Random Walk Metropolis-Hastings Research Articles

Direct Amortized Likelihood Ratio Estimation

We introduce a new amortized likelihood ratio estimator for likelihood-free simulation-based inference (SBI). Our estimator is simple to train and estimates the likelihood ratio using a single forward pass of the neural estimator. Our approach directly computes the likelihood ratio between two competing parameter sets which is different from the previous approach of comparing two neural network output values. We refer to our model as the direct neural ratio estimator (DNRE). As part of introducing the DNRE, we derive a corresponding Monte Carlo estimate of the posterior. We benchmark our new ratio estimator and compare to previous ratio estimators in the literature. We show that our new ratio estimator often outperforms these previous approaches. As a further contribution, we introduce a new derivative estimator for likelihood ratio estimators that enables us to compare likelihood-free Hamiltonian Monte Carlo (HMC) with random-walk Metropolis-Hastings (MH). We show that HMC is equally competitive, which has not been previously shown. Finally, we include a novel real-world application of SBI by using our neural ratio estimator to design a quadcopter. Code is available at https://github.com/SRI-CSL/dnre.

Learning attribute and homophily measures through random walks

We investigate the statistical learning of nodal attribute functionals in homophily networks using random walks. Attributes can be discrete or continuous. A generalization of various existing canonical models, based on preferential attachment is studied (model class mathscr {P}), where new nodes form connections dependent on both their attribute values and popularity as measured by degree. An associated model class mathscr {U} is described, which is amenable to theoretical analysis and gives access to asymptotics of a host of functionals of interest. Settings where asymptotics for model class mathscr {U} transfer over to model class mathscr {P} through the phenomenon of resolvability are analyzed. For the statistical learning, we consider several canonical attribute agnostic sampling schemes such as Metropolis-Hasting random walk, versions of node2vec (Grover and Leskovec, 2016) that incorporate both classical random walk and non-backtracking propensities and propose new variants which use attribute information in addition to topological information to explore the network. Estimators for learning the attribute distribution, degree distribution for an attribute type and homophily measures are proposed. The performance of such statistical learning framework is studied on both synthetic networks (model class mathscr {P}) and real world systems, and its dependence on the network topology, degree of homophily or absence thereof, (un)balanced attributes, is assessed.

Efficient Posterior Sampling for Bayesian Poisson Regression

ABSTRACT Poisson log-linear models are ubiquitous in many applications, and one of the most popular approaches for parametric count regression. In the Bayesian context, however, there are no sufficient specific computational tools for efficient sampling from the posterior distribution of parameters, and standard algorithms, such as random walk Metropolis-Hastings or Hamiltonian Monte Carlo algorithms, are typically used. Herein, we developed an efficient Metropolis-Hastings algorithm and importance sampler to simulate from the posterior distribution of the parameters of Poisson log-linear models under conditional Gaussian priors with superior performance with respect to the state-of-the-art alternatives. The key for both algorithms is the introduction of a proposal density based on a Gaussian approximation of the posterior distribution of parameters. Specifically, our result leverages the negative binomial approximation of the Poisson likelihood and the successful Pólya-gamma data augmentation scheme. Via simulation, we obtained that the time per independent sample of the proposed samplers is competitive with that obtained using the successful Hamiltonian Monte Carlo sampling, with the Metropolis-Hastings showing superior performance in all scenarios considered. Supplementary materials for this article are available online.

A time-domain multisource Bayesian/Markov chain Monte Carlo formulation of time-lapse seismic waveform inversion

