Adaptive Mixture Modelling Metropolis Methods for Bayesian Analysis of Non-linear State-Space Models.

We describe a strategy for Markov chain Monte Carlo analysis of non-linear, non-Gaussian state-space models involving batch analysis for inference on dynamic, latent state variables and fixed model parameters. The key innovation is a Metropolis-Hastings method for the time series of state variables based on sequential approximation of filtering and smoothing densities using normal mixtures. These mixtures are propagated through the non-linearities using an accurate, local mixture approximation method, and we use a regenerating procedure to deal with potential degeneracy of mixture components. This provides accurate, direct approximations to sequential filtering and retrospective smoothing distributions, and hence a useful construction of global Metropolis proposal distributions for simulation of posteriors for the set of states. This analysis is embedded within a Gibbs sampler to include uncertain fixed parameters. We give an example motivated by an application in systems biology. Supplemental materials provide an example based on a stochastic volatility model as well as MATLAB code.

Constraining the age of Lateglacial and early Holocene pollen zones and tephra horizons in southern Sweden with Bayesian probability methods

The sediment sequence from Hasseldala port in southeastern Sweden provides a unique Lateglacial/early Holocene record that contains five different tephra layers. Three of these have been geochemically identified as the Borrobol Tephra, the Hasseldalen Tephra and the 10-ka Askja Tephra. Twenty-eight high-resolution C-14 measurements have been obtained and three different age models based on Bayesian statistics are employed to provide age estimates for the five different tephra layers. The chrono- and pollen stratigraphic framework supports the stratigraphic position of the Borrobol Tephra as found in Sweden at the very end of the Older Dryas pollen zone and provides the first age estimates for the Askja and Hasseldalen tephras. Our results, however, highlight the limitations that arise in attempting to establish a robust, chronologically independent lacustrine sequence that can be correlated in great detail to ice core or marine records. Radiocarbon samples are prone to error and sedimentation rates in lake basins may vary considerably due to a number of factors. Any type of valid and 'realistic' age model, therefore, has to take these limitations into account and needs to include this information in its prior assumptions. As a result, the age ranges for the specific horizons at Hasseldala port are large and calendar year estimates differ according to the assumptions of the age-model. Not only do these results provide a cautionary note for overdependence on one age-model for the derivation of age estimates for specific horizons, but they also demonstrate that precise correlations to other palaeoarchives to detect leads or lags is problematic. Given the uncertainties associated with establishing age-depth models for sedimentary sequences spanning the Lateglacial period, however, this exercise employing Bayesian probability methods represents the best possible approach and provides the most statistically significant age estimates for the pollen zone boundaries and tephra horizons. Copyright (C) 2006 John Wiley & Sons, Ltd.

A comparative review of dimension reduction methods in approximate Bayesian computation

Approximate Bayesian computation (ABC) methods make use of comparisons between simulated and observed summary statistics to overcome the problem of computationally intractable likelihood functions. As the practical implementation of ABC requires computations based on vectors of summary statistics, rather than full data sets, a central question is how to derive low-dimensional summary statistics from the observed data with minimal loss of information. In this article we provide a comprehensive review and comparison of the performance of the principal methods of dimension reduction proposed in the ABC literature. The methods are split into three nonmutually exclusive classes consisting of best subset selection methods, projection techniques and regularization. In addition, we introduce two new methods of dimension reduction. The first is a best subset selection method based on Akaike and Bayesian information criteria, and the second uses ridge regression as a regularization procedure. We illustrate the performance of these dimension reduction techniques through the analysis of three challenging models and data sets.

Bayesian and multivariate methods applied to favorability quantification in recôncavo Basin and Ribeira Belt, Brazil

A methodology to define favorable areas in petroleum and mineral exploration is applied, which consists in weighting the exploratory variables, in order to characterize their importance as exploration guides. The exploration data are spatially integrated in the selected area to establish the association between variables and deposits, and the relationships among distribution, topology, and indicator pattern of all variables. Two methods of statistical analysis were compared. The first one is the Weights of Evidence Modeling, a conditional probability approach (Agterberg, 1989a), and the second one is the Principal Components Analysis (Pan, 1993). In the conditional method, the favorability estimation is based on the probability of deposit and variable joint occurrence, with the weights being defined as natural logarithms of likelihood ratios. In the multivariate analysis, the cells which contain deposits are selected as control cells and the weights are determined by eigendecomposition, being represented by the coefficients of the eigenvector related to the system's largest eigenvalue. The two techniques of weighting and complementary procedures were tested on two case studies: 1. Recôncavo Basin, Northeast Brazil (for Petroleum) and 2. Itaiacoca Formation of Ribeira Belt, Southeast Brazil (for Pb-Zn Mississippi Valley Type deposits). The applied methodology proved to be easy to use and of great assistance to predict the favorability in large areas, particularly in the initial phase of exploration programs. © 1998 International Association for Mathematical Geology.

