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A Fast Numerical Method for Max-Convolution and the Application to Efficient Max-Product Inference in Bayesian

Oliver Serang1,2

  • 11 Department of Informatik Freie Universität Berlin, Berlin, Germany .

Journal of Computational Biology : a Journal of Computational Molecular Cell Biology
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PubMed
Summary

A new O(k log k) numerical method efficiently estimates max-convolution for discrete probability distributions. This advances max-product inference, crucial for analyzing sums of random variables in various scientific fields.

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Area of Science:

  • Computer Science
  • Probability Theory
  • Applied Mathematics

Background:

  • Max-product inference, used for maximum a posteriori estimation, is computationally intensive.
  • A key bottleneck is the max-convolution problem for general discrete distributions, lacking an efficient solution.
  • Current methods do not achieve the desired O(k log k) complexity for this problem.

Purpose of the Study:

  • To develop an efficient numerical method for estimating the max-convolution of nonnegative vectors.
  • To demonstrate the application of this method for fast max-product inference on convolution trees.
  • To explore potential applications in hidden Markov models and shortest path algorithms.

Main Methods:

  • Developed an O(k log k) numerical algorithm for estimating the max-convolution of two nonnegative vectors.
  • Applied the numerical max-convolution method to perform fast max-product inference on a convolution tree.
  • Analyzed the runtime complexity for inference on sums of discrete random variables.

Main Results:

  • An efficient O(k log k) numerical method for max-convolution estimation is presented.
  • Fast max-product inference on convolution trees is achieved in O(nk log(nk)log(n)) steps.
  • Reduced runtime for Viterbi path computation in specific hidden Markov models from O(nk^2) to O(nk log k).

Conclusions:

  • The proposed numerical max-convolution method significantly improves the efficiency of max-product inference.
  • This advancement has practical implications for analyzing sums of random variables and related computational problems.
  • Potential applications include specialized hidden Markov models and the all-pairs shortest paths problem.