Seismic waveform inversion has been shown to be well-suited for identification and characterization of time-lapse changes in a reservoir. However, the subtle medium variations associated with enhanced oil recovery and/or [Formula: see text] storage problems give uncertainty quantification within such inverse approaches a heightened importance. To analyze both of these features of the time-lapse inverse problem, we formulate a Bayesian full-waveform inversion (FWI) procedure, based on a Markov chain Monte Carlo (MCMC) algorithm. The formulation uses several existing strategies, such as the use of a double-difference time-lapse FWI (DDFWI), incorporation of time-domain multisource data, and application of a local-updating target-oriented inversion. However, it incorporates these within a stochastic framework, involving computation of model covariance with an adaptive Metropolis algorithm, and a method to estimate data error statistics based on the features of time-lapse difference data that is incorporated. A random walk Metropolis-Hastings MCMC is adopted for optimization. In conventional, i.e., deterministic, DDFWI, inversions are carried out for the baseline and monitoring models; in the MCMC approach, a deterministic FWI procedure is carried out for the baseline model, and the MCMC algorithm is applied in the monitoring inversion stage; and the final time-lapse model is the difference between these. A feasibility study is carried out using synthetic 2D acoustic models and data, including time-lapse model estimation and uncertainty quantification. We compare the MCMC approach with conventional deterministic optimization DDFWI and remark on benefits derived, which appear to justify the expanded complexity and cost of a global approach. In addition to the availability of posterior distributions, which are critical for the assessment of the estimations, we observe that the MCMC approach tends to produce monitoring images with clearer edges and fewer coherent errors.

Comparison between the Hamiltonian Monte Carlo method and the Metropolis–Hastings method for coseismic fault model estimation

A rapid source fault estimation and quantitative assessment of the uncertainty of the estimated model can elucidate the occurrence mechanism of earthquakes and inform disaster damage mitigation. The Bayesian statistical method that addresses the posterior distribution of unknowns using the Markov chain Monte Carlo (MCMC) method is significant for uncertainty assessment. The Metropolis–Hastings method, especially the Random walk Metropolis–Hastings (RWMH), has many applications, including coseismic fault estimation. However, RWMH exhibits a trade-off between the transition distance and the acceptance ratio of parameter transition candidates and requires a long mixing time, particularly in solving high-dimensional problems. This necessitates a more efficient Bayesian method. In this study, we developed a fault estimation algorithm using the Hamiltonian Monte Carlo (HMC) method, which is considered more efficient than the other MCMC method, but its applicability has not been sufficiently validated to estimate the coseismic fault for the first time. HMC can conduct sampling more intelligently with the gradient information of the posterior distribution. We applied our algorithm to the 2016 Kumamoto earthquake (MJMA 7.3), and its sampling converged in 2 × 104 samples, including 1 × 103 burn-in samples. The estimated models satisfactorily accounted for the input data; the variance reduction was approximately 88%, and the estimated fault parameters and event magnitude were consistent with those reported in previous studies. HMC could acquire similar results using only 2% of the RWMH chains. Moreover, the power spectral density (PSD) of each model parameter's Markov chain showed this method exhibited a low correlation with the subsequent sample and a long transition distance between samples. These results indicate HMC has advantages in terms of chain length than RWMH, expecting a more efficient estimation for a high-dimensional problem that requires a long mixing time or a problem using nonlinear Green’s function, which has a large computational cost.Graphical

Determining passenger car equivalent (PCEs) for pretimed signalized intersections with severe motorcycle composition using Bayesian linear regression.

Pretimed signalized intersection is known as a common source of congestion, especially in urban heterogeneous traffic. Furthermore, the accuracy of saturation flow rate is found to cause efficient and vital capacity estimation, in order to ensure optimal design and operation of the signal timings. Presently, the traffic also consists of diverse vehicle presence, each with its own static and dynamic characteristics. The passenger car equivalent (PCE) in an essential unit is also used to measure heterogenous traffic into the PCU (Passenger Car Unit). Based on the collection of observed data at three targets in Banda Aceh City, this study aims to redetermine the PCEs by using Bayesian linear regression, through the Random-walk Metropolis-Hastings and Gibbs sampling. The result showed that the obtained PCE values were 0.24, 1.0, and 0.80 for motorcycle (MC), passenger car (PC), and motorized rickshaw (MR), respectively. It also showed that a significant deviation was found between new and IHCM PCEs, as the source of error was partially due to the vehicle compositions. The present traffic characteristics were also substantially different from the prevailing conditions of IHCM 1997. Therefore, the proposed PCEs enhanced the accuracy of base saturation flow prediction, provided support for traffic operation design, alleviated congestion, and reduced delay within the city, which in turn improved the estimation of signalized intersection capacity.

