We validate our models through graphical posterior predictive checks and we provide out-of-sample predictions for the second half of the season , using the first half as a training set . We use Stan to sample from the posterior distributions via Markov chain Monte Carlo . Despite the usefulness of pedigrees , however , they are often an unknown parameter and must be inferred from genetic data . In this study , we present a Bayesian method to jointly estimate pedigrees and Ne from genetic markers using Markov Chain Monte Carlo . Our method supports analysis of a large number of markers and individuals with the use of a composite likelihood , which significantly increases computational efficiency . Because of the variety of causes , identifying a lack of parameter identifiability typically requires the mathematical manipulation of models , Monte Carlo simulations , and examination of the Fisher Information Matrix . A simple test for parameter estimability is introduced , using Data Cloning , a Markov Chain Monte Carlo based algorithm . Together , Data cloning and the ANOVA based test determine if the model parameters are estimable and if so , determine their maximum likelihood estimates and provide the asymptotic standard errors . Inspired by the geometric approach to statistical model selection , we introduce a conceptually similar method to examine the global behavior of a connectionist model , by counting the number and types of response patterns it can simulate . The Markov Chain Monte Carlo-based algorithm that we constructed finds these patterns efficiently . We demonstrate the approach using two localist network models of speech perception . We develop a behavioural algorithm ( the Yun algorithm ) representing probabilistic occupant behaviour and implement this within a dynamic energy simulation tool . A core of this algorithm is the use of Markov chain and Monte Carlo methods in order to integrate probabilistic window use models into dynamic energy simulation procedures . The comparison between predicted and monitored window use patterns shows good agreement . Here we introduce SHAMAN for performing Hi-C analysis at dynamic scales , without predefined resolution , and while minimizing biases over very large datasets . Algorithmically , we devise a Markov Chain Monte Carlo-like procedure for randomizing contact matrices such that coverage and contact distance distributions are preserved . We combine this strategy with bin-free assessment of contact enrichment using a K-nearest neighbor approach . The contribution of the present paper is a rigorous proof of convergence of the nonlinear IS ( NIS ) scheme as the number of Monte Carlo samples , $ M$ , increases . Our analysis reveals that the NIS approximation errors converge to 0 almost surely and with the optimal Monte Carlo rate of $ M^{-\frac{1}{2}}$. Moreover , we prove that this is achieved even when the mean estimation error of the IWs remains constant , a property that has been termed { \it exact approximation } in the Markov chain Monte Carlo literature . We illustrate these theoretical results by means of a computer simulation example involving the estimation of the parameters of a state-space model typically used for target tracking . We present the software library marathon , which is designed to support the analysis of sampling algorithms that are based on the Markov-Chain Monte Carlo principle . The main application of this library is the computation of properties of so-called state graphs , which represent the structure of Markov chains . In this study , a trusted holistic abstraction is proposed and analytically discussed using universal scalability law and Markovian chain Monte-Carlo method . Moreover , a feedback mechanism is modeled to explain the elasticity performance of the proposed distributed system . This article discuss a class of nonparametric Bayes count functional data modelsintroduced in Canale and Dunson [ 3 ] , which are constructed through roundingreal-valued underlying processes . Computational algorithms are developed usingMarkov chain Monte Carlo and the methods are illustrated through application toasthma inhaler usage . Dropout is considerably simpler to implement than Bayesian model averaging which weights each model by its posterior probability given the training data . For complicated model classes , like feedforward neural networks , Bayesian methods typically use a Markov chain Monte Carlo method to sample models from the posterior distribution . By contrast , dropout with a probability of assumes that all the models will eventually be given equal importance in the combination but the learning of the shared weights takes this into account . This problem has been explored extensively in Bayesian optimization . For example , in the seminal works [ reference ][ reference ] , an adaptive prior is implemented via Markov Chain Monte Carlo ( MCMC ) methods . Motivated by these ideas , we incorporate an " adaptive initialization " for neural network training ( see section 2 for details ) , where we use cyclical batch size schedules to control the noise ( or temperature ) of SGD . In this paper , we model both segmentations in observation sequence and relations of segments simultaneously in our proposed integrated discriminative probabilistic framework . We propose the Metropolis-Hastings , a Markov chain Monte Carlo ( MCMC ) algorithm for approximate Bayesian inference to find the maximum a posteriori assignment of all the variables of this model . This integrated model has several advantages over previous probabilistic graphical models , and it offers a great capability of extracting implicit relations and new relation discovery for relation extraction from encyclopedic documents , and capturing sub-structures in named entities for named entity recognition . We guarantee compatibility by using as second term the substantive analysis model . We fit this model with a Gibbs sampler , and we use a Metropolis-Hastings step to accept/reject the proposed draws for the missing values , to guarantee that they are actual random draws from the desired distribution . We show with simulations how this method is up to the job and overcomes competing imputation strategies . We congratulate the Authors for an interesting and thought provoking paper that compares different MCMC strategies for the class of generalized linear Dirichlet process random effects models . In our discussion we focus on the logistic regression model presented in Section 3.2 and would like to propose two alternative sampling schemes related to manifold methods that are quite general and could be employed every time a Metropolis-Hastings step is used within an MCMC simulation and the target distribution has closed form Fisher Information ( FI ) . In particular we have implemented the same sampling scheme as presented by the Authors , but have sampled the parameters β using Simplified Manifold Metropolis Adjusted Langevin Algorithm ( S-MMALA ) and Riemann Manifold Hamiltonian Monte Carlo ( RM-HMC ) , both presented in Girolami and Calderhead ( 2011 ) . An Integrated Choice and Latent Variable model is used to bring in the role of the underlying behavioral constructs , while a Latent Class model accounts for taste heterogeneity across the two segments of our sample population . An Extended Path-Size Logit model is adopted as the choice model component and a Metropolis-Hastings based algorithm is used to generate route alternatives . Data is collected through a web-based survey designed to collect behavioral data on drivers ’ route choices , using psychometric indicators and behavioral questions on respondents ’ perceptions and attitudes . 2.4 Markov chain Monte Carlo methods . 2.5 Metropolis and Metropolis-Hastings algorithms . 2.6 Residuals and goodness of fit . Methods and Results 16 Random effects and Link functions 19 Probability Components 20 Transition Time Components 21 Combining Probability and Transition Components 22 Cross Component Correlated Random Effects 25 Estimation 26 Frequentist 26 Bayesian 28 Monte Carlo Integration 28 Markov Chains 29 Rejection Sampling 29 Metropolis and Metropolis-Hastings Algorithm 30 Gibbs sampling 30 Slice Sampling 31 Estimation Summary 32 Simulations 32 Simulation 1 Cross Component Correlated Random Effects 34 Simulation 1 Results 37 Simulation 2 Small Sample Sizes of Clusters 45 Simulation 2 Results 46 Simulation 3 Repeat Events 49 Simulation 3 Results 51 Chapter 2 Discussion 53 To overcome these two drawbacks , this paper proposes a new FastSLAM algorithm based on revised genetic resampling and square root unscented particle filter ( SR-UPF ) . Double roulette wheels as the selection operator , and fast Metropolis-Hastings ( MH ) as the mutation operator and traditional crossover are combined to form a new resampling method . Amending the particle degeneracy and keeping the particle diversity are both taken into considerations in this method . The method works by developing a function which maps individual nAFC responses into likelihood values — either supporting or conflicting with a proposed model AF shape . The aggregate of these likelihoods is then used as an objective function for optimization schemes for point estimation , or as the basis function for Metropolis Hastings-like algorithms for interval estimation , both of either parameters of the AF model or of the entire AF shape . The method is demonstrated on simulated up-down staircase data . This paper presents a parameter estimation method for the Candy model based on Monte Carlo approximation of the likelihood function . In order to produce such an approximation a Metropolis-Hastings style algorithm for simulating the Candy model is introduced . Stochastic gradient Langevin dynamics ( SGLD ) is a widely used sampler for