Structure selection
| Conjunction: P(A and B)=P(A)P(B | A). For a longer chain, each factor conditions on all relevant preceding events. Two marginal probabilities cannot usually be multiplied. |
Union: P(A or B)=P(A)+P(B)-P(A and B). If the overlap is unknown, report bounds or obtain more information. For two events the union lies between max(pA,pB) and min(1,pA+pB).
| Scenario mixture: if S1,…,Sk are disjoint and exhaustive, P(Y)=sum_j P(Sj)P(Y | Sj). If scenarios overlap, the weighted sum double-counts. If they omit a possible pathway, add an “other” branch with explicit probability. |
Dimensional estimation: express the desired quantity as compatible factors, such as active sites × requests/site/day × error probability/request. The product estimates a count expectation under stated assumptions; it does not directly give the probability of at least one error. If a Poisson count model is justified, P(count>=1)=1-exp(-lambda). State why clustering or burstiness could break that model.
Shared causes and coherence
A common driver can correlate every branch. A procurement delay may affect staffing and equipment simultaneously. Condition on the driver, or keep a joint uncertainty model. Do not attach independent random noise to all inputs and call the resulting narrow range a measured confidence interval.
Apply conjunction bounds: max(0,pA+pB-1) <= P(A and B) <= min(pA,pB). Check complements sum to one and earlier-deadline probabilities do not exceed later-deadline probabilities for a cumulative “by date” event.
Sensitivity
For Y=sa+(1-s)b:
- sensitivity to a is s;
- sensitivity to b is 1-s;
- sensitivity to s is a-b.
Use these derivatives or finite differences to locate important assumptions. Large uncertainty in an irrelevant component need not dominate the answer. A worst/best plausible-input envelope is a sensitivity analysis, not a posterior credible interval.
When simulation is useful, document distributions, parameter sources, dependence, random seed and sampling error separately from model uncertainty. The skill does not require simulation for a two-branch calculation.
Practical limits
Decomposition improves inspectability, not truth by itself. Additional factors can multiply unsupported guesses. Stop when further decomposition cannot be empirically informed or change the decision. Preserve the original question when creating subquestions.
Fermi-style dimensional estimation is conventionally associated with Enrico Fermi; no sole-inventor claim is made here for decomposition or the probability chain rule.