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In this paper we study in details the Metropolis algorithm in connection with the two mean field spin system: the Ising model and the Blume-Emery-Griffith model. It is well-known that, in the case of the ising model, the usual choice of proposal gives rise, for low temperature, to a slowly mixing Metropolis chain; that is the spectral gap decreases exponentially fast (in the dimension N of the problem) to zero. Here we show how a slight variant in the proposal chain can avoid this phenomenon, keeping the computational mean cost similar to the cost of the usual Metropolis. More precisely we prove that, with a suitable variant in the proposal, the Metropolis chain has a spectral gap which decreases polynomially in 1/N for every temperature of the target distribution. The idea rests on allowing appropriate jumps in the same energy level of the starting state, and it is strictly connected to both the "small word Markov chains" of Guan & Krone and to the "equi-energy sampling "of Kou et al.

Keywords: Latent variables; Logistic regression; Markov chain Monte Carlo; Mixed effects model; Probit link; Posterior distribution.

One of most critical issues involved in modeling binary response data is the choice of the links. In this paper, we introduce a new link based on the generalized t-distribution. There are two parameters in the generalized t-link: one parameter purely controls the heaviness of the tails of the link and the second parameter controls the scale of the link. There are two major advantages offered by the generalized t-links. First, a symmetric generalized t-link with an unknown shape parameter is much more identifiable than a Student t-link with unknown degrees of freedom and a known scale parameter. Second, skewed generalized t-links with both unknown shape and scale parameters provide much more flexible and improved skewed link regression models than the existing skewed links. Various theoretical properties and attractive features of the proposed links are examined and explored in details. An efficient Markov chain Monte Carlo algorithm is developed to sample from the posterior distribution. The Deviance Information Criteria (DIC) is used for guiding the choice of links. The proposed methodology is motivated and illustrated by prostate cancer data.

Keywords: High Order Interactions, Bayesian Modelling, Classification, Sequence Prediction

Bayesian regression and classification with high order interactions is largely infeasible because Markov chain Monte Carlo (MCMC) would need to be applied with a huge number of parameters, which typically increases exponentially with the order. In this paper we show how to make it feasible by effectively reducing the number of parameters, exploiting the fact that many interactions have the same values for all training cases. Our method represents the sum of the parameters associated with a set of patterns that have the same value for all training cases as a single ``compressed'' parameter. Using symmetric stable distributions as the priors of the original parameters, we can easily find the priors of these compressed parameters. We therefore need to deal only with a much smaller number of compressed parameters when training the model with MCMC. After training the model we can split these compressed parameters into the original ones as needed to make predictions for test cases. We show in detail how to compress parameters for logistic sequence prediction. Experiments on both simulated and real data demonstrate a huge number of parameters can indeed be reduced by our compression method.

Keywords: Importance sampling, scrambled quasirandom sequences, population genetics

Estimating parameters in population genetics is intensive computation. Importance sampling can be helpful to reduce the computational burden if the important distribution can be computed in a quick way. Scrambled quasirandom sequences can replace pseudorandom sequences in this sampling methods. While quasirandom sequences improve the convergence of many MC applications, one cannot conclude that they will automatically enhance the convergence of all MCs. In other words, enhanced convergence is not assured in all situations with the naive use of quasirandom sequences. We will theoretically and practically, through simulation studies, investigate the relationship of convergence and number of parameters and also investigate whether specific types of sampling algorithms help to achieve faster convergence using MC methods.

Keywords: Monte Carlo method, random process, energization, stochastic forces

Our knowledge of energetic particles in Saturn’s inner magnetosphere is based on observations made during the flybys of Pioneer 11, Voyager 1, Voyager 2, and recently by Cassini. The most important features of the energetic particle population in the inner Saturnian magnetosphere are: 1) the rings and the many large and small satellites inside this region reduce the population of particles whose energies are higher than 0.5 MeV to values of the order of 10^3 times less than would otherwise be present, 2) the sputtering and outgassing of the surfaces of the satellites injects particles into the system and by some physical process, particles of the resultant plasma are accelerated to energies of the order of tens of keV, 3) the radial distribution of very energetic protons Ep > tens of MeV exhibits three major peaks associated with rings and satellites, 4) a proton population Ep~1 MeV lies outside the orbit of Enceladus, 5) a proton population Ep < 0.25 MeV has an apparent origin associated with Dione, Tethys, Enceladus, E-ring, Mimas and G-ring, 6) a population of low-energy electrons is associated with the satellites. In this work I propose a mechanism to explain the energetic particle population observed in Saturn’s inner magnetosphere based on the stochastic behavior of the electric field. To simulate the stochastic electric field I employ a Monte Carlo Method taking into account the magnetic field fluctuations obtained from the observations made by Voyager 1 and Voyager 2 spacecraft. Assuming different initial conditions, like the source of charged particles and the distribution function of their velocities, I find that particles injected with very low energies ranging from 0.104 eV to 0.526 keV can be accelerated to reach much higher energies ranging from 0.944 eV to 0.547 keV after a few seconds.

