The Experts below are selected from a list of 10830 Experts worldwide ranked by ideXlab platform
Michael Betancourt - One of the best experts on this subject based on the ideXlab platform.
-
on the geometric ergodicity of Hamiltonian Monte Carlo
Bernoulli, 2019Co-Authors: Samuel Livingstone, Michael Betancourt, Simon Byrne, Mark GirolamiAbstract:We establish general conditions under which Markov chains produced by the Hamiltonian Monte Carlo method will and will not be geometrically ergodic. We consider implementations with both position-independent and position-dependent integration times. In the former case, we find that the conditions for geometric ergodicity are essentially a gradient of the log-density which asymptotically points towards the centre of the space and grows no faster than linearly. In an idealised scenario in which the integration time is allowed to change in different regions of the space, we show that geometric ergodicity can be recovered for a much broader class of tail behaviours, leading to some guidelines for the choice of this free parameter in practice.
-
a conceptual introduction to Hamiltonian Monte Carlo
arXiv: Methodology, 2017Co-Authors: Michael BetancourtAbstract:Hamiltonian Monte Carlo has proven a remarkable empirical success, but only recently have we begun to develop a rigorous understanding of why it performs so well on difficult problems and how it is best applied in practice. Unfortunately, that understanding is confined within the mathematics of differential geometry which has limited its dissemination, especially to the applied communities for which it is particularly important. In this review I provide a comprehensive conceptual account of these theoretical foundations, focusing on developing a principled intuition behind the method and its optimal implementations rather of any exhaustive rigor. Whether a practitioner or a statistician, the dedicated reader will acquire a solid grasp of how Hamiltonian Monte Carlo works, when it succeeds, and, perhaps most importantly, when it fails.
-
identifying the optimal integration time in Hamiltonian Monte Carlo
arXiv: Methodology, 2016Co-Authors: Michael BetancourtAbstract:By leveraging the natural geometry of a smooth probabilistic system, Hamiltonian Monte Carlo yields computationally efficient Markov Chain Monte Carlo estimation. At least provided that the algorithm is sufficiently well-tuned. In this paper I show how the geometric foundations of Hamiltonian Monte Carlo implicitly identify the optimal choice of these parameters, especially the integration time. I then consider the practical consequences of these principles in both existing algorithms and a new implementation called \emph{Exhaustive Hamiltonian Monte Carlo} before demonstrating the utility of these ideas in some illustrative examples.
-
a general metric for riemannian manifold Hamiltonian Monte Carlo
International Conference on Geometric Science of Information, 2013Co-Authors: Michael BetancourtAbstract:Markov Chain Monte Carlo (MCMC) is an invaluable means of inference with complicated models, and Hamiltonian Monte Carlo, in particular Riemannian Manifold Hamiltonian Monte Carlo (RMHMC), has demonstrated success in many challenging problems. Current RMHMC implementations, however, rely on a Riemannian metric that limits their application. In this paper I propose a new metric for RMHMC without these limitations and verify its success on a distribution that emulates many hierarchical and latent models.
-
generalizing the no u turn sampler to riemannian manifolds
arXiv: Methodology, 2013Co-Authors: Michael BetancourtAbstract:Hamiltonian Monte Carlo provides efficient Markov transitions at the expense of introducing two free parameters: a step size and total integration time. Because the step size controls discretization error it can be readily tuned to achieve certain accuracy criteria, but the total integration time is left unconstrained. Recently Hoffman and Gelman proposed a criterion for tuning the integration time in certain systems with their No U-Turn Sampler, or NUTS. In this paper I investigate the dynamical basis for the success of NUTS and generalize it to Riemannian Manifold Hamiltonian Monte Carlo.
Ben Calderhead - One of the best experts on this subject based on the ideXlab platform.
-
riemann manifold langevin and Hamiltonian Monte Carlo methods
Journal of The Royal Statistical Society Series B-statistical Methodology, 2011Co-Authors: Mark Girolami, Ben CalderheadAbstract:The paper proposes Metropolis adjusted Langevin and Hamiltonian Monte Carlo sampling methods defined on the Riemann manifold to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. The methods provide fully automated adaptation mechanisms that circumvent the costly pilot runs that are required to tune proposal densities for Metropolis–Hastings or indeed Hamiltonian Monte Carlo and Metropolis adjusted Langevin algorithms. This allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the Markov chain. The methodology proposed exploits the Riemann geometry of the parameter space of statistical models and thus automatically adapts to the local structure when simulating paths across this manifold, providing highly efficient convergence and exploration of the target density. The performance of these Riemann manifold Monte Carlo methods is rigorously assessed by performing inference on logistic regression models, log-Gaussian Cox point processes, stochastic volatility models and Bayesian estimation of dynamic systems described by non-linear differential equations. Substantial improvements in the time-normalized effective sample size are reported when compared with alternative sampling approaches. MATLAB code that is available from http://www.ucl.ac.uk/statistics/research/rmhmc allows replication of all the results reported.
Liam Paninski - One of the best experts on this subject based on the ideXlab platform.
-
auxiliary variable exact Hamiltonian Monte Carlo samplers for binary distributions
Neural Information Processing Systems, 2013Co-Authors: Ari Pakman, Liam PaninskiAbstract:We present a new approach to sample from generic binary distributions, based on an exact Hamiltonian Monte Carlo algorithm applied to a piecewise continuous augmentation of the binary distribution of interest. An extension of this idea to distributions over mixtures of binary and possibly-truncated Gaussian or exponential variables allows us to sample from posteriors of linear and probit regression models with spike-and-slab priors and truncated parameters. We illustrate the advantages of these algorithms in several examples in which they outperform the Metropolis or Gibbs samplers.
