scholarly journals Limit theorems for the zig-zag process

2017 ◽  
Vol 49 (3) ◽  
pp. 791-825 ◽  
Author(s):  
Joris Bierkens ◽  
Andrew Duncan
Keyword(s):  
Monte Carlo ◽  
Probability Distributions ◽  
Asymptotic Variance ◽  
Mcmc Methods ◽  
Diffusion Limit ◽  
Switching Rate ◽  
One Dimensional ◽  
Piecewise Deterministic Markov Processes ◽  
Reversible Markov Chains ◽  
The One

AbstractMarkov chain Monte Carlo (MCMC) methods provide an essential tool in statistics for sampling from complex probability distributions. While the standard approach to MCMC involves constructing discrete-time reversible Markov chains whose transition kernel is obtained via the Metropolis–Hastings algorithm, there has been recent interest in alternative schemes based on piecewise deterministic Markov processes (PDMPs). One such approach is based on the zig-zag process, introduced in Bierkens and Roberts (2016), which proved to provide a highly scalable sampling scheme for sampling in the big data regime; see Bierkenset al.(2016). In this paper we study the performance of the zig-zag sampler, focusing on the one-dimensional case. In particular, we identify conditions under which a central limit theorem holds and characterise the asymptotic variance. Moreover, we study the influence of the switching rate on the diffusivity of the zig-zag process by identifying a diffusion limit as the switching rate tends to ∞. Based on our results we compare the performance of the zig-zag sampler to existing Monte Carlo methods, both analytically and through simulations.

scholarly journals Limit theorems for sequential MCMC methods

2020 ◽  
Vol 52 (2) ◽  
pp. 377-403 ◽  
Author(s):  
Axel Finke ◽  
Arnaud Doucet ◽  
Adam M. Johansen
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Sequential Monte Carlo ◽  
Probability Distributions ◽  
Asymptotic Variance ◽  
Mcmc Methods ◽  
Time Step ◽  
Strong Law ◽  
Large Numbers

AbstractBoth sequential Monte Carlo (SMC) methods (a.k.a. ‘particle filters’) and sequential Markov chain Monte Carlo (sequential MCMC) methods constitute classes of algorithms which can be used to approximate expectations with respect to (a sequence of) probability distributions and their normalising constants. While SMC methods sample particles conditionally independently at each time step, sequential MCMC methods sample particles according to a Markov chain Monte Carlo (MCMC) kernel. Introduced over twenty years ago in [6], sequential MCMC methods have attracted renewed interest recently as they empirically outperform SMC methods in some applications. We establish an $\mathbb{L}_r$ -inequality (which implies a strong law of large numbers) and a central limit theorem for sequential MCMC methods and provide conditions under which errors can be controlled uniformly in time. In the context of state-space models, we also provide conditions under which sequential MCMC methods can indeed outperform standard SMC methods in terms of asymptotic variance of the corresponding Monte Carlo estimators.

scholarly journals A fractional generalization of the dirichlet distribution and related distributions

2021 ◽  
Vol 24 (1) ◽  
pp. 112-136
Author(s):  
Elvira Di Nardo ◽  
Federico Polito ◽  
Enrico Scalas
Keyword(s):  
Monte Carlo ◽  
Probability Density Function ◽  
Monte Carlo Simulations ◽  
Probability Density ◽  
Dirichlet Distribution ◽  
One Dimensional ◽  
Reasonable Approximation ◽  
Generalized Dirichlet Distribution ◽  
The One ◽  
Generalized Dirichlet

Abstract This paper is devoted to a fractional generalization of the Dirichlet distribution. The form of the multivariate distribution is derived assuming that the n partitions of the interval [0, Wn ] are independent and identically distributed random variables following the generalized Mittag-Leffler distribution. The expected value and variance of the one-dimensional marginal are derived as well as the form of its probability density function. A related generalized Dirichlet distribution is studied that provides a reasonable approximation for some values of the parameters. The relation between this distribution and other generalizations of the Dirichlet distribution is discussed. Monte Carlo simulations of the one-dimensional marginals for both distributions are presented.

