Monotonic Path Likelihood for Learning Full-Order Probabilities

Learning score distributions from ranking observations requires linking item-level uncertainty to the probability of the observed order. Existing objectives often rely on restricted distribution families or cutpoint-based lower bounds. Learning flexible distributions through complete ordering events therefore calls for efficient probability evaluation and differentiation. We introduce Monotonic Path Likelihood (MPL), which addresses this need through a shared bin-mass representation for continuous parametric and nonparametric discrete distributions. Under conditional independence, strictly ordered bin assignments form monotonic paths. Aggregating their probability masses through shared prefixes gives an exact grid-event likelihood in $\mathcal{O}(LT)$ time for $L$ items and $T$ bins, enabling end-to-end optimization. Our analysis establishes monotonicity under nested grid refinement and continuous approximation error bounds, and characterizes the learning signal through the path posterior. Experiments on MultiDigit MNIST, MSLR-WEB30K, Istella, and multimodal score distribution recovery show improvements in long-list ranking and distribution recovery, with competitive performance on real-world learning-to-rank benchmarks.