Variation methods in Bayesian image restoration

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The evolutionary relationships and age of Homo naledi : an assessment using dated Bayesian phylogenetic methods

Acknowledgements We wish to express our gratitude to the National Geographic Society and the National Research Foundation of South Africa for funding the discovery, recovery, and analysis of the H. naledi material. The study reported here was also made possible by grants from the Social Sciences and Humanities Research Council of Canada, the Canada Foundation for Innovation, the British Columbia Knowledge Development Fund, the Canada Research Chairs Program, Simon Fraser University, the DST/NRF Centre of Excellence in Palaeosciences (COE-Pal), as well as by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada, a Young Scientist Development Grant from the Paleontological Scientific Trust (PAST), a Baldwin Fellowship from the L.S.B. Leakey Foundation, and a Seed Grant and a Cornerstone Faculty Fellowship from the Texas A&M University College of Liberal Arts. We would like to thank the South African Heritage Resource Agency for the permits necessary to work on the Rising Star site; the Jacobs family for granting access; Wilma Lawrence, Bonita De Klerk, Merrill Van der Walt, and Justin Mukanku for their assistance during all phases of the project; Lucas Delezene for valuable discussion on the dental characters of H. naledi. We would also like to thank Peter Schmid for the preparation of the Dinaledi fossil material; Yoel Rak for explaining in detail some of the characters used in previous studies; William Kimbel for drawing our attention to the possibility that there might be a problem with Dembo et al.’s (2015) codes for the two characters related to the articular eminence; Will Stein for helpful discussion about the Bayesian analyses; Mike Lee for his comments on this manuscript; John Hawks for his support in organizing the Rising Star workshop; and the associate editor and three anonymous reviewers for their valuable comments. We are grateful to S. Potze and the Ditsong Museum, B. Billings and the School of Anatomical Sciences at the University of the Witwatersrand, and B. Zipfel and the Evolutionary Studies Institute at the University of the Witwatersrand for providing access to the specimens in their care; the University of the Witwatersrand, the Evolutionary Studies Institute, and the South African National Centre of Excellence in PalaeoSciences for hosting a number of the authors while studying the material; and the Western Canada Research Grid for providing access to the high-performance computing facilities for the Bayesian analyses. Last but definitely not least, we thank the head of the Rising Star project, Lee Berger, for his leadership and support, and for encouraging us to pursue the study reported here.

Comparison of three expert elicitation methods for logistic regression on predicting the presence of the threatened brush-tailed rock-wallaby petrogale penicillata

Numerous expert elicitation methods have been suggested for generalised linear models (GLMs). This paper compares three relatively new approaches to eliciting expert knowledge in a form suitable for Bayesian logistic regression. These methods were trialled on two experts in order to model the habitat suitability of the threatened Australian brush-tailed rock-wallaby (Petrogale penicillata). The first elicitation approach is a geographically assisted indirect predictive method with a geographic information system (GIS) interface. The second approach is a predictive indirect method which uses an interactive graphical tool. The third method uses a questionnaire to elicit expert knowledge directly about the impact of a habitat variable on the response. Two variables (slope and aspect) are used to examine prior and posterior distributions of the three methods. The results indicate that there are some similarities and dissimilarities between the expert informed priors of the two experts formulated from the different approaches. The choice of elicitation method depends on the statistical knowledge of the expert, their mapping skills, time constraints, accessibility to experts and funding available. This trial reveals that expert knowledge can be important when modelling rare event data, such as threatened species, because experts can provide additional information that may not be represented in the dataset. However care must be taken with the way in which this information is elicited and formulated.

Statistical and Econometric Methods for Transportation Data Analysis, 2nd edition

Now in its second edition, this book describes tools that are commonly used in transportation data analysis. The first part of the text provides statistical fundamentals while the second part presents continuous dependent variable models. With a focus on count and discrete dependent variable models, the third part features new chapters on mixed logit models, logistic regression, and ordered probability models. The last section provides additional coverage of Bayesian statistical modeling, including Bayesian inference and Markov chain Monte Carlo methods. Data sets are available online to use with the modeling techniques discussed.

Experimental evaluation of hotspot identification methods

Identifying crash “hotspots”, “blackspots”, “sites with promise”, or “high risk” locations is standard practice in departments of transportation throughout the US. The literature is replete with the development and discussion of statistical methods for hotspot identification (HSID). Theoretical derivations and empirical studies have been used to weigh the benefits of various HSID methods; however, a small number of studies have used controlled experiments to systematically assess various methods. Using experimentally derived simulated data—which are argued to be superior to empirical data, three hot spot identification methods observed in practice are evaluated: simple ranking, confidence interval, and Empirical Bayes. Using simulated data, sites with promise are known a priori, in contrast to empirical data where high risk sites are not known for certain. To conduct the evaluation, properties of observed crash data are used to generate simulated crash frequency distributions at hypothetical sites. A variety of factors is manipulated to simulate a host of ‘real world’ conditions. Various levels of confidence are explored, and false positives (identifying a safe site as high risk) and false negatives (identifying a high risk site as safe) are compared across methods. Finally, the effects of crash history duration in the three HSID approaches are assessed. The results illustrate that the Empirical Bayes technique significantly outperforms ranking and confidence interval techniques (with certain caveats). As found by others, false positives and negatives are inversely related. Three years of crash history appears, in general, to provide an appropriate crash history duration.

Bayesian multinomial logit: Theory and route choice example

Statisticians along with other scientists have made significant computational advances that enable the estimation of formerly complex statistical models. The Bayesian inference framework combined with Markov chain Monte Carlo estimation methods such as the Gibbs sampler enable the estimation of discrete choice models such as the multinomial logit (MNL) model. MNL models are frequently applied in transportation research to model choice outcomes such as mode, destination, or route choices or to model categorical outcomes such as crash outcomes. Recent developments allow for the modification of the potentially limiting assumptions of MNL such as the independence from irrelevant alternatives (IIA) property. However, relatively little transportation-related research has focused on Bayesian MNL models, the tractability of which is of great value to researchers and practitioners alike. This paper addresses MNL model specification issues in the Bayesian framework, such as the value of including prior information on parameters, allowing for nonlinear covariate effects, and extensions to random parameter models, so changing the usual limiting IIA assumption. This paper also provides an example that demonstrates, using route-choice data, the considerable potential of the Bayesian MNL approach with many transportation applications. This paper then concludes with a discussion of the pros and cons of this Bayesian approach and identifies when its application is worthwhile