scholarly journals Hamiltonian Monte Carlo sampling to estimate past population dynamics using the skygrid coalescent model in a Bayesian phylogenetics framework

2020 ◽  
Vol 5 ◽  
pp. 53
Author(s):  
Guy Baele ◽  
Mandev S. Gill ◽  
Philippe Lemey ◽  
Marc A. Suchard
Keyword(s):  
Monte Carlo ◽  
Population Dynamics ◽  
Monte Carlo Sampling ◽  
Estimation Methods ◽  
Model Parameters ◽  
Effective Sample Size ◽  
Hamiltonian Monte Carlo ◽  
Gradient Based ◽  
Past Population ◽  
Over Time

Nonparametric coalescent-based models are often employed to infer past population dynamics over time. Several of these models, such as the skyride and skygrid models, are equipped with a block-updating Markov chain Monte Carlo sampling scheme to efficiently estimate model parameters. The advent of powerful computational hardware along with the use of high-performance libraries for statistical phylogenetics has, however, made the development of alternative estimation methods feasible. We here present the implementation and performance assessment of a Hamiltonian Monte Carlo gradient-based sampler to infer the parameters of the skygrid model. The skygrid is a popular and flexible coalescent-based model for estimating population dynamics over time and is available in BEAST 1.10.5, a widely-used software package for Bayesian pylogenetic and phylodynamic analysis. Taking into account the increased computational cost of gradient evaluation, we report substantial increases in effective sample size per time unit compared to the established block-updating sampler. We expect gradient-based samplers to assume an increasingly important role for different classes of parameters typically estimated in Bayesian phylogenetic and phylodynamic analyses.

HMCtomo: A framework for Hamiltonian Monte Carlo sampling of Bayesian geophysical inverse problems

2020 ◽  
Author(s):  
Lars Gebraad ◽  
Andrea Zunino ◽  
Andreas Fichtner ◽  
Klaus Mosegaard
Keyword(s):  
Monte Carlo ◽  
Bayesian Inference ◽  
Inverse Problems ◽  
Large Scale ◽  
Monte Carlo Sampling ◽  
Hamiltonian Monte Carlo ◽  
Forward Models ◽  
Inference Problems ◽  
Wide Range ◽  
Gradient Based

<div>We present a framework to solve geophysical inverse problems using the Hamiltonian Monte Carlo (HMC) method, with a focus on Bayesian tomography. Recent work in the geophysical community has shown the potential for gradient-based Monte Carlo sampling for a wide range of inverse problems across several fields.</div><div> </div><div>Many high-dimensional (non-linear) problems in geophysics have readily accessible gradient information which is unused in classical probabilistic inversions. Using HMC is a way to help improve traditional Monte Carlo sampling while increasing the scalability of inference problems, allowing access to uncertainty quantification for problems with many free parameters (>10'000). The result of HMC sampling is a collection of models representing the posterior probability density function, from which not only "best" models can be inferred, but also uncertainties and potentially different plausible scenarios, all compatible with the observed data. However, the amount of tuning parameters required by HMC, as well as the complexity of existing statistical modeling software, has limited the geophysical community in widely adopting a specific tool for performing efficient large-scale Bayesian inference.</div><div> </div><div>This work attempts to make a step towards filling that gap by providing an HMC sampler tailored for geophysical inverse problems (by e.g. supplying relevant priors and visualizations) combined with a set of different forward models, ranging from elastic and acoustic wave propagation to magnetic anomaly modeling, traveltimes, etc.. The framework is coded in the didactic but performant languages Julia and Python, with the possibility for the user to combine their own forward models, which are linked to the sampler routines by proper interfaces. In this way, we hope to illustrate the usefulness and potential of HMC in Bayesian inference. Tutorials featuring an array of physical experiments are written with the aim of both showcasing Bayesian inference and successful HMC usage. It additionally includes examples on how to speed up HMC e.g. with automated tuning techniques and GPU computations.</div>

PSV-14 Using Bayesian Hamiltonian Monte Carlo and a nonlinear model for the estimation of genetic parameters for lactation curves in goats

2021 ◽  
Vol 99 (Supplement_3) ◽  
pp. 305-307
Author(s):  
Andre C Araujo ◽  
Leonardo Gloria ◽  
Paulo Abreu ◽  
Fabyano Silva ◽  
Marcelo Rodrigues ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Milk Yield ◽  
Nonlinear Model ◽  
Genetic Parameters ◽  
Model Parameters ◽  
Promising Alternative ◽  
Hamiltonian Monte Carlo ◽  
Heritability Estimates ◽  
Correlated Parameters ◽  
Wood Model

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.

