Independent Metropolis-Hastings Research Articles

A Metropolized Adaptive Subspace Algorithm for High-Dimensional Bayesian Variable Selection

A simple and efficient adaptive Markov Chain Monte Carlo (MCMC) method, called the Metropolized Adaptive Subspace (MAdaSub) algorithm, is proposed for sampling from high-dimensional posterior model distributions in Bayesian variable selection. The MAdaSub algorithm is based on an independent Metropolis-Hastings sampler, where the individual proposal probabilities of the explanatory variables are updated after each iteration using a form of Bayesian adaptive learning, in a way that they finally converge to the respective covariates’ posterior inclusion probabilities. We prove the ergodicity of the algorithm and present a parallel version of MAdaSub with an adaptation scheme for the proposal probabilities based on the combination of information from multiple chains. The effectiveness of the algorithm is demonstrated via various simulated and real data examples, including a high-dimensional problem with more than 20,000 covariates.

Topological models to infer multiphase interstellar medium properties

Context.Spectroscopic observations of high-redshift galaxies slowly reveal the same complexity of the interstellar medium (ISM) as expected from resolved observations in nearby galaxies. While providing, in principle, a wealth of diagnostics concerning galaxy evolution, star formation, or the nature and influence of compact objects, such high-zspectra are often spatially and spectrally unresolved, and inferring reliable diagnostics represents a major obstacle. Bright, nearby, unresolved galaxies observed in the optical and infrared domains provide many constraints to design methods to infer ISM properties, but they have so far been limited to deterministic methods and/or with simple topological assumptions (e.g., single 1D model).Aims.It is urgent to build upon previous ISM multiphase and multicomponent methods by using a probabilistic approach that makes it possible to derive probability density functions for relevant parameters while also enabling a large number of free parameters with potential priors. The goal is to provide a flexible statistical framework that is agnostic to the model grid and that considers either a few discrete components defined by their parameter values and/or statistical distributions of parameters. In this paper, we present a first application with the objective to infer probability distributions of several physical parameters (e.g., the mass of H0, H2, escape fraction of ionizing photons, and metallicity) for the star-forming regions of the metal-poor dwarf galaxy I Zw 18 in order to confirm the low molecular gas content and high escape fraction of ionizing photons from Hiiregions.Methods.We present a Bayesian approach to model a suite of spectral lines using a sequential Monte Carlo method provided by the Python package PyMC which combines several concepts such as tempered likelihoods, importance sampling, and independent Metropolis-Hastings chains. The algorithm, provided by the associated code MULTIGRIS, accepts a few components which can be represented as sectors around one or several stellar clusters, or continuous (e.g., power-law, normal) distributions for any given parameter. We applied this approach to a grid of models calculated with the photoionization and photodissociation code Cloudy in order to produce topological models of I Zw 18.Results.The statistical framework we present makes it possible to consider a large number of spectroscopic tracers, with the extinction and systematic uncertainties as potential additional random variables. We applied this technique to the galaxy I Zw 18 in order to reproduce and go beyond previous topological models specifically tailored to this object. While our grid is designed for global properties of low-metallicity star-forming galaxies, we were able to calculate accurate values for the metallicity, number of ionizing photons, masses of ionized and neutral hydrogen, as well as the dust mass and the dust-to-gas mass ratio in I Zw 18. We find a relatively modest amount of H2(~105M⊙) which is predominantly CO-dark and traced by C+rather than C0. Nevertheless, more than 90% of the [Cii] emission is associated with the neutral atomic gas. Our models confirm the necessity to include an X-ray source with an inferred luminosity in good agreement with direct X-ray observations. Finally, we investigate the escape fraction of ionizing photons for different energy ranges. While the escape fraction for the main Hiiregion lies around 50–65%, we show that most of the soft X-ray photons are able to escape and may play a role in the ionization and heating of the circumgalactic or intergalactic medium.Conclusions.Multicomponent ISM models associate a complex enough distribution of matter and phases with a simple enough topological description to be constrained with probabilistic frameworks. Despite ignoring effects such as reflected light, the diffuse radiation field, or ionization by several non-cospatial sources, they remain well adapted to individual Hiiregions and to star-forming galaxies dominated by one or a few Hiiregions, and the improvement due to the combination of several components largely compensates for other secondary effects.