Accelerating adaptation in the adaptive Metropolis–Hastings random walk algorithm

SummaryThe Metropolis–Hastings random walk algorithm remains popular with practitioners due to the wide variety of situations in which it can be successfully applied and the extreme ease with which it can be implemented. Adaptive versions of the algorithm use information from the early iterations of the Markov chain to improve the efficiency of the proposal. The aim of this paper is to reduce the number of iterations needed to adapt the proposal to the target, which is particularly important when the likelihood is time‐consuming to evaluate. First, the accelerated shaping algorithm is a generalisation of both the adaptive proposal and adaptive Metropolis algorithms. It is designed to remove, from the estimate of the covariance matrix of the target, misleading information from the start of the chain. Second, the accelerated scaling algorithm rapidly changes the scale of the proposal to achieve a target acceptance rate. The usefulness of these approaches is illustrated with a range of examples.

Attribute-Guided Network Sampling Mechanisms

This article introduces a novel task-independent sampler for attributed networks. The problem is important because while data mining tasks on network content are common, sampling on internet-scale networks is costly. Link-trace samplers such as Snowball sampling, Forest Fire, Random Walk, and Metropolis–Hastings Random Walk are widely used for sampling from networks. The design of these attribute-agnostic samplers focuses on preserving salient properties of network structure, and are not optimized for tasks on node content. This article has three contributions. First, we propose a task-independent, attribute aware link-trace sampler grounded in Information Theory. Our sampler greedily adds to the sample the node with the most informative (i.e., surprising) neighborhood. The sampler tends to rapidly explore the attribute space, maximally reducing the surprise of unseen nodes. Second, we prove that content sampling is an NP-hard problem. A well-known algorithm best approximates the optimization solution within 1 − 1/ e , but requires full access to the entire graph. Third, we show through empirical counterfactual analysis that in many real-world datasets, network structure does not hinder the performance of surprise based link-trace samplers. Experimental results over 18 real-world datasets reveal: surprise-based samplers are sample efficient and outperform the state-of-the-art attribute-agnostic samplers by a wide margin (e.g., 45% performance improvement in clustering tasks).

An adaptive node embedding framework for multiplex networks

Network Embedding (NE) has emerged as a powerful tool in many applications. Many real-world networks have multiple types of relations between the same entities, which are appropriate to be modeled as multiplex networks. However, at random walk-based embedding study for multiplex networks, very little attention has been paid to the problems of sampling bias and imbalanced relation types. In this paper, we propose an Adaptive Node Embedding Framework (ANEF) based on cross-layer sampling strategies of nodes for multiplex networks. ANEF is the first framework to focus on the bias issue of sampling strategies. Through metropolis hastings random walk (MHRW) and forest fire sampling (FFS), ANEF is less likely to be trapped in local structure with high degree nodes. We utilize a fixed-length queue to record previously visited layers, which can balance the edge distribution over different layers in sampled node sequence processes. In addition, to adaptively sample the cross-layer context of nodes, we also propose a node metric called Neighbors Partition Coefficient (NPC). Experiments on real-world networks in diverse fields show that our framework outperforms the state-of-the-art methods in application tasks such as cross-domain link prediction and mutual community detection.

Applying Bayesian method to investigate determinants of non performing loans of banks in Vietnam

This study was conducted to determine the factors affecting non-performing loans of commercial banks in Vietnam for the period 2007 - 2018. The study applies the Bayesian approach and the Random-walk Metropolis-Hastings algorithm to evaluate the impact of micro and macro factors on non-performing loans of commercial banks. The dependent variable is non-performing loans, which is measured by the ratio of non-performing loans divided by total outstanding loans; the independent variables in terms of bank characteristics are non-performing loans of the previous year, profitability, bank size, bak loans, and bank capital; the macro variables are inflation and GDP growth. Research data was collected from financial statements of 30 Vietnamese commercial banks and the General Statistics Office of Vietnam from 2007 to 2018. To increase the reliability and efficiency of the model as well as reasonable Bayes inference, a convergence test of the MCMC chain was performed. The result of this study shows that non-performing loans of the previous year, bank size, bank loan, bank capital, and inflation have positive impacts on bank non-performing loans. In addition, bank profitability and GDP growth rate are factors that have the opposite effects. Based on the research results, the author proposes policy implications for the decision-makers to help banks reduce non-performing loans and promote banks to operate effectively and more efficiently.

