Bayesian Model Fit Research Articles

Robust fit of Bayesian mixed effects regression models with application to colony forming unit count in tuberculosis research.

Early bactericidal activity of tuberculosis drugs is conventionally assessed using statistical regression modeling of colony forming unit (CFU) counts over time. Typically, most CFU counts deviate little from the regression curve, but gross outliers due to erroneous sputum sampling are occasionally present and can markedly influence estimates of the rate of change in CFU count, which is the parameter of interest. A recently introduced Bayesian nonlinear mixed effects regression model was adapted to offer a robust approach that accommodates both outliers and potential skewness in the data. At its most general, the proposed regression model fits the skew Student t distribution to residuals and random coefficients. Deviance information criterion statistics and compound Laplace-Metropolis marginal likelihoods were used to discriminate between alternative Bayesian nonlinear mixed effects regression models. We present a relatively easy method to calculate the marginal likelihoods required to determine compound Laplace-Metropolis marginal likelihoods, by adapting methods available in currently available statistical software. The robust methodology proposed in this paper was applied to data from 6 clinical trials. The results provide strong evidence that the distribution of CFU count is often heavy tailed and negatively skewed (suggesting the presence of outliers). Therefore, we recommend that robust regression models, such as those proposed here, should be fitted to CFU count.

Impact of prior distributions and central tendency measures on Bayesian intravoxel incoherent motion model fitting.

Bayesian model fitting has been proposed as a robust alternative for intravoxel incoherent motion (IVIM) model-fitting parameter estimation. However, consensus regarding choice of prior distribution and posterior distribution central tendency measure is needed. The aim of this study was to compare the quality of IVIM parameter estimates produced by different prior distributions and central tendency measures, and to gain knowledge about the effect of these choices. Three prior distributions (uniform, reciprocal, and lognormal) and two measures of central tendency (mean and mode) found in the literature were studied using simulations and in vivo data from a tumor mouse model. Simulations showed that the uniform and lognormal priors were superior to the reciprocal prior, especially for the parameters D and f and clinically relevant SNR levels. The choice of central tendency measure had less effect on the results, but had some effects on estimation bias. Results based on simulations and in vivo data agreed well, indicating high validity of the simulations. Choice of prior distribution and central tendency measure affects the results of Bayesian IVIM parameter estimates. This must be considered when comparing results from different studies. The best overall quality of IVIM parameter estimates was obtained using the lognormal prior. Magn Reson Med 79:1674-1683, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

Efficient occupancy model-fitting for extensive citizen-science data.

Appropriate large-scale citizen-science data present important new opportunities for biodiversity modelling, due in part to the wide spatial coverage of information. Recently proposed occupancy modelling approaches naturally incorporate random effects in order to account for annual variation in the composition of sites surveyed. In turn this leads to Bayesian analysis and model fitting, which are typically extremely time consuming. Motivated by presence-only records of occurrence from the UK Butterflies for the New Millennium data base, we present an alternative approach, in which site variation is described in a standard way through logistic regression on relevant environmental covariates. This allows efficient occupancy model-fitting using classical inference, which is easily achieved using standard computers. This is especially important when models need to be fitted each year, typically for many different species, as with British butterflies for example. Using both real and simulated data we demonstrate that the two approaches, with and without random effects, can result in similar conclusions regarding trends. There are many advantages to classical model-fitting, including the ability to compare a range of alternative models, identify appropriate covariates and assess model fit, using standard tools of maximum likelihood. In addition, modelling in terms of covariates provides opportunities for understanding the ecological processes that are in operation. We show that there is even greater potential; the classical approach allows us to construct regional indices simply, which indicate how changes in occupancy typically vary over a species’ range. In addition we are also able to construct dynamic occupancy maps, which provide a novel, modern tool for examining temporal changes in species distribution. These new developments may be applied to a wide range of taxa, and are valuable at a time of climate change. They also have the potential to motivate citizen scientists.

