scholarly journals Markov Chain Monte Carlo in a Dynamical System of Information Theoretic Particles

2021 ◽  
Author(s):  
Tokunbo Ogunfunmi ◽  
Manas Deb
Keyword(s):  
Dynamical System ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Potential Energy ◽  
Probability Density ◽  
Posterior Probability ◽  
Information Theoretic ◽  
Other Information ◽  
Posterior Probability Density

In Bayesian learning, the posterior probability density of a model parameter is estimated from the likelihood function and the prior probability of the parameter. The posterior probability density estimate is refined as more evidence becomes available. However, any non-trivial Bayesian model requires the computation of an intractable integral to obtain the probability density function (PDF) of the evidence. Markov Chain Monte Carlo (MCMC) is a well-known algorithm that solves this problem by directly generating the samples of the posterior distribution without computing this intractable integral. We present a novel perspective of the MCMC algorithm which views the samples of a probability distribution as a dynamical system of Information Theoretic particles in an Information Theoretic field. As our algorithm probes this field with a test particle, it is subjected to Information Forces from other Information Theoretic particles in this field. We use Information Theoretic Learning (ITL) techniques based on Rényi’s α-Entropy function to derive an equation for the gradient of the Information Potential energy of the dynamical system of Information Theoretic particles. Using this equation, we compute the Hamiltonian of the dynamical system from the Information Potential energy and the kinetic energy. The Hamiltonian is used to generate the Markovian state trajectories of the system.

scholarly journals Parameters Identification for Inverse Option Problems Using Markov Chain Monte Carlo Methods

2019 ◽  
Author(s):  
Yasushi Ota ◽  
Yu Jiang
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Bayesian Inference ◽  
Financial Markets ◽  
Sampling Strategy ◽  
Measured Data ◽  
Mcmc Algorithm ◽  
Unknown Parameters ◽  
Posterior Probability Density

This paper investigates the inverse option problems (IOP) in the extended Black--Scholes model arising in financial markets. We identify the volatility and the drift coefficient from the measured data in financial markets using a Bayesian inference approach, which is presented as an IOP solution. The posterior probability density function of the parameters is computed from the measured data. The statistics of the unknown parameters are estimated by a Markov Chain Monte Carlo (MCMC) algorithm, which exploits the posterior state space. The efficient sampling strategy of the MCMC algorithm enables us to solve inverse problems by the Bayesian inference technique. Our numerical results indicate that the Bayesian inference approach can simultaneously estimate the unknown trend and volatility coefficients from the measured data.

scholarly journals Bayesian Multiple Emitter Fitting using Reversible Jump Markov Chain Monte Carlo

2019 ◽  
Author(s):  
Mohamadreza Fazel ◽  
Michael J. Wester ◽  
Hanieh Mazloom-Farsibaf ◽  
Marjolein B. M. Meddens ◽  
Alexandra Eklund ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Single Molecule ◽  
Posterior Probability ◽  
Super Resolution ◽  
Image Data ◽  
Reversible Jump ◽  
Posterior Probability Distribution ◽  
Resolution Imaging

In single molecule localization-based super-resolution imaging, high labeling density or the desire for greater data collection speed can lead to clusters of overlapping emitter images in the raw super-resolution image data. We describe a Bayesian inference approach to multiple-emitter fitting that uses Reversible Jump Markov Chain Monte Carlo to identify and localize the emitters in dense regions of data. This formalism can take advantage of any prior information, such as emitter intensity and density. The output is both a posterior probability distribution of emitter locations that includes uncertainty in the number of emitters and the background structure, and a set of coordinates and uncertainties from the most probable model.

scholarly journals Nonlinear Parameter Estimation: Comparison of an Ensemble Kalman Smoother with a Markov Chain Monte Carlo Algorithm

2012 ◽  
Vol 140 (6) ◽  
pp. 1957-1974 ◽  
Author(s):  
Derek J. Posselt ◽  
Craig H. Bishop
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Parameter Uncertainty ◽  
Posterior Probability ◽  
Monte Carlo Algorithm ◽  
Parameter Estimates ◽  
Posterior Probability Distribution ◽  
Kalman Smoother ◽  
Non Gaussian

Abstract This paper explores the temporal evolution of cloud microphysical parameter uncertainty using an idealized 1D model of deep convection. Model parameter uncertainty is quantified using a Markov chain Monte Carlo (MCMC) algorithm. A new form of the ensemble transform Kalman smoother (ETKS) appropriate for the case where the number of ensemble members exceeds the number of observations is then used to obtain estimates of model uncertainty associated with variability in model physics parameters. Robustness of the parameter estimates and ensemble parameter distributions derived from ETKS is assessed via comparison with MCMC. Nonlinearity in the relationship between parameters and model output gives rise to a non-Gaussian posterior probability distribution for the parameters that exhibits skewness early and multimodality late in the simulation. The transition from unimodal to multimodal posterior probability density function (PDF) reflects the transition from convective to stratiform rainfall. ETKS-based estimates of the posterior mean are shown to be robust, as long as the posterior PDF has a single mode. Once multimodality manifests in the solution, the MCMC posterior parameter means and variances differ markedly from those from the ETKS. However, it is also shown that if the ETKS is given a multimode prior ensemble, multimodality is preserved in the ETKS posterior analysis. These results suggest that the primary limitation of the ETKS is not the inability to deal with multimodal, non-Gaussian priors. Rather it is the inability of the ETKS to represent posterior perturbations as nonlinear functions of prior perturbations that causes the most profound difference between MCMC posterior PDFs and ETKS posterior PDFs.