Farmer Perception, Environmental Awareness, and Overuse of Fertilizer in Kalpitiya: A Preliminary Investigation using Bayesian Econometrics

Overuse of fertilizer has long been identified as a major issue in many parts of the world, including Sri Lanka. Sandy regosol soil and the shallow groundwater table in Kalpitiya, has aggravated the impacts of extensive fertigation, raising doubts on the long-term socioeconomic and environmental sustainability of intensive agricultural systems of the region. In this background, this study attempts to determine the driving factors influencing the overuse of fertilizer within the farming community in Kalpitiya. Primary data collected from face-toface interviews with 107 farmers from Kalpitya area using a pre-tested questionnaire was used to estimate a Bayesian probit model with sufficiently diffuse normally distributed priors. The model was estimated using a Random Walk Metropolis-Hastings sampling method, which iterated 125,000 times and 25,000 were discarded as burn-in. Results reveal that farmers who perceive that fertilizer has no effect on soil and groundwater tend to move away from fertilizer recommendations and are overusing. Farmers who are aware and believe that ground water is contaminated in the area tend to apply the recommended amount of fertilizer. In addition, larger farmers seem to apply recommended dosages than smaller farmers. Raising awareness through proper extension services and creating economic disutility by increased fertilizer costs coupled with the introduction of organic fertilizer could be recommended to circumvent the ill effects of overuse of fertilizer.

Objective Bayesian analysis for the differential entropy of the Weibull distribution

This article addresses the estimation of differential entropy of the Weibull distribution based on different non-informative prior distributions. We first derive non-informative priors using formal rules, such as Jeffreys prior and maximal data information prior based on Fisher information and entropy, respectively. We also develop reference prior and probability matching prior for the differential entropy. Next, we investigate the effects of these priors in the Bayes estimates of the differential entropy of the Weibull distribution based on complete sample. The Bayes estimates are computed using random-walk Metropolis-Hastings algorithms. We use Monte Carlo simulation to evaluate the performance of the Bayes estimates of differential entropy in terms of bias and root mean-squared error, while highest posterior density credible intervals are assessed in terms of average width and coverage probabilities. Finally, a real data set has been analyzed for illustrative purposes.

A neural network assisted Metropolis adjusted Langevin algorithm

Abstract In this paper, we derive a Markov chain Monte Carlo (MCMC) algorithm supported by a neural network. In particular, we use the neural network to substitute derivative calculations made during a Metropolis adjusted Langevin algorithm (MALA) step with inexpensive neural network evaluations. Using a complex, high-dimensional blood coagulation model and a set of measurements, we define a likelihood function on which we evaluate the new MCMC algorithm. The blood coagulation model is a dynamic model, where derivative calculations are expensive and hence limit the efficiency of derivative-based MCMC algorithms. The MALA adaptation greatly reduces the time per iteration, while only slightly affecting the sample quality. We also test the new algorithm on a 2-dimensional example with a non-convex shape, a case where the MALA algorithm has a clear advantage over other state of the art MCMC algorithms. To assess the impact of the new algorithm, we compare the results to previously generated results of the MALA and the random walk Metropolis Hastings (RWMH).

Bayesian estimation of generalized partition of unity copulas

Abstract This paper proposes a Bayesian estimation algorithm to estimate Generalized Partition of Unity Copulas (GPUC), a class of nonparametric copulas recently introduced by [18]. The first approach is a random walk Metropolis-Hastings (RW-MH) algorithm, the second one is a random blocking random walk Metropolis-Hastings algorithm (RBRW-MH). Both approaches are Markov chain Monte Carlo methods and can cope with ˛at priors. We carry out simulation studies to determine and compare the efficiency of the algorithms. We present an empirical illustration where GPUCs are used to nonparametrically describe the dependence of exchange rate changes of the crypto-currencies Bitcoin and Ethereum.

