Bayesian Model Selection Methods Research Articles

Universal Framework of Bayesian Creep Model Selection for Steel

The creep deformation process is constructed by complex interactions of multiple factors, and the measurement of creep deformation requires enormous economic costs and a long experimental time, so there is a small amount of measurement data. In such a situation, multiple models are often proposed to explain the same experimental data. The coexistence of multiple models based on different physical assumptions makes it difficult to understand the creep deformation process. The purpose of this study is to construct a framework to compare and evaluate coexistence models based on measurement data using the Bayesian model selection framework. Basically, in the creep deformation model, basis functions, usually the exponential function eat or power function tb, corresponding to the primary creep stage and the tertiary creep stage are set, and the time series of the whole creep deformation process is regressed as the linear sum of these bases. The parameters a and b of the basis functions are parameters that must also be optimized to regress the time series data. Izuno et al. proposed the Bayesian model selection method for the creep deformation process, and it was confirmed that the model with the linear term corresponding to the secondary creep stage, called the modified theta method, is a good model compared to the model without adding the linear term. To obtain the criterion of Bayesian model selection analytically, which can reduce the computational cost greatly, Izuno et al. treated only the regression coefficients as the probabilistic parameter of the creep model. In this method, parameters a and b of the basis functions are optimized by a grid search. Because the grid search requires significant computational costs if multiple basis functions are set for each creep stage, as in the model proposed by Kimura et al., it is difficult to apply the algorithm to models with multiple bases. In addition, if parameters a and b of the basis functions correlate with the regression coefficients, the method of Izuno et al. would offer a biased evaluation of the models. To improve such an existing method, we proposed a Bayesian model selection method that treats the parameters of the basis functions as probabilistic parameters. The large computational cost to treat the parameters of the basis functions as probabilistic parameters was reduced using the exchange Monte Carlo method, which is an efficient sampling method. By applying our proposed method to the evaluation of the creep deformation model, information was obtained to understand the creep deformation process deeply. For example, it was confirmed that there exists a model in which there is a correlation between the parameter of basis function and the regression coefficients.

Signals of large scale climate drivers, hatchery enhancement, and marine factors in Yukon River Chinook salmon survival revealed with a Bayesian life history model.

Understanding how species might respond to climate change involves disentangling the influence of co-occurring environmental factors on population dynamics, and is especially problematic for migratory species like Pacific salmon that move between ecosystems. To date, debate surrounding the causes of recent declines in Yukon River Chinook salmon (Oncorhynchus tshawytscha) abundance has centered on whether factors in freshwater or marine environments control variation in survival, and how these populations at the northern extremity of the species range will respond to climate change. To estimate the effect of factors in marine and freshwater environments on Chinook salmon survival, we constructed a stage-structured assessment model that incorporates the best available data, estimates incidental marine bycatch mortality in trawl fisheries, and uses Bayesian model selection methods to quantify support for alternative hypotheses. Models fitted to two index populations of Yukon River Chinook salmon indicate that processes in the nearshore and marine environments are the most important determinants of survival. Specifically, survival declines when ice leaves the Yukon River later in the spring, increases with wintertime temperature in the Bering Sea, and declines with the abundance of globally enhanced salmon species consistent with competition at sea. In addition, we found support for density-dependent survival limitations in freshwater but not marine portions of the life cycle, increasing average survival with ocean age, and age-specific selectivity of bycatch mortality in the Bering Sea. This study underscores the utility of flexible estimation models capable of fitting multiple data types and evaluating mortality from both natural and anthropogenic sources in multiple habitats. Overall, these analyses suggest that mortality at sea is the primary driver of population dynamics, yet under warming climate Chinook salmon populations at the northern extent of the species' range may be expected to fare better than southern populations, but are influenced by foreign salmon production.

