Bayesian Applications in Auditory Research

2019 ◽  
Vol 62 (3) ◽  
pp. 577-586 ◽  
Author(s):  
Garnett P. McMillan ◽  
John B. Cannon
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Bayesian Methods ◽  
Prior Distribution ◽  
Interim Analysis ◽  
Iterative Process ◽  
Bayes Theorem ◽  
Sound Therapy ◽  
The Many ◽  
Auditory Research

Purpose This article presents a basic exploration of Bayesian inference to inform researchers unfamiliar to this type of analysis of the many advantages this readily available approach provides. Method First, we demonstrate the development of Bayes' theorem, the cornerstone of Bayesian statistics, into an iterative process of updating priors. Working with a few assumptions, including normalcy and conjugacy of prior distribution, we express how one would calculate the posterior distribution using the prior distribution and the likelihood of the parameter. Next, we move to an example in auditory research by considering the effect of sound therapy for reducing the perceived loudness of tinnitus. In this case, as well as most real-world settings, we turn to Markov chain simulations because the assumptions allowing for easy calculations no longer hold. Using Markov chain Monte Carlo methods, we can illustrate several analysis solutions given by a straightforward Bayesian approach. Conclusion Bayesian methods are widely applicable and can help scientists overcome analysis problems, including how to include existing information, run interim analysis, achieve consensus through measurement, and, most importantly, interpret results correctly. Supplemental Material https://doi.org/10.23641/asha.7822592

scholarly journals Bayesian Metric Multidimensional Scaling

2013 ◽  
Vol 21 (1) ◽  
pp. 125-140 ◽  
Author(s):  
Ryan Bakker ◽  
Keith T. Poole
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Bayesian Methods ◽  
Optimal Solution ◽  
Full Rank ◽  
Mcmc Methods ◽  
Ratio Scale ◽  
Posterior Distributions ◽  
Point Estimates ◽  
Inflection Points

In this article, we show how to apply Bayesian methods to noisy ratio scale distances for both the classical similarities problem as well as the unfolding problem. Bayesian methods produce essentially the same point estimates as the classical methods, but are superior in that they provide more accurate measures of uncertainty in the data. Identification is nontrivial for this class of problems because a configuration of points that reproduces the distances is identified only up to a choice of origin, angles of rotation, and sign flips on the dimensions. We prove that fixing the origin and rotation is sufficient to identify a configuration in the sense that the corresponding maxima/minima are inflection points with full-rank Hessians. However, an unavoidable result is multiple posterior distributions that are mirror images of one another. This poses a problem for Markov chain Monte Carlo (MCMC) methods. The approach we take is to find the optimal solution using standard optimizers. The configuration of points from the optimizers is then used to isolate a single Bayesian posterior that can then be easily analyzed with standard MCMC methods.

A two-stage Markov chain Monte Carlo method for seismic inversion and uncertainty quantification

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. R1003-R1020 ◽  
Author(s):  
Georgia K. Stuart ◽  
Susan E. Minkoff ◽  
Felipe Pereira
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Uncertainty Quantification ◽  
Bayesian Methods ◽  
Posterior Distributions ◽  
Mcmc Algorithm ◽  
Two Stage ◽  
Fine Grid ◽  
Highest Posterior Density

Bayesian methods for full-waveform inversion allow quantification of uncertainty in the solution, including determination of interval estimates and posterior distributions of the model unknowns. Markov chain Monte Carlo (MCMC) methods produce posterior distributions subject to fewer assumptions, such as normality, than deterministic Bayesian methods. However, MCMC is computationally a very expensive process that requires repeated solution of the wave equation for different velocity samples. Ultimately, a large proportion of these samples (often 40%–90%) is rejected. We have evaluated a two-stage MCMC algorithm that uses a coarse-grid filter to quickly reject unacceptable velocity proposals, thereby reducing the computational expense of solving the velocity inversion problem and quantifying uncertainty. Our filter stage uses operator upscaling, which provides near-perfect speedup in parallel with essentially no communication between processes and produces data that are highly correlated with those obtained from the full fine-grid solution. Four numerical experiments demonstrate the efficiency and accuracy of the method. The two-stage MCMC algorithm produce the same results (i.e., posterior distributions and uncertainty information, such as medians and highest posterior density intervals) as the Metropolis-Hastings MCMC. Thus, no information needed for uncertainty quantification is compromised when replacing the one-stage MCMC with the more computationally efficient two-stage MCMC. In four representative experiments, the two-stage method reduces the time spent on rejected models by one-third to one-half, which is important because most of models tried during the course of the MCMC algorithm are rejected. Furthermore, the two-stage MCMC algorithm substantially reduced the overall time-per-trial by as much as 40%, while increasing the acceptance rate from 9% to 90%.

