Counted Conditionals: Modeling Discrete Distributions From Pairwise Counts

Generative modeling requires estimating high-dimensional distributions from finite data. Counting complete configurations is impractical, but relationships between informative parts recur across examples. We ask how far discrete distributions can be modeled from empirical token-pair counts. We introduce Counted Conditionals, which separates the statistics we store from the way we use them. Rather than pooling observations as separate neighbor votes, we fit tractable joint models on narrow spatial strips from first- and second-order counts. Conditioning sums over unknown tokens within each strip, and a calibrated pool combines the component predictions. Higher-order context counts provide an optional extension. On visual- token and land-cover grids, the strip construction improves completion accuracy and log loss over direct pairwise pooling across three mask patterns. Small-data and increasing-evidence experiments examine how it uses limited training examples and partial observations. We also evaluate the normalized joint induced by applying the pooled predictor in a fixed order, separately from its marginal forecasts. Because records contribute additive statistics, updating the counts and repeating the same deterministic construction reproduces a fresh build under fixed preprocessing and reference information. These results establish a count-based approach to discrete probability modeling and clarify its predictive and computational trade-offs.