What Is Appropriate Upper Endoscopic Interval Among Dyspeptic Patients With Previously Normal Endoscopy? A Multicenter Study With Bayesian Change Point Analysis.

Background/AimsAppropriate interval for performing follow-up endoscopy among dyspeptic patients without abnormal findings on previous endoscopy is unclear. We analyzed the multicenter-collected data from the Korean Society of Neurogastroenterology and Motility.MethodsWe collected clinical data of the patients who visited the gastroenterology department and underwent 2 or more sessions of upper endoscopy during 2012–2017 at 6 university hospitals in Korea. Patients with endoscopic interval between 90 days and 760 days were included. For those with multiple endoscopic sessions, only the first 2 were analyzed. Positive outcome was defined as adenoma or cancer in the upper gastrointestinal tract. To identify the point of change and estimate the properties of the stochastic process before and after the change, we used Bayesian regression with Metropolis-Hastings algorithm.ResultsThere were 1595 patients. Mean age was 58.8 years (standard deviation, 12.8). Median interval of endoscopy was 437 days (standard deviation, 153). On follow-up endoscopy, there were 12 patients (0.75%) who had neoplasia (4 with gastric cancer and 8 with gastric adnoma). As with the prior hypothesis, we presumed the change point (CP) of increase in frequency of organic lesion as 360 days. After random-walk Metropolis-Hastings sampling with Markov-Chain Monte Carlo iterations of 5000, the CP was 560 days (95% credible interval, 139–724). Estimated average of frequency of dysplastic lesions increased by a factor of 4.4 after the estimated CP.ConclusionTo rule out dysplastic lesions among dyspeptic patients who had previously normal endoscopy, a 2-year interval could be offered as follow-up interval for repeat upper endoscopy.

Modified Hamiltonian Monte Carlo for Bayesian inference

The Hamiltonian Monte Carlo (HMC) method has been recognized as a powerful sampling tool in computational statistics. We show that performance of HMC can be significantly improved by incorporating importance sampling and an irreversible part of the dynamics into a chain. This is achieved by replacing Hamiltonians in the Metropolis test with modified Hamiltonians, and a complete momentum update with a partial momentum refreshment. We call the resulting generalized HMC importance sampler---Mix & Match Hamiltonian Monte Carlo (MMHMC). The method is irreversible by construction and further benefits from (i) the efficient algorithms for computation of modified Hamiltonians; (ii) the implicit momentum update procedure and (iii) the multi-stage splitting integrators specially derived for the methods sampling with modified Hamiltonians. MMHMC has been implemented, tested on the popular statistical models and compared in sampling efficiency with HMC, Riemann Manifold Hamiltonian Monte Carlo, Generalized Hybrid Monte Carlo, Generalized Shadow Hybrid Monte Carlo, Metropolis Adjusted Langevin Algorithm and Random Walk Metropolis-Hastings. To make a fair comparison, we propose a metric that accounts for correlations among samples and weights, and can be readily used for all methods which generate such samples. The experiments reveal the superiority of MMHMC over popular sampling techniques, especially in solving high dimensional problems.

A brief review of optimal scaling of the main MCMC approaches and optimal scaling of additive TMCMC under non-regular cases