the posterior inference with a large scale dataset . Although SGLD is designed for unbounded random variables , many practical models incorporate variables with boundaries such as non-negative ones or those in a finite interval . The stochasticity of the gradient can be mitigated by the injection of Gaussian noise , which yields the stochastic Langevin gradient method ; this method can be used for Bayesian posterior sampling . However , the performance of the stochastic Langevin gradient method depends on the mixing rate of the stochastic dynamics . In this study , we propose violating the detailed balance condition to enhance the mixing rate . In this paper , we explore a general Aggregated Gradient Langevin Dynamics framework ( AGLD ) for the Markov Chain Monte Carlo ( MCMC ) sampling . We investigate the nonasymptotic convergence of AGLD with a unified analysis for different data accessing ( e.g. random access , cyclic access and random reshuffle ) and snapshot updating strategies , under convex and nonconvex settings respectively . Our study involves modeling of a genetic toggle switch which is a set of two mutually repressing genes , acting as a cellular “ memory device ” during cell differentiation process , deciding the final cell fate . We simulate the complex switching dynamics between the multi-attractor states in these systems through direct sampling using continuous stochastic approaches Langevin dynamics and Fokker Plank equation as well as discrete stochastic methods Gillespie 's Stochastic Simulation Algorithm and numerical solution of the Chemical Master Equation . The stochastic trajectories obtained through numerical simulations not only unveil the complex dynamics leading to multi-stability , they also help explore rigorously the variations , if any , in the dynamics of transitions . We review appropriate background material from the theory of stochastic processes and provide insights that are relevant to practitioners . We propose an algorithmic framework for an extended stochastic gradient Langevin dynamics and illustrate its potential . The paper is written as a tutorial , and presents an alternative use of multi-agent learning . Two complementary global optimization algorithms are thus presented . For problems of moderate size and more akin to asset allocation problems , a deterministic Branch and Bound algorithm is developed , whereas for problems of larger size a stochastic global optimization algorithm based on Gradient Langevin Dynamics is given . We demonstrate analytically and through numerical experiments that the framework reflects the desired properties often discussed in the literature . The method is rapidly convergent ( with a quadratic convergence rate ) and allows variations for avoiding analytic computation of the nonbonded Hessian terms . The MD algorithm combines the implicit Euler scheme for integration with the Langevin dynamics formulation . The implicit scheme permits a wide range of time steps without loss of numerical stability . We propose a family of algorithms which split the classification task into two stages : representation learning and uncertainty estimation . We compare four specific instances , where uncertainty estimation is performed via either an ensemble of Stochastic Gradient Descent or Stochastic Gradient Langevin Dynamics snapshots , an ensemble of bootstrapped logistic regressions , or via a number of Monte Carlo Dropout passes . We evaluate their performance in terms of \emph{selective } classification ( risk-coverage ) , and their ability to detect out-of-distribution samples . In this paper we investigate the use of Langevin Monte Carlo methods on the probability simplex and propose a new method , Stochastic gradient Riemannian Langevin dynamics , which is simple to implement and can be applied to large scale data . We apply this method to latent Dirichlet allocation in an online mini-batch setting , and demonstrate that it achieves substantial performance improvements over the state of the art online variational Bayesian methods . In this paper , we present techniques for reducing variance in stochastic gradient Langevin dynamics , yielding novel stochastic Monte Carlo methods that improve performance by reducing the variance in the stochastic gradient . We show that our proposed method has better theoretical guarantees on convergence rate than stochastic Langevin dynamics . This is complemented by impressive empirical results obtained on a variety of real world datasets , and on four different machine learning tasks ( regression , classification , independent component analysis and mixture modeling ) . Inspired by compressed sensing , we assume all given samples are low-dimensional measurements of certain high-dimensional sparse vectors , while a continuous probability distribution exists in such high-dimensional space . Specifically , given a dictionary for sparse