Markov jump processes (MJPs) are used to model a wide range of phenomena from
disease progression to RNA path folding. However, maximum likelihood estimation
of parametric models leads to degenerate trajectories and inferential
performance is poor in 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 \emph{JUMP-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.

通过小方差渐近方法，提出了一种新的目标函数和全新的扩展伽马 - 指数过程，用于建立针对马尔可夫跳跃过程的参数估计非奇异轨迹，并开发了一种新算法 ——JUMP-means，旨在提高建模与推断的速度和重构精度。

JUMP-Means: 马尔可夫跳跃过程的小方差渐近性

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

The classical mixture of Gaussians model is related to K-means via
small-variance asymptotics: as the covariances of the Gaussians tend to zero,
the negative log-likelihood of the mixture of Gaussians model approaches the
K-means objective, and the EM algorithm approaches the K-means algorithm. Kulis
& Jordan (2012) used this observation to obtain a novel K-means-like algorithm
from a Gibbs sampler for the Dirichlet process (DP) mixture. We instead
consider applying small-variance asymptotics directly to the posterior in
Bayesian nonparametric models. This framework is independent of any specific
Bayesian inference algorithm, and it has the major advantage that it
generalizes immediately to a range of models beyond the DP mixture. To
illustrate, we apply our framework to the feature learning setting, where the
beta process and Indian buffet process provide an appropriate Bayesian
nonparametric prior. We obtain a novel objective function that goes beyond
clustering to learn (and penalize new) groupings for which we relax the mutual
exclusivity and exhaustivity assumptions of clustering. We demonstrate several
other algorithms, all of which are scalable and simple to implement. Empirical
results demonstrate the benefits of the new framework.

应用小方差渐近方法直接处理贝叶斯非参数模型的后验概率，得到一种超越聚类的特征学习目标函数，并提出一些易于实现的新算法，这些算法的效果被实验结果验证。