Big Data Bayesian Linear Regression and Variable Selection by Normal-Inverse-Gamma Summation

We introduce the normal-inverse-gamma summation operator, which combines Bayesian regression results from different data sources and leads to a simple split-and-merge algorithm for big data regressions. The summation operator is also useful for computing the marginal likelihood and facilitates Bayesian model selection methods, including Bayesian LASSO, stochastic search variable selection, Markov chain Monte Carlo model composition, etc. Observations are scanned in one pass and then the sampler iteratively combines normal-inverse-gamma distributions without reloading the data. Simulation studies demonstrate that our algorithms can efficiently handle highly correlated big data. A real-world data set on employment and wage is also analyzed.

Resolved Stellar Streams around NGC 4631 from a Subaru/Hyper Suprime-Cam Survey

Abstract We present the first results of the Subaru/Hyper Suprime-Cam survey of the interacting galaxy system, NGC 4631 and NGC 4656. From the maps of resolved stellar populations, we identify 11 dwarf galaxies (including already-known dwarfs) in the outer region of NGC 4631 and the two tidal stellar streams around NGC 4631, named Stream SE and Stream NW, respectively. This paper describes the fundamental properties of these tidal streams. Based on the tip of the red giant branch method and the Bayesian statistics, we find that Stream SE (7.10 Mpc in expected a posteriori, EAP, with 90% credible intervals of [6.22, 7.29] Mpc) and Stream NW (7.91 Mpc in EAP with 90% credible intervals of [6.44, 7.97] Mpc) are located in front of and behind NGC 4631, respectively. We also calculate the metallicity distribution of stellar streams by comparing the member stars with theoretical isochrones on the color–magnitude diagram. We find that both streams have the same stellar population based on the Bayesian model selection method, suggesting that they originated from a tidal interaction between NGC 4631 and a single dwarf satellite. The expected progenitor has a positively skewed metallicity distribution function with , with 90% credible intervals of [−1.46, −0.51]. The stellar mass of the progenitor is estimated as , with 90% credible intervals of ] based on the mass–metallicity relation for Local group dwarf galaxies. This is in good agreement with the initial stellar mass of the progenitor that was presumed in the previous N-body simulation.

The discounting model selector: Statistical software for delay discounting applications.

Original, open-source computer software was developed and validated against established delay discounting methods in the literature. The software executed approximate Bayesian model selection methods from user-supplied temporal discounting data and computed the effective delay 50 (ED50) from the best performing model. Software was custom-designed to enable behavior analysts to conveniently apply recent statistical methods to temporal discounting data with the aid of a graphical user interface (GUI). The results of independent validation of the approximate Bayesian model selection methods indicated that the program provided results identical to that of the original source paper and its methods. Monte Carlo simulation (n = 50,000) confirmed that true model was selected most often in each setting. Simulation code and data for this study were posted to an online repository for use by other researchers. The model selection approach was applied to three existing delay discounting data sets from the literature in addition to the data from the source paper. Comparisons of model selected ED50 were consistent with traditional indices of discounting. Conceptual issues related to the development and use of computer software by behavior analysts and the opportunities afforded by free and open-sourced software are discussed and a review of possible expansions of this software are provided.

On Bayesian Exponentially Embedded Family for Model Order Selection

In this paper, we derive a Bayesian model order selection rule by using the exponentially embedded family method, termed Bayesian EEF. Unlike many other Bayesian model selection methods, the Bayesian EEF can use vague proper priors and improper noninformative priors to be objective in the elicitation of parameter priors. Moreover, the penalty term of the rule is shown to be the sum of half of the parameter dimension and the estimated mutual information between parameter and observed data. This helps to reveal the EEF mechanism in selecting model orders and may provide new insights into the open problems of choosing an optimal penalty term for model order selection and choosing a good prior from information theoretic viewpoints. The important example of linear model order selection is given to illustrate the algorithms and arguments. Lastly, the Bayesian EEF that uses Jeffreys prior coincides with the EEF rule derived by frequentist strategies. This shows another interesting relationship between the frequentist and Bayesian philosophies for model selection.

