Sequential Monte Carlo Samplers with Independent Markov Chain Monte Carlo Proposals

L. F. South; A. N. Pettitt; C. C. Drovandi

doi:10.1214/18-ba1129

A sequential Bayesian approach for the estimation of the age–depth relationship of Dome Fuji ice core

Nonlinear Processes in Geophysics Discussions ◽

10.5194/npgd-2-939-2015 ◽

2015 ◽

Vol 2 (3) ◽

pp. 939-968

Author(s):

S. Nakano ◽

K. Suzuki ◽

K. Kawamura ◽

F. Parrenin ◽

T. Higuchi

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Sequential Monte Carlo ◽

Ice Core ◽

Posterior Distributions ◽

Metropolis Method ◽

Depth Relationship ◽

Relationship Of ◽

Thinning Process

Abstract. A technique for estimating the age–depth relationship in an ice core and evaluating its uncertainty is presented. The age–depth relationship is mainly determined by the accumulation of snow at the site of the ice core and the thinning process due to the horizontal stretching and vertical compression of ice layers. However, since neither the accumulation process nor the thinning process are fully understood, it is essential to incorporate observational information into a model that describes the accumulation and thinning processes. In the proposed technique, the age as a function of depth is estimated from age markers and δ18O data. The estimation is achieved using the particle Markov chain Monte Carlo (PMCMC) method, in which the sequential Monte Carlo (SMC) method is combined with the Markov chain Monte Carlo method. In this hybrid method, the posterior distributions for the parameters in the models for the accumulation and thinning processes are computed using the Metropolis method, in which the likelihood is obtained with the SMC method. Meanwhile, the posterior distribution for the age as a function of depth is obtained by collecting the samples generated by the SMC method with Metropolis iterations. The use of this PMCMC method enables us to estimate the age–depth relationship without assuming either linearity or Gaussianity. The performance of the proposed technique is demonstrated by applying it to ice core data from Dome Fuji in Antarctica.

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Limit theorems for sequential MCMC methods

Advances in Applied Probability ◽

10.1017/apr.2020.9 ◽

2020 ◽

Vol 52 (2) ◽

pp. 377-403 ◽

Cited By ~ 2

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.

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A comparison of approximate versus exact techniques for Bayesian parameter inference in nonlinear ordinary differential equation models

Royal Society Open Science ◽

10.1098/rsos.191315 ◽

2020 ◽

Vol 7 (3) ◽

pp. 191315

Author(s):

Amani A. Alahmadi ◽

Jennifer A. Flegg ◽

Davis G. Cochrane ◽

Christopher C. Drovandi ◽

Jonathan M. Keith

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Bayesian Inference ◽

Sequential Monte Carlo ◽

Simulated Data ◽

Real Data ◽

Nonlinear Ordinary Differential Equation ◽

Unknown Parameters ◽

Acceptance Probability

The behaviour of many processes in science and engineering can be accurately described by dynamical system models consisting of a set of ordinary differential equations (ODEs). Often these models have several unknown parameters that are difficult to estimate from experimental data, in which case Bayesian inference can be a useful tool. In principle, exact Bayesian inference using Markov chain Monte Carlo (MCMC) techniques is possible; however, in practice, such methods may suffer from slow convergence and poor mixing. To address this problem, several approaches based on approximate Bayesian computation (ABC) have been introduced, including Markov chain Monte Carlo ABC (MCMC ABC) and sequential Monte Carlo ABC (SMC ABC). While the system of ODEs describes the underlying process that generates the data, the observed measurements invariably include errors. In this paper, we argue that several popular ABC approaches fail to adequately model these errors because the acceptance probability depends on the choice of the discrepancy function and the tolerance without any consideration of the error term. We observe that the so-called posterior distributions derived from such methods do not accurately reflect the epistemic uncertainties in parameter values. Moreover, we demonstrate that these methods provide minimal computational advantages over exact Bayesian methods when applied to two ODE epidemiological models with simulated data and one with real data concerning malaria transmission in Afghanistan.

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Parallelised Bayesian optimisation for deep learning

10.12681/eadd/50829 ◽

2018 ◽

Author(s):