MCMC Diagnostic Approaches

2019 ◽  
pp. 212-223
Author(s):  
Therese M. Donovan ◽  
Ruth M. Mickey
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Posterior Distribution ◽  
Diagnostic Tests ◽  
Bayes Theorem ◽  
Diagnostic Tools ◽  
Bayesian Mcmc ◽  
Main Challenge ◽  
Diagnostic Approaches ◽  
Mcmc Approach

The purpose of this chapter is to illustrate some of the things that can go wrong in Markov Chain Monte Carlo (MCMC) analysis and to introduce some diagnostic tools that help identify whether the results of such an analysis can be trusted. The goal of a Bayesian MCMC analysis is to estimate the posterior distribution while skipping the integration required in the denominator of Bayes’ Theorem. The MCMC approach does this by breaking the problem into small, bite-sized pieces, allowing the posterior distribution to be built bit by bit. The main challenge, however, is that several things might go wrong in the process. Several diagnostic tests can be applied to ensure that an MCMC analysis provides an adequate estimate of the posterior distribution. Such diagnostics are required of all MCMC analyses and include tuning, burn-in, and pruning.

A Bayesian Analysis of Self-Organizing Maps

1994 ◽  
Vol 6 (5) ◽  
pp. 767-794 ◽  
Author(s):  
Stephen P. Luttrell
Keyword(s):  
Markov Chain ◽  
Bayesian Analysis ◽  
Bayesian Methods ◽  
Transition Probabilities ◽  
Bayes Theorem ◽  
Multilayer Network ◽  
Self Organizing Maps ◽  
Vector Quantizers ◽  
Self Organizing

In this paper Bayesian methods are used to analyze some of the properties of a special type of Markov chain. The forward transitions through the chain are followed by inverse transitions (using Bayes' theorem) backward through a copy of the same chain; this will be called a folded Markov chain. If an appropriately defined Euclidean error (between the original input and its “reconstruction” via Bayes' theorem) is minimized with respect to the choice of Markov chain transition probabilities, then the familiar theories of both vector quantizers and self-organizing maps emerge. This approach is also used to derive the theory of self-supervision, in which the higher layers of a multilayer network supervise the lower layers, even though overall there is no external teacher.

Bayesian Estimation of an M/M/𝑅 Queue with Heterogeneous Servers Using Markov Chain Monte Carlo Method

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
V. Deepthi ◽  
Joby K. Jose
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Monte Carlo Method ◽  
Loss Function ◽  
Prior Distribution ◽  
Arrival Rate ◽  
Heterogeneous Servers ◽  
Squared Error Loss Function ◽  
Service Rates

AbstractIn this paper, we consider the Bayesian inference of M/M/𝑅 queue with 𝑅 heterogeneous servers with service rates \mu_{1},\mu_{2},\ldots,\mu_{R}, where \mu_{1}>\mu_{2}>\cdots>\mu_{R}. Assuming multivariate gamma prior distribution for service rates and gamma prior distribution for arrival rate 𝜆, we derive the conditional posterior densities of mean arrival rate and mean service rates. We apply the Markov chain Monte Carlo method and compute the Bayes estimates and credible interval for the M/M/3 queue, as a particular case of the M/M/𝑅 queue under squared error loss function, entropy loss function and linex loss function corresponding to a different set of hyperparameters.

scholarly journals Performance of parametric Bayesian Methods for estimating the survivor function in uncensored data using Monte-Carlo simulation

2021 ◽  
Vol 11 (3) ◽  
pp. 430-436
Author(s):  
Mohammed Elamin Hassan ◽  
Fakhereldeen Elhaj Esmial Musa
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Maximum Likelihood ◽  
Bayesian Methods ◽  
Prior Distribution ◽  
Bayesian Method ◽  
The Other ◽  
Mean Square ◽  
Monte Carlo Simulation Study ◽  
Survivor Function

The paper aimed to investigate the performance of some parametric survivor function estimators based on Bayesian methodology with respect to bias and efficiency. A simulation was conducted based on Mote Carlo experiments with different sample sizes different (10, 30, 50, 75, 100). The bias and variance of mean square Error V(MSE) were selected as the basis of comparison. The methods of estimation used in this study are Maximum Likelihood, Bayesian with exponential as prior distribution and Bayesian with gamma as prior distribution. A Monte Carlo Simulation study showed that the Bayesian method with gamma as prior distribution was the best performance than the other methods. The study recommended that.