Bayesian inference for kinetic models of biotransformation using a generalized rate equation

Selecting proper rate equations for the kinetic models is essential to quantify biotransformation processes in the environment. Bayesian model selection method can be used to evaluate the candidate models. However, comparisons of all plausible models can result in high computational cost, while limiting the number of candidate models may lead to biased results. In this work, we developed an integrated Bayesian method to simultaneously perform model selection and parameter estimation by using a generalized rate equation. In the approach, the model hypotheses were represented by discrete parameters and the rate constants were represented by continuous parameters. Then Bayesian inference of the kinetic models was solved by implementing Markov Chain Monte Carlo simulation for parameter estimation with the mixed (i.e., discrete and continuous) priors. The validity of this approach was illustrated through a synthetic case and a nitrogen transformation experimental study. It showed that our method can successfully identify the plausible models and parameters, as well as uncertainties therein. Thus this method can provide a powerful tool to reveal more insightful information for the complex biotransformation processes.

Efficient nonparametric and asymptotic Bayesian model selection methods for attributed graph clustering

Attributed graph clustering, also known as community detection on attributed graphs, attracts much interests recently due to the ubiquity of attributed graphs in real life. Many existing algorithms have been proposed for this problem, which are either distance based or model based. However, model selection in attributed graph clustering has not been well addressed, that is, most existing algorithms assume the cluster number to be known a priori. In this paper, we propose two efficient approaches for attributed graph clustering with automatic model selection. The first approach is a popular Bayesian nonparametric method, while the second approach is an asymptotic method based on a recently proposed model selection criterion, factorized information criterion. Experimental results on both synthetic and real datasets demonstrate that our approaches for attributed graph clustering with automatic model selection significantly outperform the state-of-the-art algorithm.

Supplemental Material for A Comparison of Bayesian and Frequentist Model Selection Methods for Factor Analysis Models

Supplemental Material for A Comparison of Bayesian and Frequentist Model Selection Methods for Factor Analysis Models

Bayesian estimation of tension in bridge hangers using modal frequency measurements

The tension of an arch bridge hanger is estimated using a number of experimentally identified modal frequencies. The hanger is connected through metallic plates to the bridge deck and arch. Two different categories of model classes are considered to simulate the vibrations of the hanger: an analytical model based on the Euler-Bernoulli beam theory, and a high-fidelity finite element (FE) model. A Bayesian parameter estimation and model selection method is used to discriminate between models, select the best model, and estimate the hanger tension and its uncertainty. It is demonstrated that the end plate connections and boundary conditions of the hanger due to the flexibility of the deck/arch significantly affect the estimate of the axial load and its uncertainty. A fixed-end high fidelity FE model of the hanger underestimates the hanger tension by more than 20 compared to a baseline FE model with flexible supports. Simplified beam models can give fairly accurate results, close to the ones obtained from the high fidelity FE model with flexible support conditions, provided that the concept of equivalent length is introduced and/or end rotational springs are included to simulate the flexibility of the hanger ends. The effect of the number of experimentally identified modal frequencies on the estimates of the hanger tension and its uncertainty is investigated.

A mixture of g-priors for variable selection when the number of regressors grows with the sample size

We consider the problem of variable selection in linear regression using mixtures of g-priors. A number of mixtures have been proposed in the literature which work well, especially when the number of regressors p is fixed. In this paper, we propose a mixture of g-priors suitable for the case when p grows with the sample size n, more specifically when $$p=O(n^b)$$ , $$0<b<1$$ . The marginal density based on the proposed mixture has a nice approximation with a closed form expression, which makes application of the method as tractable as an information criterion-based method. The proposed method satisfies fundamental properties like model selection consistency when the true model lies in the model space, and also consistency in an appropriate sense, under misspecified models setup. The method is quite robust in the sense that the above properties are not confined to normal linear models; they continue to hold under reasonable conditions for a general class of error distributions. Finally, we compare the performance of the proposed prior theoretically with that of some other mixtures of g-priors. We also compare it with several other Bayesian methods of model selection using simulated data sets. Theoretically, as well as in simulations, it emerges that unlike most of the other methods of model selection, the proposed prior is competent enough while selecting the true model irrespective of its dimension.