scholarly journals Measuring hospital spatial wingspan by using a discrete choice model with utility-threshold

2018 ◽  
Author(s):  
Saley Issa ◽  
Ribatet Mathieu ◽  
Molinari Nicolas
Keyword(s):  
Monte Carlo ◽  
Discrete Choice ◽  
Choice Model ◽  
Discrete Choice Models ◽  
Mcmc Methods ◽  
Hospital Competition ◽  
Policy Makers ◽  
New Approach ◽  
Asthma Patients ◽  
The One

AbstractPolicy makers increasingly rely on hospital competition to incentivize patients to choose high-value care. Travel distance is one of the most important drivers of patients’ decision. The paper presents a method to numerically measure, for a given hospital, the distance beyond which no patient is expected to choose the hospital for treatment by using a new approach in discrete choice models. To illustrate, we compared 3 hospitals attractiveness related to this distance for asthma patients admissions in 2009 in Hérault (France), showing, as expected, CHU Montpellier is the one with the most important spatial wingspan. For estimation, Monte Carlo Markov Chain (MCMC) methods are used.

scholarly journals Optimal foraging and the information theory of gambling

2019 ◽  
Vol 16 (157) ◽  
pp. 20190162 ◽  
Author(s):  
Roland J. Baddeley ◽  
Nigel R. Franks ◽  
Edmund R. Hunt
Keyword(s):  
Information Theory ◽  
Monte Carlo ◽  
Optimal Foraging ◽  
Foraging Behaviour ◽  
Stochastic Dynamics ◽  
Probability Distributions ◽  
Movement Ecology ◽  
Mcmc Methods ◽  
Long Run ◽  
Resource Gradient

At a macroscopic level, part of the ant colony life cycle is simple: a colony collects resources; these resources are converted into more ants, and these ants in turn collect more resources. Because more ants collect more resources, this is a multiplicative process, and the expected logarithm of the amount of resources determines how successful the colony will be in the long run. Over 60 years ago, Kelly showed, using information theoretic techniques, that the rate of growth of resources for such a situation is optimized by a strategy of betting in proportion to the probability of pay-off. Thus, in the case of ants, the fraction of the colony foraging at a given location should be proportional to the probability that resources will be found there, a result widely applied in the mathematics of gambling. This theoretical optimum leads to predictions as to which collective ant movement strategies might have evolved. Here, we show how colony-level optimal foraging behaviour can be achieved by mapping movement to Markov chain Monte Carlo (MCMC) methods, specifically Hamiltonian Monte Carlo (HMC). This can be done by the ants following a (noisy) local measurement of the (logarithm of) resource probability gradient (possibly supplemented with momentum, i.e. a propensity to move in the same direction). This maps the problem of foraging (via the information theory of gambling, stochastic dynamics and techniques employed within Bayesian statistics to efficiently sample from probability distributions) to simple models of ant foraging behaviour. This identification has broad applicability, facilitates the application of information theory approaches to understand movement ecology and unifies insights from existing biomechanical, cognitive, random and optimality movement paradigms. At the cost of requiring ants to obtain (noisy) resource gradient information, we show that this model is both efficient and matches a number of characteristics of real ant exploration.

Quantum Monte Carlo study of the one-dimensional Hubbard model with random hopping and random potentials

1994 ◽  
Vol 50 (15) ◽  
pp. 10474-10484 ◽  
Author(s):  
A. W. Sandvik ◽  
D. J. Scalapino ◽  
P. Henelius
Keyword(s):  
Monte Carlo ◽  
Hubbard Model ◽  
Quantum Monte Carlo ◽  
Monte Carlo Study ◽  
Random Potentials ◽  
One Dimensional ◽  
The One

Quantum monte carlo study of the one-dimensional exchange interaction model

1988 ◽  
Vol 130 (4-5) ◽  
pp. 257-259 ◽  
Author(s):  
Y.C. Chen ◽  
H.H. Chen ◽  
Felix Lee
Keyword(s):  
Monte Carlo ◽  
Exchange Interaction ◽  
Quantum Monte Carlo ◽  
Interaction Model ◽  
Monte Carlo Study ◽  
One Dimensional ◽  
The One
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