-
exact Hamiltonian Monte Carlo for truncated multivariate gaussians
arXiv: Computation, 2012Co-Authors: Ari Pakman, Liam PaninskiAbstract:We present a Hamiltonian Monte Carlo algorithm to sample from multivariate Gaussian distributions in which the target space is constrained by linear and quadratic inequalities or products thereof. The Hamiltonian equations of motion can be integrated exactly and there are no parameters to tune. The algorithm mixes faster and is more efficient than Gibbs sampling. The runtime depends on the number and shape of the constraints but the algorithm is highly parallelizable. In many cases, we can exploit special structure in the covariance matrices of the untruncated Gaussian to further speed up the runtime. A simple extension of the algorithm permits sampling from distributions whose log-density is piecewise quadratic, as in the "Bayesian Lasso" model.
Juan Jose Murillofuentes - One of the best experts on this subject based on the ideXlab platform.
-
inference in deep gaussian processes using stochastic gradient Hamiltonian Monte Carlo
Neural Information Processing Systems, 2018Co-Authors: Marton Havasi, Jose Miguel Hernandezlobato, Juan Jose MurillofuentesAbstract:Deep Gaussian Processes (DGPs) are hierarchical generalizations of Gaussian Processes that combine well calibrated uncertainty estimates with the high flexibility of multilayer models. One of the biggest challenges with these models is that exact inference is intractable. The current state-of-the-art inference method, Variational Inference (VI), employs a Gaussian approximation to the posterior distribution. This can be a potentially poor unimodal approximation of the generally multimodal posterior. In this work, we provide evidence for the non-Gaussian nature of the posterior and we apply the Stochastic Gradient Hamiltonian Monte Carlo method to generate samples. To efficiently optimize the hyperparameters, we introduce the Moving Window MCEM algorithm. This results in significantly better predictions at a lower computational cost than its VI counterpart. Thus our method establishes a new state-of-the-art for inference in DGPs.
Vecchia, Claudio Dalla - One of the best experts on this subject based on the ideXlab platform.
-
Higher Order Hamiltonian Monte Carlo Sampling for Cosmological Large-Scale Structure Analysis
2021Co-Authors: Hernández-sánchez Mónica, Kitaura Francisco-shu, Ata Metin, Vecchia, Claudio DallaAbstract:We investigate higher order symplectic integration strategies within Bayesian cosmic density field reconstruction methods. In particular, we study the fourth-order discretisation of Hamiltonian equations of motion (EoM). This is achieved by recursively applying the basic second-order leap-frog scheme (considering the single evaluation of the EoM) in a combination of even numbers of forward time integration steps with a single intermediate backward step. This largely reduces the number of evaluations and random gradient computations, as required in the usual second-order case for high-dimensional cases. We restrict this study to the lognormal-Poisson model, applied to a full volume halo catalogue in real space on a cubical mesh of 1250 $h^{-1}$ Mpc side and 256$^3$ cells. Hence, we neglect selection effects, redshift space distortions, and displacements. We note that those observational and cosmic evolution effects can be accounted for in subsequent Gibbs-sampling steps within the COSMIC BIRTH algorithm. We find that going from the usual second to fourth-order in the leap-frog scheme shortens the burn-in phase by a factor of at least $\sim30$. This implies that 75-90 independent samples are obtained while the fastest second-order method converges. After convergence, the correlation lengths indicate an improvement factor of about 3.0 fewer gradient computations for meshes of 256$^3$ cells. In the considered cosmological scenario, the traditional leap-frog scheme turns out to outperform higher order integration schemes only at lower dimensional problems, e.g. meshes with 64$^3$ cells. This gain in computational efficiency can help to go towards a full Bayesian analysis of the cosmological large-scale structure for upcoming galaxy surveys.Comment: 19 pages, 12 figures, 4 tables, accepted at MNRAS, additional robust mathematical argument supported by numerical tests with longer HMC chains and a solid statistical analysi
-
Higher Order Hamiltonian Monte Carlo Sampling for Cosmological Large-Scale Structure Analysis
2020Co-Authors: Hernández-sánchez Mónica, Kitaura Francisco-shu, Ata Metin, Vecchia, Claudio DallaAbstract:We investigate higher order symplectic integration strategies within Bayesian cosmic density field reconstruction methods. In particular, we study the fourth-order discretisation of Hamiltonian equations of motion (EoM). This is achieved by recursively applying the basic second-order leap-frog scheme (considering the single evaluation of the EoM) in a combination of even numbers of forward time integration steps with a single intermediate backward step. This largely reduces the number of evaluations and random gradient computations, as required in the usual second-order case. We restrict this study to the lognormal-Poisson model, applied to a full volume halo catalogue in real space on a cubical mesh of 1250 $h^{-1}$ Mpc side and 256$^3$ cells. Hence, we neglect selection effects, redshift space distortions, and displacements. We note that those observational and cosmic evolution effects can be accounted for in subsequent Gibbs-sampling steps within the COSMIC BIRTH algorithm. We find that going from the usual second to fourth-order in the leap-frog scheme shortens the burn-in phase by a factor of at least $\sim30$. This implies that 80-100 independent samples are obtained while the fastest second-order method converges. After convergence, the correlation lengths indicate an improvement factor of $\sim2.4$ fewer gradient computations. This gain in computational efficiency can help to go towards a full Bayesian analysis of the cosmological large-scale structure for upcoming galaxy surveys.Comment: 17 pages, 10 figures, 4 table