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We present a Markov chain Monte Carlo (MCMC) methodology for sampling conditioned diffusion processes. The method builds on Langevin SPDEs defined recently in the literature which evolve on the pathspace and sample from the target diffusion in equilibrium. We show that, in this infinite-dimensional pathspace context, the choice of the numerical method for solving the SPDEs is absolutely critical for the properties of the algorithm. Analytically, we discretise the SPDEs in the algorithmic time direction to obtain proposals for a Metropolis-Hastings update. The resulting MCMC algorithm is inefficient unless one uses an IMPLICIT numerical scheme when discretising the SPDEs. Only then will the algorithm propose paths of the correct quadratic variation.

Keywords: Symmetric stable models, Monte Carlo estimation and prediction

Symmetric stable random processes are gaining ground in diverse applications which include signal processing, financial process models, spatial models, and image processing. One thinks of such processes as allowing for long tailed behaviors, but in fact they also allow for more granular and dependent structures as compared with their smoother normal domain of attraction counterparts. In particular, different realizations of such processes may exhibit markedly different behaviors, a fact which argues against merely using the available observations to obtain unconditionally evaluated predictions or estimates. The fact, due to the author, that symmetric stable processes are themselves mixtures of Gaussian processes having explicit series expansions involving model parameters, seems to have been largely ignored when attempting their statistical analysis. As will be shown, one may exploit the series representations and undertake estimation and prediction in models attracted to the symmetric stable environment using direct Monte Carlo (not MCMC), without the necessity of developing approximations of stable probability densities in multiple dimensions as some are attempting. A conditional invariance principle due to LePage, Podgorski and Ryznar may be used to establish the consistency of our MC approach.

Keywords: Gene selection, Bayes factor, Calibrating value, Multilevel model, Prior predictive distribution

A common interest in microarray data analysis is to identify genes having different expression levels between two conditions. The existing methods include using two-sampled t-statistics, a modified t-statistics (SAM), semiparametric hierarchical Bayesian models, and nonparametric permutation tests. All of these methods essentially compare two population means. In this paper, we consider using the Bayes factor to compare gene expression levels. The Bayes factor approach is quite attractive and flexible in evaluating the evidence for a gene to be differentially expressed as it allows us to compare not only two population means but also the population distributions. To facilitate the use of the Bayes factor, we propose a new calibration approach that weighs two types of error probabilities differently from the prior predictive distribution of the Bayes factor for each gene and at the same time controls overall error rates for all genes under consideration. Moreover, a novel gene selection algorithm based on the calibration of the Bayes factor is developed and the theoretical properties of the proposed method are carefully examined. Our method is shown to have smaller false discovery rate (FDR) and false non-discovery rate (FNDR) than several existing methods through simulations. Finally, a real dataset from an affymetric microarray experiment to identify genes associated with the onset of osteoblast differentiation is used to further illustrate the proposed methodology.

Keywords: Switching Regressions, Changepoint Detection, Gibbs Sampling, Material Indentation

Material indentation is a popular method for determining the mechanical properties of biomaterials. The basic premise of an indentation experiment is to physically displace the sample using an indentor that concurrently measures resistive force, in order to formulate a force-displacement curve. However, doing so requires estimating the initial contact event between the indentor and the sample--a statistical changepoint detection problem that has not been rigorously addressed in the biomaterials literature to date. Here we adopt a hierarchical Bayesian approach to contact point determination based on switching regressions, which generalizes an algorithm popular with practitioners and enables both hyperparameter estimation as well as uncertainty quantification. Results using several experimentally obtained silicone indentation data sets indicate that our approach outperforms existing techniques.

