From Bounds to a Point Estimate: What the Model Assumes

A feasible interval is the safest answer when overlap is only partly observed, but decisions sometimes require a central estimate: a planning scenario, a comparison of alternatives, or a provisional allocation. This chapter explains the subset-reach paper’s conditional-independence model and its non-negative least-squares fit. The important judgment is not whether the model produces one number—it does—but what assumptions turn that number into a prediction rather than an observed fact.

A population of activity patterns

The model begins with a different description of the audience. Instead of assigning each user directly to every observed overlap region, it divides the universe into mutually exclusive activity segments. A segment represents a pattern of high or low probability that each buying group reaches a user. One segment may contain users who are relatively likely to be reached by the first and second groups and relatively unlikely to be reached by the third. Another segment may reverse that pattern.

These segments are not claimed to be visible demographic groups, and they are not the same as platform audiences. They are latent patterns used to approximate dependence among buying groups. Within one segment, the model assumes that the probability of being reached by one group is conditionally independent of the probability of being reached by another. The union reach for that segment can then be calculated from the modeled single-group probabilities. Across the whole universe, the segment-level patterns are mixed according to their weights.

The high/low construction gives the independence assumption more flexibility than one global independence model. A single global model would apply one relationship to every user. The segmented model can represent different kinds of users: some segments make two platforms more likely to co-occur, while others make a platform more likely to appear alone. The flexibility comes from the mixture of patterns, not from observing a literal label on each person.

The observed subset reaches serve as training points. The paper requires the reach of each individual buying group and the reach of the all-groups union, while allowing additional unions to be included. Each training point becomes one row of the model’s predicted reach patterns. The unknown weights say how much of the universe is assigned to each latent activity segment. The model then chooses weights that make its predicted training reaches as close as possible to the observed training reaches.

For three buying groups, the required observations establish the three individual reaches and the reach of the full union. An additional observed pair, such as the first and second groups together, gives the model another constraint before it estimates a different pair or an incremental subset. The extra observation does not reveal the hidden people directly. It changes the shape of the training vector that the latent segment mixture must reproduce, which can make some activity-pattern combinations less plausible under the model.