On Solving Probabilistic Linear Diophantine Equations
Patrick Kreitzberg, Oliver Serang; 22(87):1−24, 2021.
Multiple methods exist for computing marginals involving a linear Diophantine constraint on random variables. Each of these extant methods has some limitation on the dimension and support or on the type of marginal computed (e.g., sum-product inference, max-product inference, maximum a posteriori, etc.). Here, we introduce the "trimmed $p$-convolution tree'" an approach that generalizes the applicability of the existing methods and achieves a runtime within a $\log$-factor or better compared to the best existing methods. A second form of trimming we call underflow/overflow trimming is introduced which aggregates events which land outside the supports for a random variable into the nearest support. Trimmed $p$-convolution trees with and without underflow/overflow trimming are used in different protein inference models. Then two different methods of approximating max-convolution using Cartesian product trees are introduced.
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