Exact convergence analysis of the independent Metropolis-Hastings algorithms

A well-known difficult problem regarding Metropolis-Hastings algorithms is to get sharp bounds on their convergence rates. Moreover, a fundamental but often overlooked problem in Markov chain theory is to study the convergence rates for different initializations. In this paper, we study the two issues mentioned above of the Independent Metropolis-Hastings (IMH) algorithms on both general and discrete state spaces. We derive the exact convergence rate and prove that the IMH algorithm’s different deterministic initializations have the same convergence rate. We get the exact convergence speed for IMH algorithms on general state spaces.

Quantifying non-communicable diseases' burden in Egypt using State-Space model.

The study aimed to model and quantify the health burden induced by four non-communicable diseases (NCDs) in Egypt, the first to be conducted in the context of a less developing county. The study used the State-Space model and adopted two Bayesian methods: Particle Filter and Particle Independent Metropolis-Hastings to model and estimate the NCDs' health burden trajectories. We drew on time-series data of the International Health Metric Evaluation, the Central Agency for Public Mobilization and Statistics (CAPMAS) Annual Bulletin of Health Services Statistics, the World Bank, and WHO data. Both Bayesian methods showed that the burden trajectories are on the rise. Most of the findings agreed with our assumptions and are in line with the literature. Previous year burden strongly predicts the burden of the current year. High prevalence of the risk factors, disease prevalence, and the disease's severity level all increase illness burden. Years of life lost due to death has high loadings in most of the diseases. Contrary to the study assumption, results found a negative relationship between disease burden and health services utilization which can be attributed to the lack of full health insurance coverage and the pattern of health care seeking behavior in Egypt. Our study highlights that Particle Independent Metropolis-Hastings is sufficient in estimating the parameters of the study model, in the case of time-constant parameters. The study recommends using state Space models with Bayesian estimation approaches with time-series data in public health and epidemiology research.

Particle MCMC With Poisson Resampling: Parallelization and Continuous Time Models

We introduce a new version of particle filter in which the number of “children” of a particle at a given time has a Poisson distribution. As a result, the number of particles is random and varies with time. An advantage of this scheme is that descendants of different particles can evolve independently. It makes easy to parallelize computations. Moreover, particle filter with Poisson resampling is readily adapted to the case when a hidden process is a continuous time, piecewise deterministic semi-Markov process. We show that the basic techniques of particle MCMC, namely particle independent Metropolis-Hastings, particle Gibbs sampler and its version with ancestor sampling, work under our Poisson resampling scheme. Our version of particle Gibbs sampler is uniformly ergodic under the same assumptions as its standard counterpart. We present simulation results which indicate that our algorithms can compete with the existing methods. Supplemental materials for this article are available online.

Expectation Propagation-Based Sampling Decoding: Enhancement and Optimization

In this paper, the paradigm of expectation propagation (EP) algorithm in large-scale MIMO detection is extended by the sampling decoding in an Markov chain Monte Carlo way to boost the approximation of the target posterior distribution. The proposed EP-based sampling decoding scheme not only theoretically addresses the inherent convergence problem of EP, but also is able to achieve the near-optimal decoding performance with the increment of Markov moves. Specifically, the EP-based independent Metropolis-Hastings (MH) is proposed to guarantee the exponential convergence to the target posterior distribution, thus bridging the EP detector and the sampling decoding as a whole. Meanwhile, the output yielded by the EP detector also provides a good initial setup for the sampling decoding, which results in a better convergence performance in the approximation. To further improve the convergence performance and the decoding efficiency, the EP-based Gibbs sampling is given, where the choice of the standard deviation of the discrete Gaussian distribution in the Markov mixing is also studied for a better decoding performance. Moreover, we extend the proposed EP-based Gibbs sampling decoding to the soft-output decoding in MIMO bit-interleaved coded modulation (BICM) systems, which enjoys a flexible decoding trade-off between performance and complexity by the number of Markov moves.

