Unsupervised Combinatorial Probabilistic Reasoning: Probabilistic Coin Change Problem

We introduce the Probabilistic Coin Change Problem (PCCP), a novel variant of the classical Combination Coin Change Problem (CCCP), motivated by a real-world scientific inverse task. The goal of CCCP is to enumerate all unordered combinations of coin denominations that sum to a given target. In PCCP, each coin type's value follows a discrete probability distribution, and the aggregate value of a combination of coins is thus stochastic. Given a set of such coin types and noisy observations of total sums, the task is to infer the most likely latent coin combination. To address the combinatorial and probabilistic complexity of PCCP, we propose DeepProReasoner (Deep Combinatorial Probabilistic Reasoning with Embedded Representations), an unsupervised, end-to-end, deep-learning framework that integrates combinatorial reasoning, latent-space modeling, and differentiable probabilistic reasoning. The model is trained using a reconstruction loss between the observed empirical distribution and a decoded probability mass function (PMF), enabling efficient gradient-based search over a continuous relaxation of the combinatorial space. We evaluate DeepProReasoner on two instances of PCCP: (1) a synthetic Candy Mix problem for ablation studies, and (2) a real-world task of molecular formula inference from ultrahigh resolution mass spectrometry (MS) data. Besides the two given instances, PCCP captures a wide range of inverse settings in biology, chemistry, environmental sciences, and medicine, where latent combinatorial structures give rise to noisy aggregate observations through stochastic processes. Our results show that Deep-ProReasoner achieves high accuracy and robustness, outperforming state-of-the-art methods.