coding , SpHMC first derives a novel likelihood evaluator of the probability distribution from the loss function of LASSO , then samples from the high-dimensional distribution using stochastic Langevin dynamics with derivatives of the logarithm likelihood and Metropolis – Hastings sampling . In addition , new samples in low-dimensional measuring spaces can be regenerated using the sampled high-dimensional vectors and the dictionary . However , the standard Langevin diffusion driven by a single temperature suffers from the tradeoff between “ global exploration ” and “ local exploitation ” , corresponding the high and low temperatures , respectively . In order to bridge such a gap , we propose to use the replica exchange Langevin diffusion for the purpose of nonconvex optimization , where two Langevin diffusions run simultaneously with positions swapping . We show that , compared with the standard Langevin diffusion , replica exchange enables us to approach the global minima faster through accelerating the convergence of Langevin diffusion . In this paper , we propose a scalable distributed Bayesian matrix factorization algorithm using stochastic gradient MCMC . Our algorithm , based on Distributed Stochastic Gradient Langevin Dynamics , can not only match the prediction accuracy of standard MCMC methods like Gibbs sampling , but at the same time is as fast and simple as stochastic gradient descent . In our experiments , we show that our algorithm can achieve the same level of prediction accuracy as Gibbs sampling an order of magnitude faster . We propose a new algorithm to obtain Bayesian posterior distribution by a hybrid deterministic-stochastic gradient Langevin dynamics . To speed up convergence and reduce computational costs , it is common to use stochastic gradient method to approximate the full gradient by sampling a subset of the large dataset . While for largely undirected motion or long observation times the dynamics can sometimes be described by isotropic random walks , usually the directional persistence due to a preference to move forward has to be accounted for , e.g. , by a correlated random walk . In this paper we generalise these descriptions to a model in terms of stochastic differential equations of Langevin type , which we use to analyse experimental search flight data of foraging bumblebees . Using parameter estimates we discuss the differences and similarities to correlated random walks . Our main result also implies the first algorithm to sample from a $ d$-dimensional log-concave distribution $ \pi_T \propto e^{-\sum_{k=0}^T f_k}$ where the $ f_k$ 's are not assumed to be strongly convex and the total number of gradient evaluations is roughly $ T\log(T)+\mathrm{poly}(d),$ as opposed to $ T\cdot \mathrm{poly}(d)$ implied by prior works . Key to our algorithm is a novel stochastic gradient Langevin dynamics Markov chain that has a carefully designed variance reduction step built-in with fixed constant batch size . Technically , lack of strong convexity is a significant barrier to the analysis , and , here , our main contribution is a martingale exit time argument showing the chain is constrained to a ball of radius roughly poly$\log(T)$ for the duration of the algorithm .
We validate our models through graphical posterior predictive checks and we provide out-of-sample predictions for the second half of the season , using the first half as a training set . We use Stan to sample from the posterior distributions via Markov chain Monte Carlo .
Despite the usefulness of pedigrees , however , they are often an unknown parameter and must be inferred from genetic data . In this study , we present a Bayesian method to jointly estimate pedigrees and Ne from genetic markers using Markov Chain Monte Carlo . Our method supports analysis of a large number of markers and individuals with the use of a composite likelihood , which significantly increases computational efficiency .
Because of the variety of causes , identifying a lack of parameter identifiability typically requires the mathematical manipulation of models , Monte Carlo simulations , and examination of the Fisher Information Matrix . A simple test for parameter estimability is introduced , using Data Cloning , a Markov Chain Monte Carlo based algorithm . Together , Data cloning and the ANOVA based test determine if the model parameters are estimable and if so , determine their maximum likelihood estimates and provide the asymptotic standard errors .
Inspired by the geometric approach to statistical model selection , we introduce a conceptually similar method to examine the global behavior of a connectionist model , by counting the number and types of response patterns it can simulate . The Markov Chain Monte Carlo-based algorithm that we constructed finds these patterns efficiently . We demonstrate the approach using two localist network models of speech perception .
