MMP.11:5.3 - Construct probabilities while keeping boundary outcomes

A report has three possible outcomes. Let w_i>=0 and let their sum W be positive. Set p_i=w_i/W. Every candidate has nonnegative probabilities summing to one. Conversely, every probability vector on these outcomes is represented by choosing w=p. Thus this construction includes zero-probability outcomes.

The weights (0,1,3) and (0,2,6) both give probabilities (0,1/4,3/4). The parameter vector is redundant even if the probability vector becomes fully determined. There is no need to distinguish those weights when the receiving question uses only the law.

A strictly positive parameterization, such as exponentiating every finite unconstrained parameter before normalization, excludes zero probabilities. It can approximate a zero closely but cannot express it with finite parameters. If the subject account rules out the first outcome, retain that zero in the construction and normalize weights for the remaining outcomes. Whether a very small nonzero value would suffice depends on the receiving question.

The constructed p is a family member, not yet an estimate from data. MMP.7 composes it with selection, rounding or other recording behavior. The appropriate statistical method then determines what the observations support. Changing the recording procedure can change that inference without changing the underlying outcome family.