Transformation based Markov Chain Monte Carlo (TMCMC) was proposed by Dutta and Bhattacharya (Statistical Methodology 16 (2014) 100–116) as an efficient alternative to the Metropolis–Hastings algorithm, especially in high dimensions. The main advantage of this algorithm is that it simultaneously updates all components of a high dimensional parameter using appropriate move types defined by deterministic transformation of a single random variable. This results in reduction in time complexity at each step of the chain and enhances the acceptance rate. In this paper, we first provide a brief review of the optimal scaling theory for various existing MCMC approaches, comparing and contrasting them with the corresponding TMCMC approaches.The optimal scaling of the simplest form of TMCMC, namely additive TMCMC, has been studied extensively for the Gaussian proposal density in Dey and Bhattacharya (2017a). Here, we discuss diffusion-based optimal scaling behavior of additive TMCMC for non-Gaussian proposal densities—in particular, uniform, Student’s $t$ and Cauchy proposals. Although we could not formally prove our diffusion result for the Cauchy proposal, simulation based results lead us to conjecture that at least the recipe for obtaining general optimal scaling and optimal acceptance rate holds for the Cauchy case as well. We also consider diffusion based optimal scaling of TMCMC when the target density is discontinuous. Such non-regular situations have been studied in the case of Random Walk Metropolis Hastings (RWMH) algorithm by Neal and Roberts (Methodology and Computing in Applied Probability 13 (2011) 583–601) using expected squared jumping distance (ESJD), but the diffusion theory based scaling has not been considered. We compare our diffusion based optimally scaled TMCMC approach with the ESJD based optimally scaled RWM with simulation studies involving several target distributions and proposal distributions including the challenging Cauchy proposal case, showing that additive TMCMC outperforms RWMH in almost all cases considered.

A Novel Hybrid-Jump-Based Sampling Method for Complex Social Networks

With the rapid development of the Internet, social media is affecting and changing people’s lives. The research of network community structure based on a large number of complex network data sets is increasingly popular. Due to the large scale of existing social network data and privacy issues, it is hard to analyze the entire network data directly. Therefore, a reliable and effective network sampling method is very important for the actual estimation of online social networks attributes. Existing network sampling methods like metropolis–hasting random walk (MHRW) can obtain unbiased sample sets from relatively large-scale social networks such as Facebook and describe the key features of the original network. Moreover, MHRW uses a proposed distribution function for sampling control, which can guarantee a well-balanced nature of resulting Markov chain. However, MHRW has the defect of partial graph over entry. In this paper, we proposed a new hybrid jump (HJ) sample by introducing an HJ strategy into MHRW during the sampling progress. First, we use a breadth-first search to obtain a data set without repeated node quickly from a list of jump nodes. Moreover, we adopted uniform sample (UNI) to get the average degree of the original network. Then, a 3-D average degree distribution model is designed to determine the optimal value of the jump parameter in HJ. Finally, we set the condition to execute the HJ strategy in each step of sampling progress. The experimental results demonstrate the performance of HJ is better than the other representation sampling methods both in strong-tie networks and weak-tie networks.

Efficient Adaptive MCMC Through Precision Estimation

ABSTRACTThe performance of Markov chain Monte Carlo (MCMC) algorithms like the Metropolis Hastings Random Walk (MHRW) is highly dependent on the choice of scaling matrix for the proposal distributions. A popular choice of scaling matrix in adaptive MCMC methods is to use the empirical covariance matrix (ECM) of previous samples. However, this choice is problematic if the dimension of the target distribution is large, since the ECM then converges slowly and is computationally expensive to use. We propose two algorithms to improve convergence and decrease computational cost of adaptive MCMC methods in cases when the precision (inverse covariance) matrix of the target density can be well-approximated by a sparse matrix. The first is an algorithm for online estimation of the Cholesky factor of a sparse precision matrix. The second estimates the sparsity structure of the precision matrix. Combining the two algorithms allows us to construct precision-based adaptive MCMC algorithms that can be used as black-box methods for densities with unknown dependency structures. We construct precision-based versions of the adaptive MHRW and the adaptive Metropolis adjusted Langevin algorithm and demonstrate the performance of the methods in two examples. Supplementary materials for this article are available online.

A Bayesian Beta-Mixture Model for Nonparametric IRT (BBM-IRT)

Item response models typically assume that the item characteristic (step) curves follow a logistic or normal cumulative distribution function, which are strictly monotone functions of person test ability. Such assumptions can be overly-restrictive for real item response data. We propose a simple and more flexible Bayesian nonparametric IRT model for dichotomous items, which constructs monotone item characteristic (step) curves by a finite mixture of beta distributions, which can support the entire space of monotone curves to any desired degree of accuracy. A simple adaptive random-walk Metropolis-Hastings algorithm is proposed to estimate the posterior distribution of the model parameters. The Bayesian IRT model is illustrated through the analysis of item response data from a 2015 TIMSS test of math performance.