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.

Why some clades have low bootstrap frequencies and high Bayesian posterior probabilities

2014 ◽  
Vol 60 (1) ◽  
pp. 41-44 ◽  
Author(s):  
Ricardo García-Sandoval
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Monte Carlo Method ◽  
Posterior Probability ◽  
Parametric Bootstrap ◽  
Sampling Procedure ◽  
Data Matrix ◽  
Posterior Probabilities ◽  
Non Parametric

Bayesian posterior probabilities are wrongly considered by many systematists as indicative of character support, and equivalent to non-parametric bootstrap frequencies. Here I argue against this view. Non-parametric bootstrap is indicative of the amount of evidence in a data matrix supporting each clade in the tree, while Bayesian posterior probabilities are not intended to represent that property. Clades with high posterior probability may not have a large amount of characters favouring them, and their frequencies are the result of the particular sampling procedure of the Bayesian Markov chain Monte Carlo method, which tends to sample very similar topologies according to their posterior probabilities. Both metrics may relate to the notion of confidence, but depict different properties.

Laser Localization of Autonomous Vehicle Using an Improved Particle Filter

2012 ◽  
Vol 629 ◽  
pp. 873-877
Author(s):  
Wen Jian Ying ◽  
Fu Chun Sun
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Monte Carlo Method ◽  
Particle Filter ◽  
Probability Density ◽  
Autonomous Vehicle ◽  
Posteriori Probability ◽  
Localization Precision ◽  
Simulation Results

This article presents an improved Rao-Blackwellized particle filter to overcome particles degeneracy phenomenon and acquire the better localization precision of the autonomous vehicle. The joint posteriori probability density is given that being correlative with the position and pose of the autonomous vehicle and the mark characters of the map. The algorithm utilizes a Markov chain Monte Carlo method with the sampling particle of the target to the resample mechanism of the Rao-Blackwellized particle filter. Simulation results show that the improved algorithm is valid.

Time-domain Markov chain Monte Carlo–based Bayesian damage detection of ballasted tracks using nonlinear ballast stiffness model

2020 ◽  
pp. 147592172096695
Author(s):  
Heung-Fai Lam ◽  
Mujib Olamide Adeagbo ◽  
Yeong-Bin Yang
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Damage Detection ◽  
Probability Density ◽  
Time Domain ◽  
Model Updating ◽  
Model Parameters ◽  
Uncertain Model ◽  
Nonlinear Stress

This article reports the development of a methodology for detecting ballast damage under a sleeper based on measured sleeper vibration following the Bayesian statistical system identification framework. To ensure the methodology is applicable under large amplitude vibration of the sleeper (e.g. under trainload), the nonlinear stress–strain behavior of railway ballast is considered. This, on one hand, significantly reduces the problem of modeling error, but, on the other hand, increases the number of uncertain model parameters. The uncertainty associated with the identified model parameters of the rail–sleeper–ballast system may be very high. To overcome this difficulty, the Markov chain Monte Carlo–based Bayesian model updating is adopted in the proposed methodology for the approximation of the posterior probability density function of uncertain model parameters. Owing to the nonlinear behavior of the system, the model updating is performed in the time domain instead of the modal domain. The applicability of the proposed damage detection methodology was first verified numerically using simulated impact hammer test data in two damaged cases perturbed with Gaussian white noise. Second, impact hammer tests of in situ sleepers in the full-scale in-door ballasted track test panel were carried out to collect data for the experimental verification of the proposed methodology. Artificial ballast damage was simulated under the target concrete sleeper by replacing normal-sized ballast particles (∼60 mm) by small-sized ballast particles (∼15 mm). The proposed methodology successfully identified the location and severity of ballast damage under the sleeper. From the calculated posterior marginal probability density functions of model parameters, one can quantify the uncertainties associated with the damage detection results. The proposed methodology is an essential step in the development of a long-term railway track health monitoring system utilizing train-induced vibration.

scholarly journals Bayesian Multiple Emitter Fitting using Reversible Jump Markov Chain Monte Carlo

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Mohamadreza Fazel ◽  
Michael J. Wester ◽  
Hanieh Mazloom-Farsibaf ◽  
Marjolein B. M. Meddens ◽  
Alexandra S. Eklund ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Single Molecule ◽  
Posterior Probability ◽  
Super Resolution ◽  
Image Data ◽  
Reversible Jump ◽  
Posterior Probability Distribution ◽  
Resolution Imaging

Abstract In single molecule localization-based super-resolution imaging, high labeling density or the desire for greater data collection speed can lead to clusters of overlapping emitter images in the raw super-resolution image data. We describe a Bayesian inference approach to multiple-emitter fitting that uses Reversible Jump Markov Chain Monte Carlo to identify and localize the emitters in dense regions of data. This formalism can take advantage of any prior information, such as emitter intensity and density. The output is both a posterior probability distribution of emitter locations that includes uncertainty in the number of emitters and the background structure, and a set of coordinates and uncertainties from the most probable model.

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