Parametric uncertainty quantification in the Rothermel model with randomised quasi-Monte Carlo methods

2015 ◽  
Vol 24 (3) ◽  
pp. 307 ◽  
Author(s):  
Yaning Liu ◽  
Edwin Jimenez ◽  
M. Yousuff Hussaini ◽  
Giray Ökten ◽  
Scott Goodrick
Keyword(s):  
Monte Carlo ◽  
Uncertainty Quantification ◽  
Monte Carlo Methods ◽  
Wildland Fire ◽  
Parametric Uncertainty ◽  
Monte Carlo Sampling ◽  
Model Parameters ◽  
Quasi Monte Carlo ◽  
Control Variate ◽  
Fuel Model

Rothermel's wildland surface fire model is a popular model used in wildland fire management. The original model has a large number of parameters, making uncertainty quantification challenging. In this paper, we use variance-based global sensitivity analysis to reduce the number of model parameters, and apply randomised quasi-Monte Carlo methods to quantify parametric uncertainties for the reduced model. The Monte Carlo estimator used in these calculations is based on a control variate approach applied to the sensitivity derivative enhanced sampling. The chaparral fuel model, selected from Rothermel's 11 original fuel models, is studied as an example. We obtain numerical results that improve the crude Monte Carlo sampling by factors as high as three orders of magnitude.

Seismic source inversion using Hamiltonian Monte Carlo and a 3-D Earth model in the Japanese Islands

2020 ◽  
Author(s):  
Saulė Simutė ◽  
Lion Krischer ◽  
Christian Boehm ◽  
Martin Vallée ◽  
Andreas Fichtner
Keyword(s):  
Monte Carlo ◽  
Moment Tensor ◽  
Seismic Source ◽  
Monte Carlo Algorithm ◽  
Earth Model ◽  
Model Parameters ◽  
Hamiltonian Monte Carlo ◽  
Double Couple ◽  
Full Moment Tensor ◽  
Japanese Islands

<p>We present a proof-of-concept catalogue of full-waveform seismic source solutions for the Japanese Islands area. Our method is based on the Bayesian inference of source parameters and a tomographically derived heterogeneous Earth model, used to compute Green’s strain tensors. We infer the full moment tensor, location and centroid time of the seismic events in the study area.</p><p>To compute spatial derivatives of Green’s functions, we use a previously derived regional Earth model (Simutė et al., 2016). The model is radially anisotropic, visco-elastic, and fully heterogeneous. It was constructed using full waveforms in the period band of 15–80 s.</p><p>Green’s strains are computed numerically with the spectral-element solver SES3D (Gokhberg & Fichtner, 2016). We exploit reciprocity, and by treating seismic stations as virtual sources we compute and store the wavefield across the domain. This gives us a strain database for all potential source-receiver pairs. We store the wavefield for more than 50 F-net broadband stations (www.fnet.bosai.go.jp). By assuming an impulse response as the source time function, the displacements are then promptly obtained by linear combination of the pre-computed strains scaled by the moment tensor elements.</p><p>With a feasible number of model parameters and the fast forward problem we infer the unknowns in a Bayesian framework. The fully probabilistic approach allows us to obtain uncertainty information as well as inter-parameter trade-offs. The sampling is performed with a variant of the Hamiltonian Monte Carlo algorithm, which we developed previously (Fichtner and Simutė, 2017). We apply an L2 misfit on waveform data, and we work in the period band of 15–80 s.</p><p>We jointly infer three location parameters, timing and moment tensor components. We present two sets of source solutions: 1) full moment tensor solutions, where the trace is free to vary away from zero, and 2) moment tensor solutions with the isotropic part constrained to be zero. In particular, we study events with significant non-double-couple component. Preliminary results of ~Mw 5 shallow to intermediate depth events indicate that proper incorporation of 3-D Earth structure results in solutions becoming more double-couple like. We also find that improving the Global CMT solutions in terms of waveform fit requires a very good 3-D Earth model and is not trivial.</p>

scholarly journals Robust and fast Monte Carlo Markov Chain sampling of diffusion MRI microstructure models

2018 ◽  
Author(s):  
R.L. Harms ◽  
A. Roebroeck
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Sample Size ◽  
Diffusion Mri ◽  
Monte Carlo Markov Chain ◽  
Point Estimate ◽  
Model Parameters ◽  
Effective Sample Size ◽  
Compartment Models ◽  
Mcmc Sampling

AbstractIn diffusion MRI analysis, advances in biophysical multi-compartment modeling have gained popularity over the conventional Diffusion Tensor Imaging (DTI), because they possess greater specificity in relating the dMRI signal to underlying cellular microstructure. Biophysical multi-compartment models require parameter estimation, typically performed using either Maximum Likelihood Estimation (MLE) or using Monte Carlo Markov Chain (MCMC) sampling. Whereas MLE provides only a point estimate of the fitted model parameters, MCMC recovers the entire posterior distribution of the model parameters given the data, providing additional information such as parameter uncertainty and correlations. MCMC sampling is currently not routinely applied in dMRI microstructure modeling because it requires adjustments and tuning specific to each model, particularly in the choice of proposal distributions, burn-in length, thinning and the number of samples to store. In addition, sampling often takes at least an order of magnitude more time than non-linear optimization. Here we investigate the performance of MCMC algorithm variations over multiple popular diffusion microstructure models to see whether a single well performing variation could be applied efficiently and robustly to many models. Using an efficient GPU-based implementation, we show that run times can be removed as a prohibitive constraint for sampling of diffusion multi-compartment models. Using this implementation, we investigated the effectiveness of different adaptive MCMC algorithms, burn-in, initialization and thinning. Finally we apply the theory of Effective Sample Size to diffusion multi-compartment models as a way of determining a relatively general target for the number of samples needed to characterize parameter distributions for different models and datasets. We conclude that robust and fast sampling is achieved in most diffusion microstructure models with the Adaptive Metropolis-Within-Gibbs (AMWG) algorithm initialized with an MLE point estimate, in which case 100 to 200 samples are sufficient as a burn-in and thinning is mostly unnecessary. As a relatively general target for the number of samples, we recommend a multivariate Effective Sample Size of 2200.