A space–time statistical climate model for hurricane intensification in the North Atlantic basin

Abstract. Climate influences on hurricane intensification are investigated by averaging hourly intensification rates over the period 1975–2014 in 8° × 8° latitude–longitude grid cells. The statistical effects of hurricane intensity and sea-surface temperature (SST), along with the climatic effects of El Niño–Southern Oscillation (ENSO), the North Atlantic Oscillation (NAO) and the Madden–Julian Oscillation (MJO), are quantified using a Bayesian hierarchical model fit to the averaged data. As expected, stronger hurricanes tend to have higher intensification rates, especially over the warmest waters. Of the three climate variables considered, the NAO has the largest effect on intensification rates after controlling for intensity and SST. The model shows an average increase in intensification rates of 0.18 [0.06, 0.31] m s−1 h−1 (95 % credible interval) for every 1 standard deviation decrease in the NAO index. Weak trade winds associated with the negative phase of the NAO might result in less vertical wind shear and thus higher mean intensification rates.

Statistically-Estimated Tree Composition for the Northeastern United States at Euro-American Settlement.

We present a gridded 8 km-resolution data product of the estimated composition of tree taxa at the time of Euro-American settlement of the northeastern United States and the statistical methodology used to produce the product from trees recorded by land surveyors. Composition is defined as the proportion of stems larger than approximately 20 cm diameter at breast height for 22 tree taxa, generally at the genus level. The data come from settlement-era public survey records that are transcribed and then aggregated spatially, giving count data. The domain is divided into two regions, eastern (Maine to Ohio) and midwestern (Indiana to Minnesota). Public Land Survey point data in the midwestern region (ca. 0.8-km resolution) are aggregated to a regular 8 km grid, while data in the eastern region, from Town Proprietor Surveys, are aggregated at the township level in irregularly-shaped local administrative units. The product is based on a Bayesian statistical model fit to the count data that estimates composition on the 8 km grid across the entire domain. The statistical model is designed to handle data from both the regular grid and the irregularly-shaped townships and allows us to estimate composition at locations with no data and to smooth over noise caused by limited counts in locations with data. Critically, the model also allows us to quantify uncertainty in our composition estimates, making the product suitable for applications employing data assimilation. We expect this data product to be useful for understanding the state of vegetation in the northeastern United States prior to large-scale Euro-American settlement. In addition to specific regional questions, the data product can also serve as a baseline against which to investigate how forests and ecosystems change after intensive settlement. The data product is being made available at the NIS data portal as version 1.0.

MODELING THE SWIFT BAT TRIGGER ALGORITHM WITH MACHINE LEARNING*

ABSTRACT To draw inferences about gamma-ray burst (GRB) source populations based on Swift observations, it is essential to understand the detection efficiency of the Swift burst alert telescope (BAT). This study considers the problem of modeling the Swift/BAT triggering algorithm for long GRBs, a computationally expensive procedure, and models it using machine learning algorithms. A large sample of simulated GRBs from Lien et al. is used to train various models: random forests, boosted decision trees (with AdaBoost), support vector machines, and artificial neural networks. The best models have accuracies of ≳97% (≲3% error), which is a significant improvement on a cut in GRB flux, which has an accuracy of 89.6% (10.4% error). These models are then used to measure the detection efficiency of Swift as a function of redshift z, which is used to perform Bayesian parameter estimation on the GRB rate distribution. We find a local GRB rate density of n 0 ∼ 0.48 − 0.23 + 0.41 Gpc − 3 yr − 1 ?> with power-law indices of n 1 ∼ 1.7 − 0.5 + 0.6 ?> and n 2 ∼ − 5.9 − 0.1 + 5.7 ?> for GRBs above and below a break point of z 1 ∼ 6.8 − 3.2 + 2.8 ?> . This methodology is able to improve upon earlier studies by more accurately modeling Swift detection and using this for fully Bayesian model fitting.