We develop a behavioural algorithm ( the Yun algorithm ) representing probabilistic occupant behaviour and implement this within a dynamic energy simulation tool . A core of this algorithm is the use of Markov chain and Monte Carlo methods in order to integrate probabilistic window use models into dynamic energy simulation procedures . The comparison between predicted and monitored window use patterns shows good agreement .
Here we introduce SHAMAN for performing Hi-C analysis at dynamic scales , without predefined resolution , and while minimizing biases over very large datasets . Algorithmically , we devise a Markov Chain Monte Carlo-like procedure for randomizing contact matrices such that coverage and contact distance distributions are preserved . We combine this strategy with bin-free assessment of contact enrichment using a K-nearest neighbor approach .
The contribution of the present paper is a rigorous proof of convergence of the nonlinear IS ( NIS ) scheme as the number of Monte Carlo samples , $ M$ , increases . Our analysis reveals that the NIS approximation errors converge to 0 almost surely and with the optimal Monte Carlo rate of $ M^{-\frac{1}{2}}$. Moreover , we prove that this is achieved even when the mean estimation error of the IWs remains constant , a property that has been termed { \it exact approximation } in the Markov chain Monte Carlo literature . We illustrate these theoretical results by means of a computer simulation example involving the estimation of the parameters of a state-space model typically used for target tracking .
We present the software library marathon , which is designed to support the analysis of sampling algorithms that are based on the Markov-Chain Monte Carlo principle . The main application of this library is the computation of properties of so-called state graphs , which represent the structure of Markov chains .
In this study , a trusted holistic abstraction is proposed and analytically discussed using universal scalability law and Markovian chain Monte-Carlo method . Moreover , a feedback mechanism is modeled to explain the elasticity performance of the proposed distributed system .
This article discuss a class of nonparametric Bayes count functional data modelsintroduced in Canale and Dunson [ 3 ] , which are constructed through roundingreal-valued underlying processes . Computational algorithms are developed usingMarkov chain Monte Carlo and the methods are illustrated through application toasthma inhaler usage .
Dropout is considerably simpler to implement than Bayesian model averaging which weights each model by its posterior probability given the training data . For complicated model classes , like feedforward neural networks , Bayesian methods typically use a Markov chain Monte Carlo method to sample models from the posterior distribution . By contrast , dropout with a probability of assumes that all the models will eventually be given equal importance in the combination but the learning of the shared weights takes this into account .
This problem has been explored extensively in Bayesian optimization . For example , in the seminal works [ reference ][ reference ] , an adaptive prior is implemented via Markov Chain Monte Carlo ( MCMC ) methods . Motivated by these ideas , we incorporate an " adaptive initialization " for neural network training ( see section 2 for details ) , where we use cyclical batch size schedules to control the noise ( or temperature ) of SGD .
In this paper , we model both segmentations in observation sequence and relations of segments simultaneously in our proposed integrated discriminative probabilistic framework . We propose the Metropolis-Hastings , a Markov chain Monte Carlo ( MCMC ) algorithm for approximate Bayesian inference to find the maximum a posteriori assignment of all the variables of this model . This integrated model has several advantages over previous probabilistic graphical models , and it offers a great capability of extracting implicit relations and new relation discovery for relation extraction from encyclopedic documents , and capturing sub-structures in named entities for named entity recognition .
We guarantee compatibility by using as second term the substantive analysis model . We fit this model with a Gibbs sampler , and we use a Metropolis-Hastings step to accept/reject the proposed draws for the missing values , to guarantee that they are actual random draws from the desired distribution . We show with simulations how this method is up to the job and overcomes competing imputation strategies .
We congratulate the Authors for an interesting and thought provoking paper that compares different MCMC strategies for the class of generalized linear Dirichlet process random effects models . In our discussion we focus on the logistic regression model presented in Section 3.2 and would like to propose two alternative sampling schemes related to manifold methods that are quite general and could be employed every time a Metropolis-Hastings step is used within an MCMC simulation and the target distribution has closed form Fisher Information ( FI ) . In particular we have implemented the same sampling scheme as presented by the Authors , but have sampled the parameters β using Simplified Manifold Metropolis Adjusted Langevin Algorithm ( S-MMALA ) and Riemann Manifold Hamiltonian Monte Carlo ( RM-HMC ) , both presented in Girolami and Calderhead ( 2011 ) .