scholarly journals Symmetrically Processed Splitting Integrators for Enhanced Hamiltonian Monte Carlo Sampling

2021 ◽  
Vol 43 (5) ◽  
pp. A3357-A3371
Author(s):  
S. Blanes ◽  
M. P. Calvo ◽  
F. Casas ◽  
J. M. Sanz-Serna
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Sampling ◽  
Hamiltonian Monte Carlo

Transdimensional seismic inversion using the reversible jump Hamiltonian Monte Carlo algorithm

Geophysics ◽  
2017 ◽  
Vol 82 (3) ◽  
pp. R119-R134 ◽  
Author(s):  
Mrinal K. Sen ◽  
Reetam Biswas
Keyword(s):  
Monte Carlo ◽  
A Priori ◽  
Seismic Inversion ◽  
Monte Carlo Algorithm ◽  
Model Parameters ◽  
Reversible Jump ◽  
Step Method ◽  
Acceptance Probability ◽  
Hamiltonian Monte Carlo ◽  
Impedance Inversion

Prestack or angle stack gathers are inverted to estimate pseudologs at every surface location for building reservoir models. Recently, several methods have been proposed to increase the resolution of the inverted models. All of these methods, however, require that the total number of model parameters be fixed a priori. We have investigated an alternate approach in which we allow the data themselves to choose model parameterization. In other words, in addition to the layer properties, the number of layers is also treated as a variable in our formulation. Such transdimensional inverse problems are generally solved by using the reversible jump Markov chain Monte Carlo (RJMCMC) approach, which is a tool for model exploration and uncertainty quantification. This method, however, has very low acceptance. We have developed a two-step method by combining RJMCMC with a fixed-dimensional MCMC called Hamiltonian Monte Carlo, which makes use of gradient information to take large steps. Acceptance probability for such a transition is also derived. We call this new method “reversible jump Hamiltonian Monte Carlo (RJHMC).” We have applied this technique to poststack acoustic impedance inversion and to prestack (angle stack) AVA inversion for estimating acoustic and shear impedance profiles. We have determined that the marginal posteriors estimated by RJMCMC and RJHMC are in good agreement. Our results demonstrate that RJHMC converges faster than RJMCMC, and it therefore can be a practical tool for inverting seismic data when the gradient can be computed efficiently.

scholarly journals Emulation-accelerated Hamiltonian Monte Carlo algorithms for parameter estimation and uncertainty quantification in differential equation models

2021 ◽  
Vol 32 (1) ◽  
Author(s):  
L. Mihaela Paun ◽  
Dirk Husmeier
Keyword(s):  
Monte Carlo ◽  
Sample Size ◽  
Posterior Distribution ◽  
Detailed Balance ◽  
Evaluation Study ◽  
Forward Model ◽  
Effective Sample Size ◽  
Hamiltonian Monte Carlo ◽  
Monte Carlo Algorithms ◽  
Efficiency Measures

AbstractWe propose to accelerate Hamiltonian and Lagrangian Monte Carlo algorithms by coupling them with Gaussian processes for emulation of the log unnormalised posterior distribution. We provide proofs of detailed balance with respect to the exact posterior distribution for these algorithms, and validate the correctness of the samplers’ implementation by Geweke consistency tests. We implement these algorithms in a delayed acceptance (DA) framework, and investigate whether the DA scheme can offer computational gains over the standard algorithms. A comparative evaluation study is carried out to assess the performance of the methods on a series of models described by differential equations, including a real-world application of a 1D fluid-dynamics model of the pulmonary blood circulation. The aim is to identify the algorithm which gives the best trade-off between accuracy and computational efficiency, to be used in nonlinear DE models, which are computationally onerous due to repeated numerical integrations in a Bayesian analysis. Results showed no advantage of the DA scheme over the standard algorithms with respect to several efficiency measures based on the effective sample size for most methods and DE models considered. These gradient-driven algorithms register a high acceptance rate, thus the number of expensive forward model evaluations is not significantly reduced by the first emulator-based stage of DA. Additionally, the Lagrangian Dynamical Monte Carlo and Riemann Manifold Hamiltonian Monte Carlo tended to register the highest efficiency (in terms of effective sample size normalised by the number of forward model evaluations), followed by the Hamiltonian Monte Carlo, and the No U-turn sampler tended to be the least efficient.

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