Predictive statistical models linking antecedent meteorological conditions and waterway bacterial contamination in urban waterways

Although the relationships between meteorological conditions and waterway bacterial contamination are being better understood, statistical models capable of fully leveraging these links have not been developed for highly urbanized settings. We present a hierarchical Bayesian regression model for predicting transient fecal indicator bacteria contamination episodes in urban waterways. Canals, creeks, and rivers of the New York City harbor system are used to examine the model. The model configuration facilitates the hierarchical structure of the underlying system with weekly observations nested within sampling sites, which in turn were nested inside of the harbor network. Models are compared using cross-validation and a variety of Bayesian and classical model fit statistics. The uncertainty of predicted enterococci concentration values is reflected by sampling from the posterior predictive distribution. Issuing predictions with the uncertainty reasonably reflected allows a water manager or a monitoring agency to issue warnings that better reflect the underlying risk of exposure. A model using only antecedent meteorological conditions is shown to correctly classify safe and unsafe levels of enterococci with good accuracy. The hierarchical Bayesian regression approach is most valuable where transient fecal indicator bacteria contamination is problematic and drainage network data are scarce.

Fast processing of diel oxygen curves: Estimating stream metabolism with BASE (BAyesian Single‐station Estimation)

AbstractThe measurement of stream metabolism (gross primary production and respiration) has become more feasible with the availability of more reliable dissolved oxygen (DO) probes. Such metabolic measurements offer important opportunities in fundamental and applied research, especially in relating stream metabolic responses to human and other pressures. The accurate determination of the reaeration coefficient is one challenge for making reliable ecological inferences from DO measurements made over many diel periods (i.e., months or years). We outline three methods for calculating atmospheric reaeration but concentrate on the use of statistical estimation to simultaneously estimate reaeration and metabolic rates using Bayesian model fitting. While there are existing programs (ModelMaker and Bayesian Metabolic Model [BaMM]), these are either slow or unable to be used easily for fitting multiple days of metabolic data (one to many months). Our implementation, BAyesian Single‐station Estimation (BASE), uses freely available software (R and OpenBUGS), includes a batch mode that can fit data for many days, and provides visual and statistical measures of “goodness‐of‐fit.” We compare the results of the BASE, ModelMaker, and BaMM programs.

Distribution, Abundance, and Habitat Associations of a Large Bivalve (Panopea generosa) in a Eutrophic Fjord Estuary

ABSTRACT Marine bivalves are important ecosystem constituents and frequently support valuable fisheries. In many nearshore areas, human disturbance—including declining habitat and water quality—can affect the distribution and abundance of bivalve populations, and complicate ecosystem and fishery management assessments. Infaunal bivalves, in particular, are frequently cryptic and difficult to detect; thus, assessing potential impacts on their populations requires suitable, scalable methods for estimating abundance and distribution. In this study, population size of a common benthic bivalve (the geoduck Panopea generosa) is estimated with a Bayesian habitat-based model fit to scuba and tethered camera data in Hood Canal, a fjord basin in Washington state. Densities declined more than two orders of magnitude along a north—south gradient, concomitant with patterns of deepwater dissolved oxygen, and intensity and duration of seasonal hypoxia. Across the basin, geoducks were most abundant in loose, unconsolidat...

Inference of perceptual priors from path dynamics of passive self-motion.

The monitoring of one's own spatial orientation depends on the ability to estimate successive self-motion cues accurately. This process has become to be known as path integration. A feature of sequential cue estimation, in general, is that the history of previously experienced stimuli, or priors, biases perception. Here, we investigate how during angular path integration, the prior imparted by the displacement path dynamics affects the translation of vestibular sensations into perceptual estimates. Subjects received successive whole-body yaw rotations and were instructed to report their position within a virtual scene after each rotation. The overall movement trajectory either followed a parabolic path or was devoid of explicit dynamics. In the latter case, estimates were biased toward the average stimulus prior and were well captured by an optimal Bayesian estimator model fit to the data. However, the use of parabolic paths reduced perceptual uncertainty, and a decrease of the average size of bias and thus the weight of the average stimulus prior were observed over time. The produced estimates were, in fact, better accounted for by a model where a prediction of rotation magnitude is inferred from the underlying path dynamics on each trial. Therefore, when passively displaced, we seem to be able to build, over time, from sequential vestibular measurements an internal model of the vehicle's movement dynamics. Our findings suggest that in ecological conditions, vestibular afference can be internally predicted, even when self-motion is not actively generated by the observer, thereby augmenting both the accuracy and precision of displacement perception.