An Integrated Choice and Latent Variable model is used to bring in the role of the underlying behavioral constructs , while a Latent Class model accounts for taste heterogeneity across the two segments of our sample population . An Extended Path-Size Logit model is adopted as the choice model component and a Metropolis-Hastings based algorithm is used to generate route alternatives . Data is collected through a web-based survey designed to collect behavioral data on drivers ’ route choices , using psychometric indicators and behavioral questions on respondents ’ perceptions and attitudes .
2.4 Markov chain Monte Carlo methods . 2.5 Metropolis and Metropolis-Hastings algorithms . 2.6 Residuals and goodness of fit .
Methods and Results 16 Random effects and Link functions 19 Probability Components 20 Transition Time Components 21 Combining Probability and Transition Components 22 Cross Component Correlated Random Effects 25 Estimation 26 Frequentist 26 Bayesian 28 Monte Carlo Integration 28 Markov Chains 29 Rejection Sampling 29 Metropolis and Metropolis-Hastings Algorithm 30 Gibbs sampling 30 Slice Sampling 31 Estimation Summary 32 Simulations 32 Simulation 1 Cross Component Correlated Random Effects 34 Simulation 1 Results 37 Simulation 2 Small Sample Sizes of Clusters 45 Simulation 2 Results 46 Simulation 3 Repeat Events 49 Simulation 3 Results 51 Chapter 2 Discussion 53
To overcome these two drawbacks , this paper proposes a new FastSLAM algorithm based on revised genetic resampling and square root unscented particle filter ( SR-UPF ) . Double roulette wheels as the selection operator , and fast Metropolis-Hastings ( MH ) as the mutation operator and traditional crossover are combined to form a new resampling method . Amending the particle degeneracy and keeping the particle diversity are both taken into considerations in this method .
The method works by developing a function which maps individual nAFC responses into likelihood values — either supporting or conflicting with a proposed model AF shape . The aggregate of these likelihoods is then used as an objective function for optimization schemes for point estimation , or as the basis function for Metropolis Hastings-like algorithms for interval estimation , both of either parameters of the AF model or of the entire AF shape . The method is demonstrated on simulated up-down staircase data .
This paper presents a parameter estimation method for the Candy model based on Monte Carlo approximation of the likelihood function . In order to produce such an approximation a Metropolis-Hastings style algorithm for simulating the Candy model is introduced .
Stochastic gradient Langevin dynamics ( SGLD ) is a widely used sampler for the posterior inference with a large scale dataset . Although SGLD is designed for unbounded random variables , many practical models incorporate variables with boundaries such as non-negative ones or those in a finite interval .
The stochasticity of the gradient can be mitigated by the injection of Gaussian noise , which yields the stochastic Langevin gradient method ; this method can be used for Bayesian posterior sampling . However , the performance of the stochastic Langevin gradient method depends on the mixing rate of the stochastic dynamics . In this study , we propose violating the detailed balance condition to enhance the mixing rate .
In this paper , we explore a general Aggregated Gradient Langevin Dynamics framework ( AGLD ) for the Markov Chain Monte Carlo ( MCMC ) sampling . We investigate the nonasymptotic convergence of AGLD with a unified analysis for different data accessing ( e.g. random access , cyclic access and random reshuffle ) and snapshot updating strategies , under convex and nonconvex settings respectively .
Our study involves modeling of a genetic toggle switch which is a set of two mutually repressing genes , acting as a cellular “ memory device ” during cell differentiation process , deciding the final cell fate . We simulate the complex switching dynamics between the multi-attractor states in these systems through direct sampling using continuous stochastic approaches Langevin dynamics and Fokker Plank equation as well as discrete stochastic methods Gillespie 's Stochastic Simulation Algorithm and numerical solution of the Chemical Master Equation . The stochastic trajectories obtained through numerical simulations not only unveil the complex dynamics leading to multi-stability , they also help explore rigorously the variations , if any , in the dynamics of transitions .
