Comparison of the computational cost of a Monte Carlo and deterministic algorithm for computing bilinear forms of matrix powers

In this paper we consider bilinear forms of matrix polynomials and show that these polynomials can be used to construct solutions for the problems of solving systems of linear algebraic equations, matrix inversion and finding extremal eigenvalues. An almost Optimal Monte Carlo (MAO) algorithm for computing bilinear forms of matrix polynomials is presented. Results for the computational costs of a balanced algorithm for computing the bilinear form of a matrix power is presented, i.e., an algorithm for which probability and systematic errors are of the same order, and this is compared with the computational cost for a corresponding deterministic method.

A cluster computing implementation of a Monte Carlo simulation of a particle growth mechanism

Clusters of computers can be used together to provide a powerful computing resource. Large Monte Carlo simulations, such as those used to model particle growth, are computationally intensive and take considerable time to execute on conventional workstations. By spreading the work of the simulation across a cluster of computers, the elapsed execution time can be greatly reduced. Thus a user has apparently the performance of a supercomputer by using the spare cycles on other workstations.

Monte Carlo phase diagram for a polydisperse diblock copolymer melt

The phase diagram for an AB diblock copolymer melt with polydisperse A blocks and monodisperse B blocks is evaluated using lattice-based Monte Carlo simulations. Experiments on this system have shown that the A-block polydispersity shifts the order-order transitions (OOTs) towards higher A-monomer content, while the order-disorder transition (ODT) moves towards higher temperatures when the A blocks form the minority domains and lower temperatures when the A blocks form the matrix. Although self-consistent field theory (SCFT) correctly accounts for the change in the OOTs, it incorrectly predicts the ODT to shift towards higher temperatures at all diblock copolymer compositions. In contrast, our simulations predict the correct shifts for both the OOTs and the ODT. This implies that polydispersity amplifies the fluctuation-induced correction to the mean-field ODT, which we attribute to a reduction in packing frustration. Consistent with this explanation, polydispersity is found to enhance the stability of the perforated-lamellar phase.

The influence of the carbon surface chemical composition on Dubinin-Astakhov equation parameters calculated from SF(6) adsorption data-grand canonical Monte Carlo simulation

Using grand canonical Monte Carlo simulation we show, for the first time, the influence of the carbon porosity and surface oxidation on the parameters of the Dubinin-Astakhov (DA) adsorption isotherm equation. We conclude that upon carbon surface oxidation, the adsorption decreases for all carbons studied. Moreover, the parameters of the DA model depend on the number of surface oxygen groups. That is why in the case of carbons containing surface polar groups, SF(6) adsorption isotherm data cannot be used for characterization of the porosity.

Likelihood-free estimation of model evidence

Statistical methods of inference typically require the likelihood function to be computable in a reasonable amount of time. The class of “likelihood-free” methods termed Approximate Bayesian Computation (ABC) is able to eliminate this requirement, replacing the evaluation of the likelihood with simulation from it. Likelihood-free methods have gained in efficiency and popularity in the past few years, following their integration with Markov Chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC) in order to better explore the parameter space. They have been applied primarily to estimating the parameters of a given model, but can also be used to compare models. Here we present novel likelihood-free approaches to model comparison, based upon the independent estimation of the evidence of each model under study. Key advantages of these approaches over previous techniques are that they allow the exploitation of MCMC or SMC algorithms for exploring the parameter space, and that they do not require a sampler able to mix between models. We validate the proposed methods using a simple exponential family problem before providing a realistic problem from human population genetics: the comparison of different demographic models based upon genetic data from the Y chromosome.

Bayesian parameter estimation for latent Markov random fields and social networks

Undirected graphical models are widely used in statistics, physics and machine vision. However Bayesian parameter estimation for undirected models is extremely challenging, since evaluation of the posterior typically involves the calculation of an intractable normalising constant. This problem has received much attention, but very little of this has focussed on the important practical case where the data consists of noisy or incomplete observations of the underlying hidden structure. This paper specifically addresses this problem, comparing two alternative methodologies. In the first of these approaches particle Markov chain Monte Carlo (Andrieu et al., 2010) is used to efficiently explore the parameter space, combined with the exchange algorithm (Murray et al., 2006) for avoiding the calculation of the intractable normalising constant (a proof showing that this combination targets the correct distribution in found in a supplementary appendix online). This approach is compared with approximate Bayesian computation (Pritchard et al., 1999). Applications to estimating the parameters of Ising models and exponential random graphs from noisy data are presented. Each algorithm used in the paper targets an approximation to the true posterior due to the use of MCMC to simulate from the latent graphical model, in lieu of being able to do this exactly in general. The supplementary appendix also describes the nature of the resulting approximation.

