computer. But it may be helpful to the reader to step through a numerical example just to see the nature of the calculation. 

The 1st step is to partition the p-boxes of the addends into sets of intervals and associated probability masses. The p-box for A can be partitioned into the following 3 interval-mass pairs  

We are now ready to combine the 2 p-boxes. To do so, we construct the Cartesian product of the 2 collections of interval-mass pairs in the matrix shown in Table 6.2. 

It turns out that, given the variability and incertitude in the inputs, this is the best possible p-box for the sum A + B. This means that we could not tighten the bounds in any way and still have it include all possible distributions that could arise as a sum of distributions from inside the p-boxes of the inputs. 

Cartesian product of 2 collections of interval-mass pairs 

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