We review appropriate background material from the theory of stochastic processes and provide insights that are relevant to practitioners . We propose an algorithmic framework for an extended stochastic gradient Langevin dynamics and illustrate its potential . The paper is written as a tutorial , and presents an alternative use of multi-agent learning .
Two complementary global optimization algorithms are thus presented . For problems of moderate size and more akin to asset allocation problems , a deterministic Branch and Bound algorithm is developed , whereas for problems of larger size a stochastic global optimization algorithm based on Gradient Langevin Dynamics is given . We demonstrate analytically and through numerical experiments that the framework reflects the desired properties often discussed in the literature .
The method is rapidly convergent ( with a quadratic convergence rate ) and allows variations for avoiding analytic computation of the nonbonded Hessian terms . The MD algorithm combines the implicit Euler scheme for integration with the Langevin dynamics formulation . The implicit scheme permits a wide range of time steps without loss of numerical stability .
We propose a family of algorithms which split the classification task into two stages : representation learning and uncertainty estimation . We compare four specific instances , where uncertainty estimation is performed via either an ensemble of Stochastic Gradient Descent or Stochastic Gradient Langevin Dynamics snapshots , an ensemble of bootstrapped logistic regressions , or via a number of Monte Carlo Dropout passes . We evaluate their performance in terms of \emph{selective } classification ( risk-coverage ) , and their ability to detect out-of-distribution samples .
In this paper we investigate the use of Langevin Monte Carlo methods on the probability simplex and propose a new method , Stochastic gradient Riemannian Langevin dynamics , which is simple to implement and can be applied to large scale data . We apply this method to latent Dirichlet allocation in an online mini-batch setting , and demonstrate that it achieves substantial performance improvements over the state of the art online variational Bayesian methods .
In this paper , we present techniques for reducing variance in stochastic gradient Langevin dynamics , yielding novel stochastic Monte Carlo methods that improve performance by reducing the variance in the stochastic gradient . We show that our proposed method has better theoretical guarantees on convergence rate than stochastic Langevin dynamics . This is complemented by impressive empirical results obtained on a variety of real world datasets , and on four different machine learning tasks ( regression , classification , independent component analysis and mixture modeling ) .
Inspired by compressed sensing , we assume all given samples are low-dimensional measurements of certain high-dimensional sparse vectors , while a continuous probability distribution exists in such high-dimensional space . Specifically , given a dictionary for sparse coding , SpHMC first derives a novel likelihood evaluator of the probability distribution from the loss function of LASSO , then samples from the high-dimensional distribution using stochastic Langevin dynamics with derivatives of the logarithm likelihood and Metropolis – Hastings sampling . In addition , new samples in low-dimensional measuring spaces can be regenerated using the sampled high-dimensional vectors and the dictionary .
However , the standard Langevin diffusion driven by a single temperature suffers from the tradeoff between “ global exploration ” and “ local exploitation ” , corresponding the high and low temperatures , respectively . In order to bridge such a gap , we propose to use the replica exchange Langevin diffusion for the purpose of nonconvex optimization , where two Langevin diffusions run simultaneously with positions swapping . We show that , compared with the standard Langevin diffusion , replica exchange enables us to approach the global minima faster through accelerating the convergence of Langevin diffusion .
In this paper , we propose a scalable distributed Bayesian matrix factorization algorithm using stochastic gradient MCMC . Our algorithm , based on Distributed Stochastic Gradient Langevin Dynamics , can not only match the prediction accuracy of standard MCMC methods like Gibbs sampling , but at the same time is as fast and simple as stochastic gradient descent . In our experiments , we show that our algorithm can achieve the same level of prediction accuracy as Gibbs sampling an order of magnitude faster .
We propose a new algorithm to obtain Bayesian posterior distribution by a hybrid deterministic-stochastic gradient Langevin dynamics . To speed up convergence and reduce computational costs , it is common to use stochastic gradient method to approximate the full gradient by sampling a subset of the large dataset .
