2,190 research outputs found
First passage process of a Markov additive process, with applications to reflection problems
In this paper we consider the first passage process of a spectrally negative
Markov additive process (MAP). The law of this process is uniquely
characterized by a certain matrix function, which plays a crucial role in
fluctuation theory. We show how to identify this matrix using the theory of
Jordan chains associated with analytic matrix functions. Importantly, our
result also provides us with a technique, which can be used to derive various
further identities. We then proceed to show how to compute the stationary
distribution associated with a one-sided reflected (at zero) MAP for both the
spectrally positive and spectrally negative cases as well as for the two sided
reflected Markov-modulated Brownian motion; these results can be interpreted in
terms of queues with MAP input.Comment: 16 page
Stochastic Gradient Hamiltonian Monte Carlo
Hamiltonian Monte Carlo (HMC) sampling methods provide a mechanism for
defining distant proposals with high acceptance probabilities in a
Metropolis-Hastings framework, enabling more efficient exploration of the state
space than standard random-walk proposals. The popularity of such methods has
grown significantly in recent years. However, a limitation of HMC methods is
the required gradient computation for simulation of the Hamiltonian dynamical
system-such computation is infeasible in problems involving a large sample size
or streaming data. Instead, we must rely on a noisy gradient estimate computed
from a subset of the data. In this paper, we explore the properties of such a
stochastic gradient HMC approach. Surprisingly, the natural implementation of
the stochastic approximation can be arbitrarily bad. To address this problem we
introduce a variant that uses second-order Langevin dynamics with a friction
term that counteracts the effects of the noisy gradient, maintaining the
desired target distribution as the invariant distribution. Results on simulated
data validate our theory. We also provide an application of our methods to a
classification task using neural networks and to online Bayesian matrix
factorization.Comment: ICML 2014 versio
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