Keywords: Bayesian Monte Carlo

Fixation of advantageous mutations is an important evolutionary force driving the accelerated protein diversification. However, the standard phylogenetic approach to infer positive selection is based on relative rate of nonsynonymous to synonymous substitutions, and requires the knowledge of DNA sequences, hence precludes its application to family of remotely related sequences where saturated substitutions occur. In this study, we develop a new method to detect positive selection directly from amino acid sequences by treating codon usage as hidden parameters. For a given amino acid sequence set and a phylogenetic tree, we use a reversible continuous time Markov process as our evolutionary model. This model has fewer parameters than normal amino acid evolutionary model, with only transition/transversion rate ratio, nonsynonymous/synonymous rate ratio (omega), and codon usage. Similar to earlier work, we assume that omega is a random variable with different probabilities to take a set of discrete values. Those with omega > 1 model sites under positive selection. We use the Bayesian Monte Carlo method to estimate model parameters, as it allows implementation of complex model of sequence evolution. Here unobserved DNA sequences are sampled from protein sequences based on distributions parametrized by codon usages, based on the fact that both protein sequences and the native protein-encoding DNA sequences have the same phylogenetic tree. The object is that sampled DNA sequences should fit the same phylogenetic tree as well as the native DNA sequences. Data set of beta-globin sequences from vertebrates is used to verify our model. We are able to detect all eight positive selection sites, which were originally reported using native nucleotide sequences. Our work shows that although nonsynonymous/synonymous rate ratio is defined at codon level, it can be used to detect selective pressures of amino acid sequences by our implicit codon-based model.

Keywords: Battery management, adaptive modulation. Markov decision process, reinforcement learning, sparse sampling method

We treat the problem of designing the optimal transmission scheme that is adapted to the battery state, the channel and buffer conditions, and the incoming traffic rate. We assume that the battery states can be tracked at every time slot, so that the problem is formulated as a large-scale Markov decision process (MDP). An efficient sparse sampling method is employed to obtain a solution. Simulation results are provided to demonstrate that the proposed schemes can considerably increase the lifetime of the battery-powered wireless systems while satisfying QoS constraints.

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Quite often we use Monte Carlo to "solve" many instances of a single problem that differ only in parameter values. We introduce a new strategy called Structured Database Monte Carlo (SDMC) that utilizes information obtained at one parameter to design effective variance reduction techniques at neighboring parameters. We induce a linear order (structure) on the underlying probability space (database) using sampled values at a nominal parameter. This structure is approximately maintained when the parameter is perturbed, enabling design of generic variance reduction algorithms based on, for example, stratification, control variate, and importance sampling methods for estimation in the neighborhood of the nominal parameter. Experimental results for examples in finance and physics are provided.

Keywords: Reversible Jump, MCMC, Hematopoiesis, stem cell, goodness of fit

Hematopoiesis is the development of blood cells from hematopoietic stem cells (HSC). A HSC could divide into two HSC or differentiate into a blood progenitor cell. We are interested in testing the hypothesis that HSC also divides asymmetrically into one HSC cell and one progenitor cell. Data were collected from a bone marrow transplantation experiments done on hybrid cats. We build a hierarchical model and fit it using reversible jump Metropolis-Hasting embedded in a Gibbs sampler. Testing is done based on the posterior distribution of the parameters and a Goodness-of-Fit statistic.

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We propose a position weight matrix-based deterministic sequential Monte Carlo (DSMC)method to locate conserved motifs in a given set of sequences. We also propose another DSMC algorithm to align the motifs for cases where insertions/deletions occur in instances of the motifs, which cannot be satisfactorily done using other multiple alignment and motif discovery algorithms.

Keywords: Markov chain Monte Carlo, Bayesian variable selection, Adaptation Scheme, Multimodality

We describe adaptive Markov chain Monte Carlo (MCMC) methods for sampling posterior distributions arising from Bayesian variable selection problems. Point mass mixture priors are commonly used in Bayesian variable selection problems in regression. However, for generalized linear and nonlinear models where the conditional densities cannot be obtained directly, the resulting mixture posterior may be difficult to sample using standard MCMC methods. In particular, random-walk Metropolis-Hastings chains and Metropolized independence chains can be poorly mixing due to multimodality. We provide a novel approach to this problem via an adaptation scheme. First, a generic adaptive MCMC method is introduced which automatically tunes the parameters of the proposal density during simulation and attempts to 'learn' the best parameter values. By using proposals from a family of mixture distributions, the chain is able to sample efficiently from the multimodal target distribution. Then the adaptive independence sampler for variable selection is proposed, using a proposal density obtained from a mixture of a point mass and Gaussian mixture distribution. Under the adaptation strategy, the mixing weight of the point mass component in the proposal adapts to approximate the posterior inclusion probability of its associated variable, while the Gaussian mixture distribution approximates the non-zero component of the coefficient's posterior distribution, enabling the efficient mixing between models with and without the variable included. The resulting sampling scheme performs parameter estimation and variable selection simultaneously. The convergence and ergodicity of these algorithms can be guaranteed with a careful design of the adaptation strategy. The algorithm is applied to several problems including a logistic regression model, a random field model from statistical biophysics, and a sparse kernel regression. A simulation study indicates that this adaptive independent sampler outperforms traditional MH algorithms.