A Metropolis-Hastings Sampling of Subtrees in Graphs

This article presents two methods to sample uniform subtrees from graphs using Metropolis-Hastings algorithms. One method is an independent Metropolis-Hastings and the other one is a type of add-and-delete MCMC.
 In addition to the theoretical contributions, we present simulation studies which confirm the theoretical convergence results on our methods by monitoring the convergence of our Markov chains to the equilibrium distribution.

Survival function model estimation for parkinson disease using independent metropolis-hastings algorithm with uniform proposal distribution in bayesian inference

In medicine study, one of the things that is interesting enough to be studied is “time-to-event”. In general, time-to-event is used in doing survival analysis, such as analysis of Parkinson disease. Parkinson disease is one of the diseases which affects dopamine producer in brain area that is called by substantia nigra. The symptom of Parkinson disease is measured specifically by stages that are called by Hoehn and Yahr stages. This stages are distributed on integers between 0 to 5 with stage 0 is stage that does not have big impact and stage 5 is the most severe level. In this study, the survival function will be constructed from the time that the patient has the Hoehn and Yahr stages at A until increase to stage B with A < B. With A = 1, 2 and B = 3, 4, 5, overall it will be estimated six graphs of survival function. The process of construction survival function is using the Independent Metropolis-Hastings algorithm in Markov Chain Monte Carlo Methods on Bayesian Inference with uniform proposal distribution and the results are compared with Kaplan-Meier estimator for survival function. The result that is obtained through this algorithm is more represents the actual survival function if it is compared with Kaplan-Meier estimator, although there are so many censored data in the dataset.

Markov Chain Monte Carlo Method for Estimating Implied Volatility in Option Pricing

Using market covered European call option prices, the Independence Metropolis-Hastings Sampler algorithm for estimating Implied volatility in option pricing was proposed. This algorithm has an acceptance criteria which facilitate accurate approximation of this volatility from an independent path in the Black Scholes Model, from a set of finite data observation from the stock market. Assuming the underlying asset indeed follow the geometric brownian motion, inverted version of the Black Scholes model was used to approximate this Implied Volatility which was not directly seen in the real market: for which the BS model assumes the volatility to be a constant. Moreover, it is demonstrated that, the Implied Volatility from the options market tends to overstate or understate the actual expectation of the market. In addition, a 3-month market Covered European call option data, from 30 different stock companies was acquired from Optionistic.Com, which was used to estimate the Implied volatility. This accurately approximate the actual expectation of the market with low standard errors ranging between 0.0035 to 0.0275.

Bayesian Computational Methods for Sampling from the Posterior Distribution of a Bivariate Survival Model, Based on AMH Copula in the Presence of Right-Censored Data.

In this paper, we study the performance of Bayesian computational methods to estimate the parameters of a bivariate survival model based on the Ali–Mikhail–Haq copula with marginal distributions given by Weibull distributions. The estimation procedure was based on Monte Carlo Markov Chain (MCMC) algorithms. We present three version of the Metropolis–Hastings algorithm: Independent Metropolis–Hastings (IMH), Random Walk Metropolis (RWM) and Metropolis–Hastings with a natural-candidate generating density (MH). Since the creation of a good candidate generating density in IMH and RWM may be difficult, we also describe how to update a parameter of interest using the slice sampling (SS) method. A simulation study was carried out to compare the performances of the IMH, RWM and SS. A comparison was made using the sample root mean square error as an indicator of performance. Results obtained from the simulations show that the SS algorithm is an effective alternative to the IMH and RWM methods when simulating values from the posterior distribution, especially for small sample sizes. We also applied these methods to a real data set.

Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms

We study convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms (Andrieu and Roberts [Ann. Statist. 37 (2009) 697-725]). We find that the asymptotic variance of the pseudo-marginal algorithm is always at least as large as that of the marginal algorithm. We show that if the marginal chain admits a (right) spectral gap and the weights (normalised estimates of the target density) are uniformly bounded, then the pseudo-marginal chain has a spectral gap. In many cases, a similar result holds for the absolute spectral gap, which is equivalent to geometric ergodicity. We consider also unbounded weight distributions and recover polynomial convergence rates in more specific cases, when the marginal algorithm is uniformly ergodic or an independent Metropolis-Hastings or a random-walk Metropolis targeting a super-exponential density with regular contours. Our results on geometric and polynomial convergence rates imply central limit theorems. We also prove that under general conditions, the asymptotic variance of the pseudo-marginal algorithm converges to the asymptotic variance of the marginal algorithm if the accuracy of the estimators is increased.

Joint Bayesian Analysis of Parameters and States in Nonlinear, Non-Gaussian State Space Models

We propose a new methodology for designing flexible proposal densities for the joint posterior density of parameters and states in a nonlinear, non-Gaussian state space model. We show that a highly efficient Bayesian procedure emerges when these proposal densities are used in an independent Metropolis-Hastings algorithm or in importance sampling. Our method provides a computationally more efficient alternative to several recently proposed algorithms. We present extensive simulation evidence for stochastic intensity and stochastic volatility models based on Ornstein-Uhlenbeck processes. For our empirical study, we analyse the performance of our methods for corporate default panel data and stock index returns.(This paper is an updated version of the paper that appeared earlier as Barra, I., Hoogerheide, L.F., Koopman, S.J., and Lucas, A. (2013) Joint Independent Metropolis-Hastings Methods for Nonlinear Non-Gaussian State Space Models. TI Discussion Paper 13-050/III. Amsterdam: Tinbergen Institute.)

Comparison of asymptotic variances of inhomogeneous Markov chains with application to Markov chain Monte Carlo methods

In this paper, we study the asymptotic variance of sample path averages for inhomogeneous Markov chains that evolve alternatingly according to two different $\pi$-reversible Markov transition kernels $P$ and $Q$. More specifically, our main result allows us to compare directly the asymptotic variances of two inhomogeneous Markov chains associated with different kernels $P_i$ and $Q_i$, $i\in\{0,1\}$, as soon as the kernels of each pair $(P_0,P_1)$ and $(Q_0,Q_1)$ can be ordered in the sense of lag-one autocovariance. As an important application, we use this result for comparing different data-augmentation-type Metropolis-Hastings algorithms. In particular, we compare some pseudo-marginal algorithms and propose a novel exact algorithm, referred to as the random refreshment algorithm, which is more efficient, in terms of asymptotic variance, than the Grouped Independence Metropolis-Hastings algorithm and has a computational complexity that does not exceed that of the Monte Carlo Within Metropolis algorithm.

Comment on Article by Schmidl et al.

The authors develop a novel Markov chain method (ACIMH) that is designed to sample efficiently by adapting to the features of the target distribution, learning from the samples obtained previously. Their approach is based on independence MetropolisHastings (IMH), and takes the proposal distribution to be an estimate of the target distribution. Its appeal is that the efficiency of IMH is controlled by how close the proposal density is to the target density. If a very accurate estimate of the target density can be obtained, then the samples obtained by the Markov chain are nearly independent, leading to near-optimal accuracy of Monte Carlo approximations. The authors’ estimate of the target density is based on D-vine copulas, an extremely flexible class of models that has not been used previously for this purpose. Other authors have proposed similar “adaptive” IMH methods based on alternative estimates of the target density, in particular mixture distributions such as normal or t mixtures (Andrieu and Moulines 2006; Andrieu and Thoms 2008). Continued development of efficient general-purpose sampling algorithms like these is critical as we create robust software packages for Bayesian statistics that will encourage its widespread use. D-vine copulas use a factorization approach that may scale more effectively with dimension than the mixture model approach. ACIMH factorizes the target density as

Joint Independent Metropolis-Hastings Methods for Nonlinear Non-Gaussian State Space Models

We propose a new methodology for the Bayesian analysis of nonlinear nonGaussian state space models with a Gaussian time-varying signal, where the signal is a function of a possibly high-dimensional state vector. The novelty of our approach is the development of proposal densities for the joint posterior density of parameter and state vectors: a mixture of Student’s t-densities as the marginal proposal density for the parameter vector, and a Gaussian density as the conditional proposal density for the signal given the parameter vector. We argue that a highly efficient procedure emerges when these proposal densities are used in an independent Metropolis-Hastings algorithm. A particular feature of our approach is that smoothed estimates of the states and an estimate of the marginal likelihood are obtained directly as an output of the algorithm. Our methods are computationally efficient and produce more accurate estimates when compared to recently proposed alternatives. We present extensive simulation evidence for stochastic volatility and stochastic intensity models. For our empirical study, we analyse the performance of our method for stock return data and corporate default panel data. JEL classification: C11, C15, C22, C32, C58.