Bayesian reference condition models achieve comparable or superior performance to existing standard techniques

AbstractDespite the existence of many approaches to reference-condition modeling, Bayesian statistical methods have not been used. We assessed whether a hybrid approach that combined features of existing reference-condition approaches with Bayesian model fitting and assessment of test sites could provide superior results to existing established methods. We used 4 Bayesian models of increasing complexity to develop and test reference-condition models for 5 biotic endpoints across 3 data sets. Our best models were comparable or superior to standard approaches (Benthic Assessment of Sediment, Australian River Assessment System) using the same data. Those of our models with the simplest endpoint (species richness) performed best. On average, those models with the simplest model structures also performed best, but differences in performance among models of different complexity were small. All models performed poorly at detecting the lower levels of simulated impact in the test data. However, these impacts were...

Assessing fit in Bayesian models for spatial processes

Gaussian random fields are frequently used to model spatial and spatial–temporal data, particularly in geostatistical settings. As much of the attention of the statistics community has been focused on defining and estimating the mean and covariance functions of these processes, little effort has been devoted to developing goodness‐of‐fit tests to allow users to assess the models' adequacy. We describe a general goodness‐of‐fit test and related graphical diagnostics for assessing the fit of Bayesian Gaussian process models using pivotal discrepancy measures. Our method is applicable for both regularly and irregularly spaced observation locations on planar and spherical domains. The essential idea behind our method is to evaluate pivotal quantities defined for a realization of a Gaussian random field at parameter values drawn from the posterior distribution. Because the nominal distribution of the resulting pivotal discrepancy measures is known, it is possible to quantitatively assess model fit directly from the output of Markov chain Monte Carlo algorithms used to sample from the posterior distribution on the parameter space. We illustrate our method in a simulation study and in two applications. Copyright © 2014 John Wiley & Sons, Ltd.

A novel Bayesian method for inferring and interpreting the dynamics of adaptive landscapes from phylogenetic comparative data.

Our understanding of macroevolutionary patterns of adaptive evolution has greatly increased with the advent of large-scale phylogenetic comparative methods. Widely used Ornstein-Uhlenbeck (OU) models can describe an adaptive process of divergence and selection. However, inference of the dynamics of adaptive landscapes from comparative data is complicated by interpretational difficulties, lack of identifiability among parameter values and the common requirement that adaptive hypotheses must be assigned a priori. Here, we develop a reversible-jump Bayesian method of fitting multi-optima OU models to phylogenetic comparative data that estimates the placement and magnitude of adaptive shifts directly from the data. We show how biologically informed hypotheses can be tested against this inferred posterior of shift locations using Bayes Factors to establish whether our a priori models adequately describe the dynamics of adaptive peak shifts. Furthermore, we show how the inclusion of informative priors can be used to restrict models to biologically realistic parameter space and test particular biological interpretations of evolutionary models. We argue that Bayesian model fitting of OU models to comparative data provides a framework for integrating of multiple sources of biological data-such as microevolutionary estimates of selection parameters and paleontological timeseries-allowing inference of adaptive landscape dynamics with explicit, process-based biological interpretations.

Meta-analyses using information reweighting: An application to online advertising

Because technology-enabled marketing research has led to information arriving at a rapid pace, methods in marketing that allow for coherent, sequential and fast information integration are needed. We propose in this research a new approach to information integration: Information Reweighted Priors (IRPs). It is a sample reweighting approach which utilizes the output from a Bayesian model fit using Markov Chain Monte Carlo, with no restrictions on the likelihood, prior distributions, or data structure; hence a general purpose tool. We demonstrate the approach with simulated datasets and an online advertising dataset with external information obtained from i) previous advertising studies in the industry from a major online advertising portal, ii) past academic studies of online adverting and iii) out-of-sample summaries of the dataset.