Statistical Approach on Grading the Student Achievement via Normal Mixture Modeling

2012 ◽  
Author(s):  
Zairul Nor Deana Md. Desa ◽  
Ismail Mohamad ◽  
Zarina Mohd. Khalid ◽  
Hanafiah Md. Zin
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Standard Deviation ◽  
Bayesian Methods ◽  
Mixture Distribution ◽  
Coefficient Of Determination ◽  
Normal Mixture ◽  
Posterior Density ◽  
Letter Grades

Kajian dijalankan untuk membanding keputusan yang didapati daripada tiga kaedah penggredan terhadap pencapaian pelajar. Kaedah konvensional yang popular adalah kaedah Skala Tegak. Pendekatan statistik yang menggunakan kaedah Sisihan Piawai dan kaedah Bayesian bersyarat dipertimbangkan untuk memberi gred. Dalam model Bayesian, dianggapkan bahawa data adalah mengikut taburan Normal Tergabung di mana setiap gred adalah dipisahkan secara berasingan oleh parameter; min dan kadar bandingan dari taburan Normal Tergabung. Masalah yang timbul adalah sukar untuk menganggarkan ketumpatan posterior bagi parameter tersebut secara analitik. Satu penyelesaiannya adalah dengan menggunakan pendekatan Markov Chain Monte Carlo iaitu melalui algoritma pensampelan Gibbs. Kaedah Skala Tegak, kaedah Sisihan Piawai dan kaedah Bayesian bersyarat diaplikasikan untuk markah mentah peperiksaan bagi dua kumpulan pelajar. Pencapaian ketiga–tiga kaedah dibandingkan melalui nilai Kehilangan Kelas Neutral, Kehilangan Kelas Tidak Tegas dan Pekali Penentuan. Didapati keputusan dari kaedah Bayesian bersyarat menunjukkan penggredan yang lebih baik berbanding kaedah Skala Tegak dan kaedah Sisihan Piawai. Kata kunci: Kaedah penggredan, pengukuran pendidikan, Skala Tegak, kaedah Sisihan Piawai, Normal Tergabung, Markov Chain Monte Carlo, pensampelan Gibbs The purpose of this study is to compare results obtained from three methods of assigning letter grades to students’ achievement. The conventional and the most popular method to assign grades is the Straight Scale method (SS). Statistical approaches which used the Standard Deviation (GC) and conditional Bayesian methods are considered to assign the grades. In the conditional Bayesian model, we assume the data to follow the Normal Mixture distribution where the grades are distinctively separated by the parameters: means and proportions of the Normal Mixture distribution. The problem lies in estimating the posterior density of the parameters which is analytically intractable. A solution to this problem is using the Markov Chain Monte Carlo approach namely Gibbs sampler algorithm. The Straight Scale, Standard Deviation and Conditional Bayesian methods are applied to the examination raw scores of two sets of students. The performances of these methods are measured using the Neutral Class Loss, Lenient Class Loss and Coefficient of Determination. The results showed that Conditional Bayesian outperformed the Conventional Methods of assigning grades. Key words: Grading methods, educational measurement, Straight Scale, Standard Deviation method, Normal Mixture, Markov Chain Monte Carlo, Gibbs sampling

scholarly journals Bayesian methods and estimation of nonlinear dynamical systems

2019 ◽  
Author(s):  
Κωνσταντίνος Καλούδης
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Bayesian Methods ◽  
Dirichlet Process ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Dynamic Noise ◽  
Process Reconstruction ◽  
Global Stable Manifold