Λυκούργος Κεκεμπάνος

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Deep Learning ◽

Markov Chain Monte Carlo ◽

Importance Sampling ◽

Message Passing ◽

Message Passing Interface ◽

Sequential Monte Carlo ◽

Particle Filters

Η εκπαίδευση σε βαθιά νευρωνικά δίκτυα (ΒΝΔ) είναι μια απαραίτητη διαδικασία στη μηχανική μάθηση. Η διαδικασία εκπαίδευσης των ΒΝΔ στοχεύει στη βελτιστοποίηση των τιμών των παραμέτρων του δικτύου, που συχνά βασίζεται στην παράγωγο των λογαριθμικών πιθανοτήτων των παραμέτρων. Ως εκ τούτου, είναι πολύ πιθανό η διαδικασία βελτιστοποίησης να βρει τοπικές βέλτιστες τιμές αντί για καθολικές. Επιπλέον, οι συμβατικές προσεγγίσεις που χρησιμοποιούνται για αυτή τη διαδικασία, όπως οι μέθοδοι Μαρκοβιανής αλυσίδας Μόντε Κάρλο, όχι μόνο προσφέρουν μη βέλτιστη απόδοση χρόνου εκτέλεσης, αλλά επίσης αποτρέπουν την αποτελεσματική παραλληλοποίηση λόγω εγγενών εξαρτήσεων στη διαδικασία. Σε αυτή τη διατριβή, εξετάζουμε μια εναλλακτική προσέγγιση στις μεθόδους Μαρκοβιανής αλυσίδας Μόντε Κάρλο (Markov Chain Monte Carlo, MCMC), τον δειγματολήπτη ακολουθιακών Μόντε Κάρλο (Sequential Monte Carlo, SMC), ο οποίος γενικεύει τα φίλτρα σωματιδίων (particle filters). Πιο συγκεκριμένα, η διατριβή εστιάζει στη βελτίωση της απόδοσης και της ακρίβειας των μεθόδων SMC, ιδιαίτερα στο πλαίσιο της πλήρους Μπεϋζιανής μάθησης. Σε αυτό το πλαίσιο, η διατριβή προτείνει μια νέα μέθοδο εκπαίδευσης νευρωνικών δικτύων χρησιμοποιώντας τις μεθόδους σημαντικής δειγματοληψίας (μέθοδος importance sampling) και επαναδειγματοληψίας. Η αρχική σύγκριση των δύο μεθόδων αποκαλύπτει ότι η προτεινόμενη μεθοδολογία είναι χειρότερη τόσο στην ακρίβεια όσο και στην απόδοση. Αυτό οδήγησε την έρευνα να επικεντρωθεί στην βελτίωση της απόδοσης και ακρίβειας της προτεινόμενης μεθοδολογίας. Η ανάλυση απόδοσης ξεκίνησε με την εφαρμογή μιας νέας προτεινόμενης, παράλληλης και πλήρως κατανεμημένης μεθοδολογίας επαναδειγματοληψίας, με βελτιωμένη χρονική πολυπλοκότητα από την αρχική προσέγγιση χρησιμοποιώντας δύο πλαίσια MapReduce, το Hadoop και το Spark. Τα αποτελέσματα δείχνουν ότι το Spark είναι έως και 25 φορές ταχύτερο από το Hadoop, ενώ στο Spark η νέα προτεινόμενη μεθοδολογία είναι έως και 10 φορές ταχύτερη από την αρχική μέθοδο. Ωστόσο, παρατηρείται ότι η εφαρμογή του ίδιου αλγορίθμου στο Message Passing Interface (MPI) παρέχει σημαντικά καλύτερους χρόνους εκτέλεσης και είναι πιο κατάλληλος για τον προτεινόμενο αλγόριθμο. Η ανάλυση ακρίβειας ξεκίνησε με πειράματα που δείχνουν ότι ο βασικός δειγματολήπτης SMC παρέχει χειρότερη ακρίβεια από τους εναλλακτικούς ή ανταγωνιστικούς αλγόριθμους MCMC. Τρεις διαφορετικές στρατηγικές εφαρμόζονται στον βασικό δειγματολήπτη SMC παρέχοντας καλύτερη ακρίβεια. Η ανάλυση επεκτείνεται για να συμπεριλάβει ανταγωνιστικούς αλγόριθμους. Η εξαντλητική αξιολόγηση δείχνει ότι η προτεινόμενη προσέγγιση προσφέρει ανώτερη απόδοση και ακρίβεια.

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A sequential Bayesian approach for the estimation of the age–depth relationship of the Dome Fuji ice core

Nonlinear Processes in Geophysics ◽

10.5194/npg-23-31-2016 ◽

2016 ◽

Vol 23 (1) ◽

pp. 31-44 ◽

Cited By ~ 2

Author(s):

Shin'ya Nakano ◽

Kazue Suzuki ◽

Kenji Kawamura ◽

Frédéric Parrenin ◽

Tomoyuki Higuchi

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Sequential Monte Carlo ◽

Accumulation Rate ◽

Ice Core ◽

Metropolis Method ◽

Depth Relationship ◽

Relationship Of ◽

Thinning Process

Abstract. A technique for estimating the age–depth relationship in an ice core and evaluating its uncertainty is presented. The age–depth relationship is determined by the accumulation of snow at the site of the ice core and the thinning process as a result of the deformation of ice layers. However, since neither the accumulation rate nor the thinning process is fully known, it is essential to incorporate observational information into a model that describes the accumulation and thinning processes. In the proposed technique, the age as a function of depth is estimated by making use of age markers and δ18O data. The age markers provide reliable age information at several depths. The data of δ18O are used as a proxy of the temperature for estimating the accumulation rate. The estimation is achieved using the particle Markov chain Monte Carlo (PMCMC) method, which is a combination of the sequential Monte Carlo (SMC) method and the Markov chain Monte Carlo method. In this hybrid method, the posterior distributions for the parameters in the models for the accumulation and thinning process are computed using the Metropolis method, in which the likelihood is obtained with the SMC method, and the posterior distribution for the age as a function of depth is obtained by collecting the samples generated by the SMC method with Metropolis iterations. The use of this PMCMC method enables us to estimate the age–depth relationship without assuming either linearity or Gaussianity. The performance of the proposed technique is demonstrated by applying it to ice core data from Dome Fuji in Antarctica.

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