Bayesian selection of graphical regulatory models

We define a new class of coloured graphical models, called regulatory graphs. These graphs have their own distinctive formal semantics and can directly represent typical qualitative hypotheses about regulatory processes like those described by various biological mechanisms. They admit an embellishment into classes of probabilistic statistical models and so standard Bayesian methods of model selection can be used to choose promising candidate explanations of regulation. Regulation is modelled by the existence of a deterministic relationship between the longitudinal series of observations labelled by the receiving vertex and the donating one. This class contains longitudinal cluster models as a degenerate graph. Edge colours directly distinguish important features of the mechanism like inhibition and excitation and graphs are often cyclic. With appropriate distributional assumptions, because the regulatory relationships map onto each other through a group structure, it is possible to define a conditional conjugate analysis. This means that even when the model space is huge it is nevertheless feasible, using a Bayesian MAP search, to a discover regulatory network with a high Bayes Factor score. We also show that, like the class of Bayesian Networks, regulatory graphs also admit a formal but distinctive causal algebra. The topology of the graph then represents collections of hypotheses about the predicted effect of controlling the process by tearing out message passers or forcing them to transmit certain signals. We illustrate our methods on a microarray experiment measuring the expression of thousands of genes as a longitudinal series where the scientific interest lies in the circadian regulation of these plants.

Comparison of Bayesian predictive methods for model selection

The goal of this paper is to compare several widely used Bayesian model selection methods in practical model selection problems, highlight their differences and give recommendations about the preferred approaches. We focus on the variable subset selection for regression and classification and perform several numerical experiments using both simulated and real world data. The results show that the optimization of a utility estimate such as the cross-validation (CV) score is liable to finding overfitted models due to relatively high variance in the utility estimates when the data is scarce. This can also lead to substantial selection induced bias and optimism in the performance evaluation for the selected model. From a predictive viewpoint, best results are obtained by accounting for model uncertainty by forming the full encompassing model, such as the Bayesian model averaging solution over the candidate models. If the encompassing model is too complex, it can be robustly simplified by the projection method, in which the information of the full model is projected onto the submodels. This approach is substantially less prone to overfitting than selection based on CV-score. Overall, the projection method appears to outperform also the maximum a posteriori model and the selection of the most probable variables. The study also demonstrates that the model selection can greatly benefit from using cross-validation outside the searching process both for guiding the model size selection and assessing the predictive performance of the finally selected model.

Bayesian model selection methods in modeling small area colon cancer incidence

PurposeMany types of cancer have an underlying spatial incidence distribution. Spatial model selection methods can be useful when determining the linear predictor that best describes incidence outcomes. MethodsIn this article, we examine the applications and benefits of using two different types of spatial model selection techniques, Bayesian model selection and Bayesian model averaging, in relation to colon cancer incidence in the state of Georgia, United States. ResultsBoth methods produce useful results that lead to the determination that median household income and percent African American population are important predictors of colon cancer incidence in the Northern counties of the state, whereas percent persons below poverty level and percent African American population are important in the Southern counties. ConclusionsOf the two presented methods, Bayesian model selection appears to provide more succinct results, but applying the two in combination offers even more useful information into the spatial preferences of the alternative linear predictors.

Price discovery under crisis: uncovering the determinant factors of prices using efficient Bayesian model selection methods

We seek to uncover the determinants of consumer prices in Greece by considering a large set of potential factors and accounting for delayed effects. To accomplish this, we rely on recently presented Bayesian model selection methods which are efficiently adapted in the present context. Consumer prices drop significantly during the periods of discount sales, especially in the last 2 years; VAT changes are mostly absorbed (above 50 %) by the producers, while a drop in retail sales affects prices with a significant lag (of 6 months). Evidence on the ways several other factors affect prices is also obtained.

