Randomized Hamiltonian Monte Carlo as scaling limit of the bouncy particle sampler and dimension-free convergence rates

Abstract Hamiltonian Monte Carlo (HMC) is an algorithm of the Markov Chain Monte Carlo (MCMC) method that uses dynamics to propose samples that follow a target distribution. This algorithm enables more effective and consistent exploration of the probability interval and is more sensitive to correlated parameters. Therefore, Bayesian-HMC is a promising alternative to estimate individual parameters of complex functions such as nonlinear models, especially when using small datasets. Our objective was to estimate genetic parameters for milk traits defined based on nonlinear model parameters predicted using the Bayesian-HMC algorithm. A total of 64,680 milk yield test-day records from 2,624 first, second, and third lactations of Saanen and Alpine goats were used. First, the Wood model was fitted to the data. Second, lactation persistency (LP), peak time (PT), peak yield (PY), and total milk yield [estimated from zero to 50 (TMY50), 100(TMY100), 150(TMY150), 200(TMY200), 250(TMY250), and 300(TMY300) days-in-milk] were predicted for each animal and parity based on the output of the first step (the individual phenotypic parameters of the Wood model). Thereafter, these predicted phenotypes were used for estimating genetic parameters for each trait. In general, the heritability estimates across lactations ranged from 0.10 to 0.20 for LP, 0.04 to 0.07 for PT, 0.26 to 0.27 for PY, and 0.21 to 0.28 for TMY (considering the different intervals). Lower heritabilities were obtained for the nonlinear function parameters (A, b and l) compared to its predicted traits (except PT), especially for the first and second lactations (range: 0.09 to 0.18). Higher heritability estimates were obtained for the third lactation traits. To our best knowledge, this study is the first attempt to use the HMC algorithm to fit a nonlinear model in animal breeding. The two-step method proposed here allowed us to estimate genetic parameters for all traits evaluated.

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Hamiltonian Monte Carlo calculation in two-dimensional SU(2) QCD with fermions

Physical Review D ◽

10.1103/physrevd.31.1453 ◽

1985 ◽

Vol 31 (6) ◽

pp. 1453-1459 ◽

Cited By ~ 5

Author(s):

Jen-Fa Min ◽

Joel A. Shapiro ◽

Thomas A. DeGrand

Keyword(s):

Monte Carlo ◽

Monte Carlo Calculation ◽

Two Dimensional ◽

Hamiltonian Monte Carlo

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Hybrid (Hamiltonian) Monte Carlo

SpringerBriefs in Physics - Markov Chain Monte Carlo Methods in Quantum Field Theories ◽

10.1007/978-3-030-46044-0_7 ◽

2020 ◽

pp. 59-69

Author(s):

Anosh Joseph

Keyword(s):

Monte Carlo ◽

Hamiltonian Monte Carlo

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MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2021029 ◽

2021 ◽

Author(s):

Dong T.P. Nguyen ◽

Dirk Nuyens

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Monte Carlo ◽

Convergence Rates ◽

Higher Order ◽

Elliptic Pdes ◽

Infinite Dimensional ◽

Quasi Monte Carlo ◽

Higher Order Convergence ◽

Element Method

We introduce the \emph{multivariate decomposition finite element method} (MDFEM) for elliptic PDEs with lognormal diffusion coefficients, that is, when the diffusion coefficient has the form $a=\exp(Z)$ where $Z$ is a Gaussian random field defined by an infinite series expansion $Z(\bsy) = \sum_{j \ge 1} y_j \, \phi_j$ with $y_j \sim \calN(0,1)$ and a given sequence of functions $\{\phi_j\}_{j \ge 1}$. We use the MDFEM to approximate the expected value of a linear functional of the solution of the PDE which is an infinite-dimensional integral over the parameter space. The proposed algorithm uses the \emph{multivariate decomposition method} (MDM) to compute the infinite-dimensional integral by a decomposition into finite-dimensional integrals, which we resolve using \emph{quasi-Monte Carlo} (QMC) methods, and for which we use the \emph{finite element method} (FEM) to solve different instances of the PDE. We develop higher-order quasi-Monte Carlo rules for integration over the finite-di\-men\-si\-onal Euclidean space with respect to the Gaussian distribution by use of a truncation strategy. By linear transformations of interlaced polynomial lattice rules from the unit cube to a multivariate box of the Euclidean space we achieve higher-order convergence rates for functions belonging to a class of \emph{anchored Gaussian Sobolev spaces} while taking into account the truncation error. These cubature rules are then used in the MDFEM algorithm. Under appropriate conditions, the MDFEM achieves higher-order convergence rates in term of error versus cost, i.e., to achieve an accuracy of $O(\epsilon)$ the computational cost is $O(\epsilon^{-1/\lambda-\dd/\lambda}) = O(\epsilon^{-(p^* + \dd/\tau)/(1-p^*)})$ where $\epsilon^{-1/\lambda}$ and $\epsilon^{-\dd/\lambda}$ are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with $\dd = d \, (1+\ddelta)$ for some $\ddelta \ge 0$ and $d$ the physical dimension, and $0 < p^* \le (2 + \dd/\tau)^{-1}$ is a parameter representing the sparsity of $\{\phi_j\}_{j \ge 1}$.

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