A Practical Sequential Stopping Rule for High-Dimensional Markov Chain Monte Carlo

2016 ◽  
Vol 25 (3) ◽  
pp. 684-700 ◽  
Author(s):  
Lei Gong ◽  
James M. Flegal
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Stopping Rule ◽  
High Dimensional

scholarly journals Efficiency and robustness in Monte Carlo sampling of 3-D geophysical inversions with Obsidian v0.1.2: Setting up for success

2019 ◽  
Author(s):  
Richard Scalzo ◽  
David Kohn ◽  
Hugo Olierook ◽  
Gregory Houseman ◽  
Rohitash Chandra ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Adaptive Sampling ◽  
Sampling Methods ◽  
South Australia ◽  
Monte Carlo Sampling ◽  
Dimensional Structure ◽  
High Dimensional ◽  
Dimensional Parameter

Abstract. The rigorous quantification of uncertainty in geophysical inversions is a challenging problem. Inversions are often ill-posed and the likelihood surface may be multimodal; properties of any single mode become inadequate uncertainty measures, and sampling methods become inefficient for irregular posteriors or high-dimensional parameter spaces. We explore the influences of different choices made by the practitioner on the efficiency and accuracy of Bayesian geophysical inversion methods that rely on Markov chain Monte Carlo sampling to assess uncertainty, using a multi-sensor inversion of the three-dimensional structure and composition of a region in the Cooper Basin of South Australia as a case study. The inversion is performed using an updated version of the Obsidian distributed inversion software. We find that the posterior for this inversion has complex local covariance structure, hindering the efficiency of adaptive sampling methods that adjust the proposal based on the chain history. Within the context of a parallel-tempered Markov chain Monte Carlo scheme for exploring high-dimensional multi-modal posteriors, a preconditioned Crank-Nicholson proposal outperforms more conventional forms of random walk. Aspects of the problem setup, such as priors on petrophysics or on 3-D geological structure, affect the shape and separation of posterior modes, influencing sampling performance as well as the inversion results. Use of uninformative priors on sensor noise can improve inversion results by enabling optimal weighting among multiple sensors even if noise levels are uncertain. Efficiency could be further increased by using posterior gradient information within proposals, which Obsidian does not currently support, but which could be emulated using posterior surrogates.

scholarly journals Integration with an adaptive harmonic mean algorithm

2020 ◽  
Vol 35 (24) ◽  
pp. 1950142
Author(s):  
Allen Caldwell ◽  
Philipp Eller ◽  
Vasyl Hafych ◽  
Rafael Schick ◽  
Oliver Schulz ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Probability Distribution ◽  
Marginal Likelihood ◽  
Harmonic Mean ◽  
High Dimensional ◽  
Test Cases ◽  
Multiple Test ◽  
Mathematical Properties

Numerically estimating the integral of functions in high dimensional spaces is a nontrivial task. A oft-encountered example is the calculation of the marginal likelihood in Bayesian inference, in a context where a sampling algorithm such as a Markov Chain Monte Carlo provides samples of the function. We present an Adaptive Harmonic Mean Integration (AHMI) algorithm. Given samples drawn according to a probability distribution proportional to the function, the algorithm will estimate the integral of the function and the uncertainty of the estimate by applying a harmonic mean estimator to adaptively chosen regions of the parameter space. We describe the algorithm and its mathematical properties, and report the results using it on multiple test cases.

scholarly journals Efficient posterior sampling for high-dimensional imbalanced logistic regression

Biometrika ◽  
2020 ◽  
Vol 107 (4) ◽  
pp. 1005-1012 ◽  
Author(s):  
Deborshee Sen ◽  
Matthias Sachs ◽  
Jianfeng Lu ◽  
David B Dunson
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Imbalanced Data ◽  
Simulated Data ◽  
Computational Time ◽  
High Dimensional ◽  
Cancer Data ◽  
Monte Carlo Algorithms ◽  
Using Data

Summary Classification with high-dimensional data is of widespread interest and often involves dealing with imbalanced data. Bayesian classification approaches are hampered by the fact that current Markov chain Monte Carlo algorithms for posterior computation become inefficient as the number $p$ of predictors or the number $n$ of subjects to classify gets large, because of the increasing computational time per step and worsening mixing rates. One strategy is to employ a gradient-based sampler to improve mixing while using data subsamples to reduce the per-step computational complexity. However, the usual subsampling breaks down when applied to imbalanced data. Instead, we generalize piecewise-deterministic Markov chain Monte Carlo algorithms to include importance-weighted and mini-batch subsampling. These maintain the correct stationary distribution with arbitrarily small subsamples and substantially outperform current competitors. We provide theoretical support for the proposed approach and demonstrate its performance gains in simulated data examples and an application to cancer data.

Markov-Chain Monte-Carlo methods and non-identifiabilities

2018 ◽  
Vol 24 (3) ◽  
pp. 203-214 ◽  
Author(s):  
Christian Müller ◽  
Fabian Weysser ◽  
Thomas Mrziglod ◽  
Andreas Schuppert
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Likelihood Function ◽  
Quality Criterion ◽  
Quality Criteria ◽  
High Dimensional ◽  
Mcmc Methods ◽  
Effective Sample Size ◽  
Sample Quality

Abstract We consider the problem of sampling from high-dimensional likelihood functions with large amounts of non-identifiabilities via Markov-Chain Monte-Carlo algorithms. Non-identifiabilities are problematic for commonly used proposal densities, leading to a low effective sample size. To address this problem, we introduce a regularization method using an artificial prior, which restricts non-identifiable parts of the likelihood function. This enables us to sample the posterior using common MCMC methods more efficiently. We demonstrate this with three MCMC methods on a likelihood based on a complex, high-dimensional blood coagulation model and a single series of measurements. By using the approximation of the artificial prior for the non-identifiable directions, we obtain a sample quality criterion. Unlike other sample quality criteria, it is valid even for short chain lengths. We use the criterion to compare the following three MCMC variants: The Random Walk Metropolis Hastings, the Adaptive Metropolis Hastings and the Metropolis adjusted Langevin algorithm.

Motion Redirection Based on Markov Chain Monte Carlo Particle Filter

2014 ◽  
Vol 668-669 ◽  
pp. 1086-1089
Author(s):  
Jin Bao Song ◽  
Long Ye ◽  
Qin Zhang ◽  
Jian Ping Chai
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Particle Filter ◽  
Monte Carlo Sampling ◽  
Space Time ◽  
High Dimensional ◽  
Observation Model ◽  
Key Frame ◽  
The Subject

This paper aims at the difficulty that lack of observation model and high-dimensional sampling in video tooning, proposes a method based on key frame matching and dual-directional Markov chain Monte Carlo sampling of video motion redirection. At first, after extracting the key frame of a given video, By affine transformation and linear superposition, the subject initializes the video’s space-time parameters and forms the observation model; Secondly, in each space-time, based on the bi-directional Markov property of each frame, This paper proposed a dual-directional Markov chain Monte Carlo sampling particle filter structure and takes full advantage of the relationship of the front and back frame of the parameters to estimate motion redirection parameters. At the same time, for high-dimensional sampling problem, the subject according to the directional parameters’ correlation implements classification of skeleton parameters-morphological parameters-physical parameters, proposes a hierarchical genetic strategy to optimize the output parameters and improves the efficiency of the algorithm. The research of this paper will produce an efficient and prominent animation expressive video motion redirection method and play an important role on video animation of the development.

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