Η παρούσα διατριβή αφορά τη διάδραση μεταξύ Μπεϋζιανής στατιστικής και μη γραμμικών δυναμικών συστημάτων. Ειδικότερα, ο βασικός στόχος της διατριβής είναι η ανάπτυξη νέων μεθόδων Markov Chain Monte Carlo (MCMC) με εφαρμογές στο ευρύτερο πεδίο της μη γραμμικής δυναμικής. Το κίνητρο για την ανάπτυξη τέτοιων μεθόδων, αφορά την διάκριση της διαδικασίας μοντελοποίησης σε δύο βασικά διαδραστικά μέρη: το αιτιοκρατικό (ντετερμινιστικό) μέρος και τη στοχαστική διαδικασία θορύβου. Μέσω μιας τέτοιου είδους μοντελοποίησης, επιτυγχάνεται η σύλληψη μιας ευρείας συλλογής φαινομένων, αξιοποιώντας την πολυπλοκότητα της δυναμικής συμπεριφοράς λόγω του μη γραμμικού μέρους και τα νέα χαρακτηριστικά που αναδεικνύονται λόγω της εμπλοκής των στοχαστικών διαταραχών. Οι προτεινόμενες στατιστικές μέθοδοι είναι μη παραμετρικές και βασίζονται στη χρήση τυχαίων μέτρων πιθανότητας με γεωμετρικά βάρη (Geometric stick breaking process (GSB)) ως εκ των προτέρων κατανομές στο χώρο των μέτρων πιθανότητας. Μια σημαντική πτυχή των προτεινόμενων μεθόδων είναι η επίτευξη της χαλάρωσης μιας πολύ συχνής υπόθεσης στη βιβλιογραφία: της κανονικότητας της διαδικασίας θορύβου. Στα δύο πρώτα Κεφάλαια γίνεται αναφορά σε βασικές έννοιες της Μπεϋζιανής στατιστικής και της θεωρίας των δυναμικών συστημάτων. Στο Κεφάλαιο 3 κατασκεύαζουμε ένα μη παραμετρικό Μπεϋζιανό μοντέλο κατάλληλο για αναδόμηση των δυναμικών εξισώσεων και πρόγνωση μελλοντικών τιμών από παρατηρηθείσες χρονοσειρές μολυσμένες με προσθετικό δυναμικό θόρυβο: το μοντέλο geometric stick-breaking reconstruction (GSBR). Το GSBR μοντέλο βασίζεται στο τυχαίο μέτρο με γεωμετρικά βάρη (GSB), ενώ γίνεται επίσης παρουσίαση του αντίστοιχου μοντέλου Dirichlet process reconstruction (DPR) βασισμένου στο τυχαίο μέτρο DP, καθώς και η μεταξύ τους σύγκριση. Η μεθοδολογία επεκτείνεται ώστε να γίνει εφικτή η μοντελοποίηση χρησιμοποιώντας αυθαίρετο πεπερασμένο πλήθος όρων χρονικών υστερήσεων (lags), καθώς και στην πολυδιάστατη περίπτωση μέσω της άπειρης μίξης πολυδιάστατων κανονικών πυρήνων με άγνωστους πίνακες αποκρίσεων, χρησιμοποιώντας ως μέτρο μίξης το τυχαίο μέτρο GSB και μέτρο βάσης (base measure) μια κατανομή Wishart. Στο Κεφάλαιο 4, προτείνεται μια μη παραμετρική Μπεϋζιανή μεθοδολογία βασιζόμενη επίσης στο τυχαίο μέτρο GSB, με σκοπό τη μείωση δυναμικού θορύβου σε διαθέσιμα δεδομένα μη γραμμικών χρονοσειρών με προσθετικό θορυβο. Το μοντέλο Dynamic Noise Reduction Replicator (DNRR) επιτυγχάνει μεγάλη ακρίβεια στην αναδόμηση των δυναμικών εξισώσεων, ώστε να αναπαράγει την υποκείμενη δυναμική σε περιβάλλον ασθενέστερου δυναμικού θορύβου. Μέσω της εφαρμογής του DNRR είναι δυνατή η σύνδεση των περιοχών υψηλών αποκλίσεων από τον ντετερμινισμό με τις περιοχές των πρωταρχικών ομοκλινικών εφαπτομενικοτήτων του υποκείμενου ντετερμινιστικού συστήματος. Συσχετίζοντας τα στοχαστικά δυναμικά συστήματα με τα αντίστοιχα ντετερμινιστικά τους μέρη, στο Κεφάλαιο 5 παρουσιάζεται μία επέκταση του μοντέλου GSBR, με σκοπό τη στοχαστική προσέγγιση της ολικής ευσταθούς πολλαπλότητας (global stable manifold), με χρήση μεθόδου MCMC. Ειδικότερα, γίνεται παρουσίαση του οπισθοδρομικού (backward) GSBR μοντέλου BGSBR, μέσω του οποίου επιτυγχάνεται πρόβλεψη σε αντεστραμμένο χρόνο. Με κατάλληλες πολλαπλές εφαρμογές του BGSBR χρησιμοποιώντας υποσύνολα των διαθέσιμων δεδομένων, δείχνουμε ότι η ένωση των στηριγμάτων των περιθώριων κατανομών για τις διάφορες αρχικές συνθήκες παρέχουν μια στοχαστική προσέγγιση της ευσταθούς πολλαπλότητας του υποκείμενου ντετερμινιστικού συστήματος. Η μεθοδολογία είναι εφαρμόσιμη τόσο σε αντιστρέψιμες όσο και σε μη αντιστρέψιμες απεικονίσεις. Στο Κεφάλαιο 6 γίνεται σύνοψη των αποτελεσμάτων των προηγούμενων Κεφαλαίων και αναφορά σε θέματα για μελλοντική έρευνα, τα οποία προέκυψαν κατά τη διάρκεια εκπόνησης της παρούσας Διατριβής.

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