While for largely undirected motion or long observation times the dynamics can sometimes be described by isotropic random walks , usually the directional persistence due to a preference to move forward has to be accounted for , e.g. , by a correlated random walk . In this paper we generalise these descriptions to a model in terms of stochastic differential equations of Langevin type , which we use to analyse experimental search flight data of foraging bumblebees . Using parameter estimates we discuss the differences and similarities to correlated random walks .
Our main result also implies the first algorithm to sample from a $ d$-dimensional log-concave distribution $ \pi_T \propto e^{-\sum_{k=0}^T f_k}$ where the $ f_k$ 's are not assumed to be strongly convex and the total number of gradient evaluations is roughly $ T\log(T)+\mathrm{poly}(d),$ as opposed to $ T\cdot \mathrm{poly}(d)$ implied by prior works . Key to our algorithm is a novel stochastic gradient Langevin dynamics Markov chain that has a carefully designed variance reduction step built-in with fixed constant batch size . Technically , lack of strong convexity is a significant barrier to the analysis , and , here , our main contribution is a martingale exit time argument showing the chain is constrained to a ball of radius roughly poly$\log(T)$ for the duration of the algorithm .
Bayesian hierarchical models for predicting individual performance in soccer Joint Estimation of Pedigrees and Effective Population Size Using Markov Chain Monte Carlo An ANOVA test for parameter estimability using data cloning with application to statistical inference for dynamic systems An MCMC-Based Method of Comparing Connectionist Models in Cognitive Science Thermal performance of a naturally ventilated building using a combined algorithm of probabilistic occupant behaviour and deterministic heat and mass balance models SHAMAN: bin-free randomization, normalization and screening of Hi-C matrices Analysis of a nonlinear importance sampling scheme for Bayesian parameter estimation in state-space models Marathon: An Open Source Software Library for the Analysis of Markov-Chain Monte Carlo Algorithms A Holistic Abstraction to Ensure Trusted Scaling and Memory Speed Trusted Analytics A Bayesian nonparametric model for count functional data Improving neural networks by preventing co-adaptation of feature detectors Parameter Re-Initialization through Cyclical Batch Size Schedules An integrated discriminative probabilistic approach to information extraction Multilevel Multiple Imputation in presence of interactions, non-linearities and random slopes Discussion of the paper: “Sampling schemes for generalized linear Dirichlet process random effects models” by M. Kyung, J. Gill, and G. Casella Frequent versus occasional drivers: A hybrid route choice model Disease Mapping with WinBUGS and MLwiN Flowgraph Models for Clustered Multistate Time to Event Data An Improved FastSLAM Algorithm Based on Revised Genetic Resampling and SR-UPF A likelihood-based method for point and interval estimation on auditory filter data Parameter estimation by a Markov chain Monte Carlo technique for the Candy model On Transformations in Stochastic Gradient MCMC Stochastic gradient method with accelerated stochastic dynamics Aggregated Gradient Langevin Dynamics Noise Treatment in Models of Genetic Switches The sharp, the flat and the shallow: Can weakly interacting agents learn to escape bad minima? Optimising portfolio diversification and dimensionality. New Approaches to Potential Energy Minimization and Molecular Dynamics Algorithms On Last-Layer Algorithms for Classification: Decoupling Representation from Uncertainty Estimation Stochastic Gradient Riemannian Langevin Dynamics on the Probability Simplex Variance Reduction in Stochastic Gradient Langevin Dynamics SpHMC: Spectral Hamiltonian Monte Carlo Accelerating Nonconvex Learning via Replica Exchange Langevin diffusion Large-Scale Distributed Bayesian Matrix Factorization using Stochastic Gradient MCMC Hybrid deterministic-stochastic gradient Langevin dynamics for Bayesian learning Constructing a Stochastic Model of Bumblebee Flights from Experimental Data Online Sampling from Log-Concave Distributions
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NIPS ArXiv ArXiv CIKM AAAI NIPS NIPS AAAI ICLR (Poster)
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