# JUMP-Means: Small-Variance Asymptotics for Markov Jump Processes

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Jonathan Huggins,
Karthik Narasimhan,
Ardavan Saeedi,
Vikash Mansinghka
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Proceedings of the 32nd International Conference on Machine Learning, PMLR 37:693-701, 2015.

#### Abstract

Markov jump processes (MJPs) are used to model a wide range of phenomenon from disease progression to RNA path folding. However, existing methods suffer from a number of shortcomings: degenerate trajectories in the case of ML estimation of parametric models and poor inferential performance in the case of nonparametric models. We take a small-variance asymptotics (SVA) approach to overcome these limitations. We derive the small-variance asymptotics for parametric and nonparametric MJPs for both directly observed and hidden state models. In the parametric case we obtain a novel objective function which leads to non-degenerate trajectories. To derive the nonparametric version we introduce the gamma-gamma process, a novel extension to the gamma-exponential process. We propose algorithms for each of these formulations, which we call \emphJUMP-means. Our experiments demonstrate that JUMP-means is competitive with or outperforms widely used MJP inference approaches in terms of both speed and reconstruction accuracy.

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