A marginal probability measures how often one feature value occurs without considering the values of other features. It comes from a table margin: divide a feature value's count by the total number of rows. This separates values that are individually rare from combinations that are unusual despite having ordinary marginal frequencies. section 2.4.2
Use marginal probabilities to assess whether a rare combination is rare for an ordinary reason or is unusual in its joint behavior:
- Compute each feature value's marginal fraction from its row or column total. section 2.4.2
- Estimate an expected combination count by multiplying the relevant marginal fractions and the dataset size, under the independence-based estimate. section 2.4.2 section 3.1.1
- Compare observed and expected counts to distinguish combinations of rare values from combinations that are unexpectedly sparse or frequent. section 2.4.2 section 13.3.3
- Apply the comparison to missing or unusually low combinations, including pairs or triples when the dataset is sufficiently large. section 12.5
Separating marginal rarity from combination rarity
In a table with 975,257 rows, value A occurs 1,513 times and value W occurs 3,014 times, giving marginal fractions of about 0.001 and 0.003. Their observed combination occurs twice, while multiplying those fractions predicts about 4.6 rows. The combination is rare, but both component values are already rare. By contrast, a combination of individually common values can be flagged when its observed count is far below its marginally expected count.
Marginal probabilities also support algorithmic detectors. COPOD uses empirical cumulative distributions and an empirical copula to estimate tail probabilities, while CountsOutlierDetector can check marginal probabilities when scoring sparse categorical subspaces. These methods improve interpretation but do not make every flagged combination meaningful automatically. section 6.4 section 13.3.3