Hierarchical analysis of production efficiency in a coastal trawl fishery

We present, pedagogically, the Bayesian approach to composed error models under alternative, hierarchical characterizations; demonstrate, briefly, the Bayesian approach to model comparison using recent advances in Markov Chain Monte Carlo (MCMC) methods; and illustrate, empirically, the value of these techniques to natural resource economics and coastal fisheries management, in particular. The Bayesian approach to fisheries efficiency analysis is interesting for at least three reasons. First, it is a robust and highly flexible alternative to commonly applied, frequentist procedures, which dominate the literature. Second,the Bayesian approach is extremely simple to implement, requiring only a modest addition to most natural-resource economist tool-kits. Third, despite its attractions, applications of Bayesian methodology in coastal fisheries management are few.

Bayesian ranking and slection of fishing boat efficiencies

The steadily accumulating literature on technical efficiency in fisheries attests to the importance of efficiency as an indicator of fleet condition and as an object of management concern. In this paper, we extend previous work by presenting a Bayesian hierarchical approach that yields both efficiency estimates and, as a byproduct of the estimation algorithm, probabilistic rankings of the relative technical efficiencies of fishing boats. The estimation algorithm is based on recent advances in Markov Chain Monte Carlo (MCMC) methods— Gibbs sampling, in particular—which have not been widely used in fisheries economics. We apply the method to a sample of 10,865 boat trips in the US Pacific hake (or whiting) fishery during 1987–2003. We uncover systematic differences between efficiency rankings based on sample mean efficiency estimates and those that exploit the full posterior distributions of boat efficiencies to estimate the probability that a given boat has the highest true mean efficiency.

Linguistic evidence supports date for Homeric epics

The Homeric epics are among the greatest masterpieces of literature, but when they were produced is not known with certainty. Here we apply evolutionary-linguistic phylogenetic statistical methods to differences in Homeric, Modern Greek and ancient Hittite vocabulary items to estimate a date of approximately 710–760 BCE for these great works. Our analysis compared a common set of vocabulary items among the three pairs of languages, recording for each item whether the words in the two languages were cognate – derived from a shared ancestral word – or not. We then used a likelihood-based Markov chain Monte Carlo procedure to estimate the most probable times in years separating these languages given the percentage of words they shared, combined with knowledge of the rates at which different words change. Our date for the epics is in close agreement with historians' and classicists' beliefs derived from historical and archaeological sources.

Theory and generalization of Monte Carlo models of the BOLD signal source

The dependency of the blood oxygenation level dependent (BOLD) signal on underlying hemodynamics is not well understood. Building a forward biophysical model of this relationship is important for the quantitative estimation of the hemodynamic changes and neural activity underlying functional magnetic resonance imaging (fMRI) signals. We have developed a general model of the BOLD signal which can model both intra- and extravascular signals for an arbitrary tissue model across a wide range of imaging parameters. The model of the BOLD signal was instantiated as a look-up-table (LuT), and was verified against concurrent fMRI and optical imaging measurements of activation induced hemodynamics. Magn Reson Med, 2008. © 2008 Wiley-Liss, Inc.

Monte Carlo field-theoretic simulations for melts of symmetric diblock copolymer

Monte Carlo field-theoretic simulations (MCFTS) are performed on melts of symmetric diblock copolymer for invariant polymerization indexes extending down to experimentally relevant values of N̅ ∼ 10^4. The simulations are performed with a fluctuating composition field, W_−(r), and a pressure field, W_+(r), that follows the saddle-point approximation. Our study focuses on the disordered-state structure function, S(k), and the order−disorder transition (ODT). Although shortwavelength fluctuations cause an ultraviolet (UV) divergence in three dimensions, this is readily compensated for with the use of an effective Flory−Huggins interaction parameter, χ_e. The resulting S(k) matches the predictions of renormalized one-loop (ROL) calculations over the full range of χ_eN and N̅ examined in our study, and agrees well with Fredrickson−Helfand (F−H) theory near the ODT. Consistent with the F−H theory, the ODT is discontinuous for finite N̅ and the shift in (χ_eN)_ODT follows the predicted N̅^−1/3 scaling over our range of N̅.

Bayesian model comparison with un-normalised likelihoods

Models for which the likelihood function can be evaluated only up to a parameter-dependent unknown normalizing constant, such as Markov random field models, are used widely in computer science, statistical physics, spatial statistics, and network analysis. However, Bayesian analysis of these models using standard Monte Carlo methods is not possible due to the intractability of their likelihood functions. Several methods that permit exact, or close to exact, simulation from the posterior distribution have recently been developed. However, estimating the evidence and Bayes’ factors for these models remains challenging in general. This paper describes new random weight importance sampling and sequential Monte Carlo methods for estimating BFs that use simulation to circumvent the evaluation of the intractable likelihood, and compares them to existing methods. In some cases we observe an advantage in the use of biased weight estimates. An initial investigation into the theoretical and empirical properties of this class of methods is presented. Some support for the use of biased estimates is presented, but we advocate caution in the use of such estimates.

