Fast Exact Inference for Recursive Cardinality Models
This work addresses a bottleneck in probabilistic inference for machine learning researchers and practitioners, offering incremental improvements in efficiency for specific model classes.
The paper tackles the problem of efficient marginalization and sampling for cardinality potential models, which were previously not thoroughly addressed, and presents an algorithm for computing marginal probabilities and drawing exact joint samples in O(Dlog^2 D) time, along with introducing Recursive Cardinality models for more general applications.
Cardinality potentials are a generally useful class of high order potential that affect probabilities based on how many of D binary variables are active. Maximum a posteriori (MAP) inference for cardinality potential models is well-understood, with efficient computations taking O(DlogD) time. Yet efficient marginalization and sampling have not been addressed as thoroughly in the machine learning community. We show that there exists a simple algorithm for computing marginal probabilities and drawing exact joint samples that runs in O(Dlog2 D) time, and we show how to frame the algorithm as efficient belief propagation in a low order tree-structured model that includes additional auxiliary variables. We then develop a new, more general class of models, termed Recursive Cardinality models, which take advantage of this efficiency. Finally, we show how to do efficient exact inference in models composed of a tree structure and a cardinality potential. We explore the expressive power of Recursive Cardinality models and empirically demonstrate their utility.