Keywords: tempering; mixing; convergence

Multimodal posterior distributions are commonly encountered in Bayesian statistics. However, one standard tool for sampling from a posterior distribution, namely Markov Chain Monte Carlo, can become stuck in local modes. Parallel tempering is a modification of MCMC that is often used in hopes of circumventing bottlenecks between modes. We obtain a general bound on the convergence rate of parallel tempering that reflects its ability to move between the modes of the distribution. This bound leads to a set of sufficient conditions for rapid mixing of parallel tempering. We use these conditions to show the rapid mixing of parallel tempering on a number of multimodal distributions, including a symmetric mixture of normal distributions in R^M and a weighted mixture of normal distributions in R^M with equal covariance matrices. We also illustrate the failure of these conditions on several multimodal distributions and prove that parallel tempering is torpidly mixing on these distributions.

Keywords: data hiding, additive attacks, amplitude scaling, desynchronization, Markov chain Monte Carlo (MCMC)

Currently, watermarking algorithms have become a real trend in the field of copyright protection. A watermark is certain imperceptive information embedded into a host signal according to a procedure which must achieve satisfatctory rate-distortion-robustness performance. In our study, we are tackling with the problem of reliably recovering the watermark in the case of additive independent random noise, amplitude scaling and desynchronization attacking scenarios. For this task we are employing conventional and adaptive MCMC techniques at the decoder both for watermark detection and channel identification purposes.

Keywords: IEEE 802.11 DCF, Sequential Monte Carlo.

We propose an enhanced version of the IEEE 802.11 DCF that employs an adaptive estimator of the number of competing terminals based on sequential Monte Carlo methods. The algorithm uses a Bayesian approach, optimizing the backoff parameters of the DCF based on the predictive distribution of the number of competing terminals. We show that our algorithm is simple yet highly accurate even at small time scales. We implement our proposed new DCF in the ns-2 simulator and show that it outperforms existing methods. We also show that its accuracy can be used to improve the results of the protocol even when the terminals are not in saturation mode.

Keywords: Monte Carlo methods, control variates, variance reduction, importance sampling, Markov chain Monte Carlo

Importance sampling and Markov chain Monte Carlo methods are widely used tools employed to estimate functionals of a probability distribution that may be difficult to sample from directly. Given additional information about the distribution or functional of interest, it is often possible to employ variance reduction techniques such as the well-known method of control variates. However, as implemented in practice, this method essentially reduces the empirical sample variance, and is not robust to coefficient estimation error as the number of control variate functions increases. Here we propose two extensions that robustify the control variates method---diagonal and variable loading---and show how to realize them via an iterative implementation that significantly reduces computational cost. These methods are validated using test cases that clearly demonstrate the shortcomings of traditional control variates techniques.

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We examine properties of Replica-Exchange Molecular Dynamics (REMD) for molecular simulation and Bayesian posterior integration. We show that several integrators in common use for REMD simulation either fail to be volume preserving or are not ergodic. As a result, convergence of time-averages to ensemble averages does not hold even asymptotically for these algorithms. We show that for measure-preserving integrators, the addition of a hybrid Monte Carlo acceptance step restores ergodicity and leads to convergence in the limit. We demonstrate these issues on some small examples including simulating from a Gaussians mixture distribution and from the canonical ensemble of alanine dipeptide.

Under certain regularity conditions, it is known that a Markov chain will converge to stationarity. However, it's the rate of this convergence that holds considerable weight in the application of MCMC methods. A Markov chain that converges to its stationary distribution at a geometric rate is said to be geometrically ergodic. A major consequence of geometric ergodicity is the existence of a Markov chain central limit theorem and consistent estimates of the corresponding asymptotic variance. These estimates provide guidance on how long to run the chain so that our estimates are sufficiently credible, a matter usually approached in an ad hoc fashion. In practice, geometric ergodicity can be established by constructing drift and minorization conditions for the Markov chain. In this paper, we use these techniques to establish geometric ergodicity for a block Gibbs sampler for a Bayesian hierarchical version of the random intercepts model.

Keywords: Bayesian Parameter Estimation, CARMA Models, Adaptive MCMC

In this paper we present adaptive MCMC methods for estimating the parameters of the continuous-time autoregressive moving average (CARMA) model driven by Browian motions. In this challenging framework, we do not naively adopt static proposal distributions, but instead we automatically tune the associated parameters of the proposal densities according to the Kullback-Leibler distance between proposal distribution and target distribution. The parameters are then estimated using a Bayesian Monte Carlo scheme, and employing a Kalman filter to marginalize the variance of the state process. An efficient exploration of the parameter space is obtained through a novel reparameterization in terms of an equivalent mechanical system, and the adaptive MCMC sampling method achieves fast mixing over this reparameterization. Simulations demonstrate the effectivity and efficiency of the proposed adaptive sampling method for CARMA models up to very high orders.