Anthropogenic land-use signals propagate through stream food webs in a California, USA, watershed

Human development of watersheds can change aquatic ecosystems via multiple pathways. For instance, human rural development may add nutrients to ecosystems. We used naturally occurring stable isotopes in stream food webs to investigate how land use affects stream ecosystems across a gradient of land development in the San Lorenzo watershed, California. Road density was used as a proxy for land development. We found that streams in watersheds with higher road densities had elevated concentrations of phosphate and nitrate. Furthermore, algal δ15N values increased as a function of nitrate concentration, but saturated at approximately 6‰. This saturating pattern was consistent with a two-source mixing model with anthropogenic and watershed sources, fit using Bayesian model fitting. In sites that had >2.6 km roads km−2, anthropogenic sources of N were estimated to represent >90% of the N pool. This anthropogenic N signal was propagated to stream consumers: rainbow trout (Oncorhynchus mykiss), signal crayfish (Pacifasticus leniusculus), and benthic invertebrate δ15N were positively correlated with algal δ15N. Even relatively low density rural human land use may have substantial impacts on nutrient cycling of stream ecosystems.

Bayesian lensing shear measurement

We derive an estimator of weak gravitational lensing shear from background galaxy images that avoids noise-induced biases through a rigorous Bayesian treatment of the measurement. The derived shear estimator disposes with the assignment of ellipticities to individual galaxies that is typical of previous approaches to galaxy lensing. Shear estimates from the mean of the Bayesian posterior are unbiased in the limit of large number of background galaxies, regardless of the noise level on individual galaxies. The Bayesian formalism requires a prior describing the (noiseless) distribution of the target galaxy population over some parameter space; this prior can be constructed from low-noise images of a subsample of the target population, attainable from long integrations of a fraction of the survey field. We find two ways to combine this exact treatment of noise with rigorous treatment of the effects of the instrumental point-spread function and sampling. The Bayesian model fitting (BMF) method assigns a likelihood of the pixel data to galaxy models (e.g. Sersic ellipses). The Bayesian Fourier domain (BFD) method compresses the pixel data to a small set of weighted moments calculated after PSF correction in Fourier space. A numerical test using a simplified model of a biased galaxy measurement process demonstrates that the Bayesian formalism recovers applied shears to $<1$ part in $10^3$ accuracy and provides accurate uncertainty estimates. BFD is the first shear measurement algorithm that is model-free and requires no approximations or ad hoc assumptions in correcting for the effects of PSF, noise, or sampling on the galaxy images. These algorithms are good candidates for attaining the part-per-thousand shear inference required for hemisphere-scale weak gravitational lensing surveys. (abridged)

Inference for Size Demography From Point Pattern Data Using Integral Projection Models

Population dynamics with regard to evolution of traits has typically been studied using matrix projection models (MPMs). Recently, to work with continuous traits, integral projection models (IPMs) have been proposed. Imitating the path with MPMs, IPMs are handled first with a fitting stage, then with a projection stage. Fitting these models has so far been done only with individual-level transition data. These data are used to estimate the demographic functions (survival, growth, fecundity) that comprise the kernel of the IPM specification. Then, the estimated kernel is iterated from an initial trait distribution to project steady state population behavior under this kernel. When trait distributions are observed over time, such an approach does not align projected distributions with these observed temporal benchmarks. The contribution here, focusing on size distributions, is to address this issue. Our concern is that the above approach introduces an inherent mismatch in scales. The redistribution kernel in the IPM proposes a mechanistic description of population level redistribution. A kernel of the same functional form, fitted to data at the individual level, would provide a mechanistic model for individual-level processes. Resulting parameter estimates and the associated estimated kernel are at the wrong scale and do not allow population-level interpretation. Our approach views the observed size distribution at a given time as a point pattern over a bounded interval. We build a three-stage hierarchical model to infer about the dynamic intensities used to explain the observed point patterns. This model is driven by a latent deterministic IPM and we introduce uncertainty by having the operating IPM vary around this deterministic specification. Further uncertainty arises in the realization of the point pattern given the operating IPM. Fitted within a Bayesian framework, such modeling enables full inference about all features of the model. Such dynamic modeling, optimized by fitting data observed over time, is better suited to projection. Exact Bayesian model fitting is very computationally challenging; we offer approximate strategies to facilitate computation. We illustrate with simulated data examples as well as well as a set of annual tree growth data from Duke Forest in North Carolina. A further example shows the benefit of our approach, in terms of projection, compared with the foregoing individual level fitting.

