scholarly journals Efficient strategy for the Markov chain Monte Carlo in high-dimension with heavy-tailed target probability distribution

Bernoulli ◽  
2018 ◽  
Vol 24 (4B) ◽  
pp. 3711-3750 ◽  
Author(s):  
Kengo Kamatani
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Probability Distribution ◽  
High Dimension ◽  
Efficient Strategy ◽  
Target Probability ◽  
Heavy Tailed

scholarly journals Measuring Stellar Radial Velocity using Markov Chain Monte Carlo(MCMC) Method

2013 ◽  
Vol 9 (S298) ◽  
pp. 441-441
Author(s):  
Yihan Song ◽  
Ali Luo ◽  
Yongheng Zhao
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Probability Distribution ◽  
Radial Velocity ◽  
Mcmc Methods ◽  
Mcmc Method ◽  
Stellar Spectra ◽  
Mcmc Simulation ◽  
Template Fitting

AbstractStellar radial velocity is estimated by using template fitting and Markov Chain Monte Carlo(MCMC) methods. This method works on the LAMOST stellar spectra. The MCMC simulation generates a probability distribution of the RV. The RV error can also computed from distribution.

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.

Introduction to Simulation and MCMC Methods

2011 ◽  
pp. 182-217 ◽  
Author(s):  
Siddhartha Chib
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Marginal Likelihood ◽  
Mcmc Methods ◽  
Posterior Distributions ◽  
Extensive Discussion ◽  
Block Algorithm ◽  
Target Probability ◽  
Acceptance Condition

The purpose of this article is to provide an overview of Monte Carlo methods for generating variates from a target probability distribution that are based on Markov chains. These methods, called Markov chain Monte Carlo (MCMC) methods, are widely used to summarize complicated posterior distributions in Bayesian statistics and econometrics. This article begins with an intuitive explanation of the ideas and concepts that underlie popular algorithms such as the Metropolis-Hastings algorithm and multi-block algorithm. It provides the concept of a source or proposal density, which is used to supply a randomization step or an acceptance condition to determine if the candidate draw should be accepted. It is important to assess the performance of the sampling algorithm to determine the rate of mixing. Finally, this article offers an extensive discussion of marginal likelihood calculation using posterior simulator output.

scholarly journals Joining forces of Bayesian and frequentist methodology: a study for inference in the presence of non-identifiability

2013 ◽  
Vol 371 (1984) ◽  
pp. 20110544 ◽  
Author(s):  
Andreas Raue ◽  
Clemens Kreutz ◽  
Fabian Joachim Theis ◽  
Jens Timmer
Keyword(s):  
Experimental Data ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Probability Distribution ◽  
Cell Biology ◽  
Posterior Probability ◽  
Monte Carlo Sampling ◽  
Posterior Probability Distribution ◽  
Model Predictions

Increasingly complex applications involve large datasets in combination with nonlinear and high-dimensional mathematical models. In this context, statistical inference is a challenging issue that calls for pragmatic approaches that take advantage of both Bayesian and frequentist methods. The elegance of Bayesian methodology is founded in the propagation of information content provided by experimental data and prior assumptions to the posterior probability distribution of model predictions. However, for complex applications, experimental data and prior assumptions potentially constrain the posterior probability distribution insufficiently. In these situations, Bayesian Markov chain Monte Carlo sampling can be infeasible. From a frequentist point of view, insufficient experimental data and prior assumptions can be interpreted as non-identifiability. The profile-likelihood approach offers to detect and to resolve non-identifiability by experimental design iteratively. Therefore, it allows one to better constrain the posterior probability distribution until Markov chain Monte Carlo sampling can be used securely. Using an application from cell biology, we compare both methods and show that a successive application of the two methods facilitates a realistic assessment of uncertainty in model predictions.

scholarly journals Another look at the treatment of data uncertainty in Markov chain Monte Carlo inversion and other probabilistic methods

2020 ◽  
Vol 222 (1) ◽  
pp. 388-405
Author(s):  
F J Tilmann ◽  
H Sadeghisorkhani ◽  
A Mauerberger
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Standard Deviation ◽  
Measurement Error ◽  
Probability Distribution ◽  
Unknown Parameter ◽  
Model Space ◽  
Data Uncertainty ◽  
Uncertainty Distribution