Efficient MCMC for Binomial Logit Models

This article deals with binomial logit models where the parameters are estimated within a Bayesian framework. Such models arise, for instance, when repeated measurements are available for identical covariate patterns. To perform MCMC sampling, we rewrite the binomial logit model as an augmented model which involves some latent variables called random utilities. It is straightforward, but inefficient, to use the individual random utility model representation based on the binary observations reconstructed from each binomial observation. Alternatively, we present in this article a new method to aggregate the random utilities for each binomial observation. Based on this aggregated representation, we have implemented an independence Metropolis-Hastings sampler, an auxiliary mixture sampler, and a novel hybrid auxiliary mixture sampler. A comparative study on five binomial datasets shows that the new aggregation method leads to a superior sampler in terms of efficiency compared to previously published data augmentation samplers.

Sequentially interacting Markov chain Monte Carlo methods

Sequential Monte Carlo (SMC) is a methodology for sampling approximately from a sequence of probability distributions of increasing dimension and estimating their normalizing constants. We propose here an alternative methodology named Sequentially Interacting Markov Chain Monte Carlo (SIMCMC). SIMCMC methods work by generating interacting non-Markovian sequences which behave asymptotically like independent Metropolis-Hastings (MH) Markov chains with the desired limiting distributions. Contrary to SMC, SIMCMC allows us to iteratively improve our estimates in an MCMC-like fashion. We establish convergence results under realistic verifiable assumptions and demonstrate its performance on several examples arising in Bayesian time series analysis.

An Inflated Multivariate Integer Count Hurdle Model: An Application to Bid and Ask Quote Dynamics

In this paper we develop a model for the conditional inflated multivariate density of integer count variables with domain Z^n. Our modelling framework is based on a copula approach and can be used for a broad set of applications where the primary characteristics of the data are: (i) discrete domain, (ii) the tendency to cluster at certain outcome values and (iii) contemporaneous dependence. These kind of properties can be found for high or ultra-high frequent data describing the trading process on financial markets. We present a straightforward method of sampling from such an inflated multivariate density through the application of an Independence Metropolis-Hastings sampling algorithm.We demonstrate the power of our approach by modelling the conditional bivariate density of bid and ask quote changes in a high frequency setup. We show how to derive the implied conditional discrete density of the bid-ask spread, taking quote clusterings (at multiples of 5 ticks) into account.

Adaptive Independent Metropolis-Hastings by Fast Estimation of Mixtures of Normals

We construct an adaptive independent Metropolis-Hastings sampler that uses a mixture of normals as a proposal distribution. To take full advantage of the potential of adaptive sampling our algorithm updates the mixture of normals frequently, starting early in the chain. The algorithm is built for speed and reliability and its sampling performance is evaluated with real and simulated examples.Our article outlines conditions for adaptive sampling to hold and gives a readily accessible proof that under these conditions the sampling scheme generates iterates that converge to the target distribution.

On the geometric ergodicity of Metropolis-Hastings algorithms

Under a compactness assumption, we show that a φ-irreducible and aperiodic Metropolis-Hastings chain is geometrically ergodic if and only if its rejection probability is bounded away from unity. In the particular case of the independence Metropolis-Hastings algorithm, we obtain that the whole spectrum of the induced operator is contained in (and in many cases equal to) the essential range of the rejection probability of the chain as conjectured by [Liu, J.S., 1996, Metropolized independent sampling with comparaisons to rejection sampling and importance sampling. Statistics and Computing, 6, 113–119.].