Using the functional response of a consumer to predict biotic resistance to invasive prey

Predators sometimes provide biotic resistance against invasions by nonnative prey. Understanding and predicting the strength of biotic resistance remains a key challenge in invasion biology. A predator's functional response to nonnative prey may predict whether a predator can provide biotic resistance against nonnative prey at different prey densities. Surprisingly, functional responses have not been used to make quantitative predictions about biotic resistance. We parameterized the functional response of signal crayfish (Pacifastacus leniusculus) to invasive New Zealand mud snails (Potamopyrgus antipodarum; NZMS) and used this functional response and a simple model of NZMS population growth to predict the probability of biotic resistance at different predator and prey densities. Signal crayfish were effective predators of NZMS, consuming more than 900 NZMS per predator in a 12-h period, and Bayesian model fitting indicated their consumption rate followed a type 3 functional response to NZMS density. Based on this functional response and associated parameter uncertainty, we predict that NZMS will be able to invade new systems at low crayfish densities (< 0.2 crayfish/m2) regardless of NZMS density. At intermediate to high crayfish densities (> 0.2 crayfish/m2), we predict that low densities of NZMS will be able to establish in new communities; however, once NZMS reach a threshold density of -2000 NZMS/m2, predation by crayfish will drive negative NZMS population growth. Further, at very high densities, NZMS overwhelm predation by crayfish and invade. Thus, interacting thresholds of propagule pressure and predator densities define the probability of biotic resistance. Quantifying the shape and uncertainty of predator functional responses to nonnative prey may help predict the outcomes of invasions.

A Bayesian approach to calibrating period-luminosity relations of RR Lyrae stars in the mid-infrared

A Bayesian approach to calibrating period-luminosity (PL) relations has substantial benefits over generic least-squares fits. In particular, the Bayesian approach takes into account the full prior distribution of the model parameters, such as the a priori distances, and refits these parameters as part of the process of settling on the most highly-constrained final fit. Additionally, the Bayesian approach can naturally ingest data from multiple wavebands and simultaneously fit the parameters of PL relations for each waveband in a procedure that constrains the parameter posterior distributions so as to minimize the scatter of the final fits appropriately in all wavebands. Here we describe the generalized approach to Bayesian model fitting and then specialize to a detailed description of applying Bayesian linear model fitting to the mid-infrared PL relations of RR Lyrae variable stars. For this example application we quantify the improvement afforded by using a Bayesian model fit. We also compare distances previously predicted in our example application to recently published parallax distances measured with the Hubble Space Telescope and find their agreement to be a vindication of our methodology. Our intent with this article is to spread awareness of the benefits and applicability of this Bayesian approach and encourage future PL relation investigations to consider employing this powerful analysis method.

Bayesian modeling for genetic anticipation in presence of mutational heterogeneity: a case study in Lynch syndrome.

Genetic anticipation, described by earlier age of onset (AOO) and more aggressive symptoms in successive generations, is a phenomenon noted in certain hereditary diseases. Its extent may vary between families and/or between mutation subtypes known to be associated with the disease phenotype. In this article, we posit a Bayesian approach to infer genetic anticipation under flexible random effects models for censored data that capture the effect of successive generations on AOO. Primary interest lies in the random effects. Misspecifying the distribution of random effects may result in incorrect inferential conclusions. We compare the fit of four-candidate random effects distributions via Bayesian model fit diagnostics. A related statistical issue here is isolating the confounding effect of changes in secular trends, screening, and medical practices that may affect time to disease detection across birth cohorts. Using historic cancer registry data, we borrow from relative survival analysis methods to adjust for changes in age-specific incidence across birth cohorts. Our motivating case study comes from a Danish cancer register of 124 families with mutations in mismatch repair (MMR) genes known to cause hereditary nonpolyposis colorectal cancer, also called Lynch syndrome (LS). We find evidence for a decrease in AOO between generations in this article. Our model predicts family-level anticipation effects that are potentially useful in genetic counseling clinics for high-risk families.