SUMMARY In probabilistic Bayesian inversions, data uncertainty is a crucial parameter for quantifying the uncertainties and correlations of the resulting model parameters or, in transdimensional approaches, even the complexity of the model. However, in many geophysical inference problems it is poorly known. Therefore, it is common practice to allow the data uncertainty itself to be a parameter to be determined. Although in principle any arbitrary uncertainty distribution can be assumed, Gaussian distributions whose standard deviation is then the unknown parameter to be estimated are the usual choice. In this special case, the paper demonstrates that a simple analytical integration is sufficient to marginalise out this uncertainty parameter, reducing the complexity of the model space without compromising the accuracy of the posterior model probability distribution. However, it is well known that the distribution of geophysical measurement errors, although superficially similar to a Gaussian distribution, typically contains more frequent samples along the tail of the distribution, so-called outliers. In linearized inversions these are often removed in subsequent iterations based on some threshold criterion, but in Markov chain Monte Carlo (McMC) inversions this approach is not possible as they rely on the likelihood ratios, which cannot be formed if the number of data points varies between the steps of the Markov chain. The flexibility to define the data error probability distribution in McMC can be exploited in order to account for this pattern of uncertainties in a natural way, without having to make arbitrary choices regarding residual thresholds. In particular, we can regard the data uncertainty distribution as a mixture between a Gaussian distribution, which represent valid measurements with some measurement error, and a uniform distribution, which represents invalid measurements. The relative balance between them is an unknown parameter to be estimated alongside the standard deviation of the Gauss distribution. For each data point, the algorithm can then assign a probability to be an outlier, and the influence of each data point will be effectively downgraded according to its probability to be an outlier. Furthermore, this assignment can change as the McMC search is exploring different parts of the model space. The approach is demonstrated with both synthetic and real tomography examples. In a synthetic test, the proposed mixed measurement error distribution allows recovery of the underlying model even in the presence of 6 per cent outliers, which completely destroy the ability of a regular McMC or linear search to provide a meaningful image. Applied to an actual ambient noise tomography study based on automatically picked dispersion curves, the resulting model is shown to be much more consistent for different data sets, which differ in the applied quality criteria, while retaining the ability to recover strong anomalies in selected parts of the model.

scholarly journals Markov Chain Monte Carlo for Computing Rare-Event Probabilities for a Heavy-Tailed Random Walk

2014 ◽  
Vol 51 (2) ◽  
pp. 359-376 ◽  
Author(s):  
Thorbjörn Gudmundsson ◽  
Henrik Hult
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Random Walk ◽  
Markov Chain Monte Carlo ◽  
Conditional Distribution ◽  
Rare Event ◽  
High Threshold ◽  
Full Generality ◽  
Normalizing Constant ◽  
Heavy Tailed

In this paper a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalizing constant. Using the MCMC methodology, a Markov chain is simulated, with the aforementioned conditional distribution as its invariant distribution, and information about the normalizing constant is extracted from its trajectory. The algorithm is described in full generality and applied to the problem of computing the probability that a heavy-tailed random walk exceeds a high threshold. An unbiased estimator of the reciprocal probability is constructed whose normalized variance vanishes asymptotically. The algorithm is extended to random sums and its performance is illustrated numerically and compared to existing importance sampling algorithms.

scholarly journals Ergodicity of Markov chain Monte Carlo with reversible proposal

2017 ◽  
Vol 54 (2) ◽  
pp. 638-654 ◽  
Author(s):  
K. Kamatani
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Random Walk ◽  
Markov Chain Monte Carlo ◽  
Geometric Ergodicity ◽  
Suitable Transformation ◽  
Ergodic Properties ◽  
Random Walk Metropolis ◽  
Heavy Tailed ◽  
Random Walk Metropolis Algorithm

Abstract We describe the ergodic properties of some Metropolis–Hastings algorithms for heavy-tailed target distributions. The results of these algorithms are usually analyzed under a subgeometric ergodic framework, but we prove that the mixed preconditioned Crank–Nicolson (MpCN) algorithm has geometric ergodicity even for heavy-tailed target distributions. This useful property comes from the fact that, under a suitable transformation, the MpCN algorithm becomes a random-walk Metropolis algorithm.

scholarly journals An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

Entropy ◽  
2018 ◽  
Vol 20 (2) ◽  
pp. 110 ◽  
Author(s):  
Yosra Marnissi ◽  
Emilie Chouzenoux ◽  
Amel Benazza-Benyahia ◽  
Jean-Christophe Pesquet
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
High Dimension ◽  
Auxiliary Variable ◽  
Variable Method ◽  
Monte Carlo Algorithms

scholarly journals Markov Chain Monte Carlo for Computing Rare-Event Probabilities for a Heavy-Tailed Random Walk

2014 ◽  
Vol 51 (02) ◽  
pp. 359-376 ◽  
Author(s):  
Thorbjörn Gudmundsson ◽  
Henrik Hult
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Random Walk ◽  
Markov Chain Monte Carlo ◽  
Conditional Distribution ◽  
Rare Event ◽  
High Threshold ◽  
Full Generality ◽  
Normalizing Constant ◽  
Heavy Tailed

In this paper a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalizing constant. Using the MCMC methodology, a Markov chain is simulated, with the aforementioned conditional distribution as its invariant distribution, and information about the normalizing constant is extracted from its trajectory. The algorithm is described in full generality and applied to the problem of computing the probability that a heavy-tailed random walk exceeds a high threshold. An unbiased estimator of the reciprocal probability is constructed whose normalized variance vanishes asymptotically. The algorithm is extended to random sums and its performance is illustrated numerically and compared to existing importance sampling algorithms.

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