Quantifying uncertainty in critical loads: (B) acidity mass balance critical loads on a sensitive site

This paper reports an uncertainty analysis of critical loads for acid deposition for a site in southern England, using the Steady State Mass Balance Model. The uncertainty bounds, distribution type and correlation structure for each of the 18 input parameters was considered explicitly, and overall uncertainty estimated by Monte Carlo methods. Estimates of deposition uncertainty were made from measured data and an atmospheric dispersion model, and hence the uncertainty in exceedance could also be calculated. The uncertainties of the calculated critical loads were generally much lower than those of the input parameters due to a "compensation of errors" mechanism - coefficients of variation ranged from 13% for CLmaxN to 37% for CL(A). With 1990 deposition, the probability that the critical load was exceeded was > 0.99; to reduce this probability to 0.50, a 63% reduction in deposition is required; to 0.05, an 82% reduction. With 1997 deposition, which was lower than that in 1990, exceedance probabilities declined and uncertainties in exceedance narrowed as deposition uncertainty had less effect. The parameters contributing most to the uncertainty in critical loads were weathering rates, base cation uptake rates, and choice of critical chemical value, indicating possible research priorities. However, the different critical load parameters were to some extent sensitive to different input parameters. The application of such probabilistic results to environmental regulation is discussed.

Estimating uncertainty in terrestrial critical loads and their exceedances at four sites in the UK

Critical loads are the basis for policies controlling emissions of acidic substances in Europe and elsewhere. They are assessed by several elaborate and ingenious models, each of which requires many parameters, and have to be applied on a spatially-distributed basis. Often the values of the input parameters are poorly known, calling into question the validity of the calculated critical loads. This paper attempts to quantify the uncertainty in the critical loads due to this "parameter uncertainty", using examples from the UK. Models used for calculating critical loads for deposition of acidity and nitrogen in forest and heathland ecosystems were tested at four contrasting sites. Uncertainty was assessed by Monte Carlo methods. Each input parameter or variable was assigned a value, range and distribution in an objective a fashion as possible. Each model was run 5000 times at each site using parameters sampled from these input distributions. Output distributions of various critical load parameters were calculated. The results were surprising. Confidence limits of the calculated critical loads were typically considerably narrower than those of most of the input parameters. This may be due to a "compensation of errors" mechanism. The range of possible critical load values at a given site is however rather wide, and the tails of the distributions are typically long. The deposition reductions required for a high level of confidence that the critical load is not exceeded are thus likely to be large. The implication for pollutant regulation is that requiring a high probability of non-exceedance is likely to carry high costs. The relative contribution of the input variables to critical load uncertainty varied from site to site: any input variable could be important, and thus it was not possible to identify variables as likely targets for research into narrowing uncertainties. Sites where a number of good measurements of input parameters were available had lower uncertainties, so use of in situ measurement could be a valuable way of reducing critical load uncertainty at particularly valuable or disputed sites. From a restricted number of samples, uncertainties in heathland critical loads appear comparable to those of coniferous forest, and nutrient nitrogen critical loads to those of acidity. It was important to include correlations between input variables in the Monte Carlo analysis, but choice of statistical distribution type was of lesser importance. Overall, the analysis provided objective support for the continued use of critical loads in policy development. (c) 2007 Elsevier B.V. All rights reserved.

Evolutionary history of vegetative reproduction in Porpidia s. l. (lichen-forming ascomycota)

The evolutionary history of gains and losses of vegetative reproductive propagules (soredia) in Porpidia s.l., a group of lichen-forming ascomycetes, was clarified using Bayesian Markov chain Monte Carlo (MCMC) approaches to monophyly tests and a combined MCMC and maximum likelihood approach to ancestral character state reconstructions. The MCMC framework provided confidence estimates for the reconstructions of relationships and ancestral character states, which formed the basis for tests of evolutionary hypotheses. Monophyly tests rejected all hypotheses that predicted any clustering of reproductive modes in extant taxa. In addition, a nearest-neighbor statistic could not reject the hypothesis that the vegetative reproductive mode is randomly distributed throughout the group. These results show that transitions between presence and absence of the vegetative reproductive mode within Porpidia s.l. occurred several times and independently of each other. Likelihood reconstructions of ancestral character states at selected nodes suggest that - contrary to previous thought - the ancestor to Porpidia s.l. already possessed the vegetative reproductive mode. Furthermore, transition rates are reconstructed asymmetrically with the vegetative reproductive mode being gained at a much lower rate than it is lost. A cautious note has to be added, because a simulation study showed that the ancestral character state reconstructions were highly dependent on taxon sampling. However, our central conclusions, particularly the higher rate of change from vegetative reproductive mode present to absent than vice versa within Porpidia s.l., were found to be broadly independent of taxon sampling. [Ancestral character state reconstructions; Ascomycota, Bayesian inference; hypothesis testing; likelihood; MCMC; Porpidia; reproductive systems]