Bayesian inference of chemical kinetic models from proposed reactions

Bayesian inference provides a natural framework for combining experimental data with prior knowledge to develop chemical kinetic models and quantify the associated uncertainties, not only in parameter values but also in model structure. Most existing applications of Bayesian model selection methods to chemical kinetics have been limited to comparisons among a small set of models, however. The significant computational cost of evaluating posterior model probabilities renders traditional Bayesian methods infeasible when the model space becomes large. We present a new framework for tractable Bayesian model inference and uncertainty quantification using a large number of systematically generated model hypotheses. The approach involves imposing point-mass mixture priors over rate constants and exploring the resulting posterior distribution using an adaptive Markov chain Monte Carlo method. The posterior samples are used to identify plausible models, to quantify rate constant uncertainties, and to extract key diagnostic information about model structure—such as the reactions and operating pathways most strongly supported by the data. We provide numerical demonstrations of the proposed framework by inferring kinetic models for catalytic steam and dry reforming of methane using available experimental data.

Bayesian high-dimensional screening via MCMC

We explore the theoretical and numerical properties of a fully Bayesian model selection method in the context of sparse high-dimensional settings, i.e., p≫n, where p is the number of covariates and n is the sample size. Our method consists of (1) a hierarchical Bayesian model with a novel prior placed over the model space which includes a hyperparameter tn controlling the model size and (2) an efficient MCMC algorithm for automatic and stochastic search of the models. Our theory shows that, when specifying tn correctly, the proposed method yields selection consistency, i.e., the posterior probability of the true model asymptotically approaches one; when tn is misspecified, the selected model is still asymptotically nested in the true model. The theory also reveals insensitivity of the selection result with respect to the choice of tn. In implementations, a reasonable prior is further assumed on tn. Our approach conducts selection, estimation and even inference in a unified framework. No additional prescreening or dimension reduction step is needed. Two novel g-priors are proposed to make our approach more flexible. The numerical advantages of the proposed approach are demonstrated through comparisons with sure independence screening (SIS).

Evidence for substructure in Ursa Minor dwarf spheroidal galaxy using a Bayesian object detection method

We present a method for identifying localized secondary populations in stellar velocity data using Bayesian statistical techniques. We apply this method to the dwarf spheroidal galaxy Ursa Minor and find two secondary objects in this satellite of the Milky Way. One object is kinematically cold with a velocity dispersion of $4.25 \pm 0.75\ \kms$ and centered at $(9.1\arcmin \pm 1.5, 7.2\arcmin \pm 1.2)$ in relative RA and DEC with respect to the center of Ursa Minor. The second object has a large velocity offset of $-12.8^{+1.75}_{-1.5}\ \kms$ compared to Ursa Minor and centered at $(-14.0\arcmin^{+2.4}_{-5.8}, -2.5\arcmin^{+0.4}_{-1.0})$. The kinematically cold object has been found before using a smaller data set but the prediction that this cold object has a velocity dispersion larger than $2.0\ \kms$ at 95% C.L. differs from previous work. We use two and three component models along with the information criteria and Bayesian evidence model selection methods to argue that Ursa Minor has one or two localized secondary populations. The significant probability for a large velocity dispersion in each secondary object raises the intriguing possibility that each has its own dark matter halo, that is, it is a satellite of a satellite of the Milky Way.

Errata: A survey of Bayesian predictive methods for model assessment, selection and comparison

Errata: A survey of Bayesian predictive methods for model assessment, selection and comparison

Effective Parameter Dimension via Bayesian Model Selection in the Inverse Acoustic Scattering Problem

We address a prototype inverse scattering problem in the interface of applied mathematics, statistics, and scientific computing. We pose the acoustic inverse scattering problem in a Bayesian inference perspective and simulate from the posterior distribution using MCMC. The PDE forward map is implemented using high performance computing methods. We implement a standard Bayesian model selection method to estimate an effective number of Fourier coefficients that may be retrieved from noisy data within a standard formulation.