베이지안 로지스틱 회귀모형에서의 추론에 대한 연구

A review and comparison of four commonly used Bayesian and maximum likelihood model selection tools

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.

Parameter estimation and model selection for a class of hysteretic systems using Bayesian inference

The dynamic nature of the ocean environment presents numerous challenges to parameter estimation in ocean acoustics. We present a variational Bayesian method which selects an optimal model for parameter estimation in ocean acoustics using Gaussian process (GP) regression. GP model selection is based on Bayesian model selection, which treats the parameters and kernel selection of the model as hyperparameters and uses Bayesian inference to compute a model’s likelihood with respect to its hyperparameters. However, in Bayesian model selection, the integrals over parameter space are often intractable and require approximation or sampling methods such as Markov chain Monte Carlo. GP renders these integrals tractable by treating hyperparameters as multivariate Gaussian distributions, which greatly speeds up the optimization. Furthermore, model selection using GPs allows for straightforward evaluation of uncertainty in the selected model’s likelihood. In this study, we compare results of acoustic parameter estimation from Bayesian and GP model selection using an ocean acoustic propagation model.

This article deals with the problem of testing for the equality of the shape parameters in two inverse Weibull distributions. We propose Bayesian hypothesis testing procedures for the equality of the shape parameters under the noninformative prior. The noninformative prior is usually improper which yields a calibration problem that makes the Bayes factor to be defined up to a multiplicative constant. So we propose the default Bayesian hypothesis testing procedures based on the fractional Bayes factor and the intrinsic Bayes factors under the reference priors. Simulation study and an example are provided.

SummaryWe examine philosophical problems and sampling deficiencies that are associated with current Bayesian hypothesis testing methodology, paying particular attention to objective Bayes methodology. Because the prior densities that are used to define alternative hypotheses in many Bayesian tests assign non-negligible probability to regions of the parameter space that are consistent with null hypotheses, resulting tests provide exponential accumulation of evidence in favour of true alternative hypotheses, but only sublinear accumulation of evidence in favour of true null hypotheses. Thus, it is often impossible for such tests to provide strong evidence in favour of a true null hypothesis, even when moderately large sample sizes have been obtained. We review asymptotic convergence rates of Bayes factors in testing precise null hypotheses and propose two new classes of prior densities that ameliorate the imbalance in convergence rates that is inherited by most Bayesian tests. Using members of these classes, we obtain analytic expressions for Bayes factors in linear models and derive approximations to Bayes factors in large sample settings.

In this study, three types of Bayesian coincident-index models were designed based on modification from the so-called Stock-Watson (S-W) index, i.e. the established economic coincident-index. The S-W index approach is commonly used to estimate parameters using sufficient time series data in composite index. However, the S-W approach in small time series data sets, irregular observations and unbalanced data across time is not well-studied. Essentially, all the problem situations are easily handled under the proposed class of Bayesian models. Therefore, we were motivated to design a new Bayesian S-W coincident-index with application in river water quality index. The river water quality data has been chosen due to the availability of small, unbalanced data sets and irregular observations at particular time in selected sampling sites. We tested all the new algorithms which are based on the three coincident-index model i.e. model with no lag, model with first order lag in the water quality variables and model with a lag for unobserved water quality index. We used Bayesian model comparison criterion to choose the best coincident-index model. Deviance Information Criteria (DIC) was performed as a criterion for selecting models since its application to a variety of data-diagnostic models was not well-discussed. Sensitivity model analyses for all models were also reported. The results showed that the model with σ prior 2 was the most appropriate for the Langat river water quality data in the selected sampling sites. The results also showed that that the third model with σ prior 2 for all sampling sites was the most adequate.

ObjectiveA putative relationship between markers for the serotonin system and the personality scale self-transcendence (ST) and its subscale spiritual acceptance (SA) has been demonstrated in a previous PET study of 5-HT1A receptor binding in healthy control subjects. The results could however not be replicated in a subsequent PET study at an independent centre. In this study, we performed a replication of our original study in a larger sample using Bayesian hypothesis testing to evaluate relative evidence both for and against this hypothesis.MethodsRegional 5-HT1A receptor binding potential (BPND) was examined in 50 healthy male subjects using PET with the radioligand [11C]WAY100635. 5-HT1Aavailability was calculated using the simplified reference tissue model (SRTM) yielding regional BPND. ST and SA were measured using the Temperament and Character Inventory (TCI) questionnaire. Correlations between ST/SA scores and 5-HT1ABPND in frontal cortex, hippocampus and raphe nuclei were examined by calculation of default correlation Bayes factors (BFs) and replication BFs.ResultsThere were no significant correlations between 5-HT1A receptor binding and ST/SA scores. Rather, five of six replication BFs provided moderate to strong evidence for no association between 5-HT1A availability and ST/SA, while the remaining BF provided only weak evidence.ConclusionWe could not replicate our previous findings of an association between 5-HT1A availability and the personality trait ST/SA. Rather, the Bayesian analysis provided evidence for a lack of correlation. Further research should focus on whether other components of the serotonin system may be related to ST or SA. This study also illustrates how Bayesian hypothesis testing allows for greater flexibility and more informative conclusions than traditional p-values, suggesting that this approach may be advantageous for analysis of molecular imaging data.

Capture-Recapture models estimate unknown population sizes. Eight standard closed population models exist, allowing for time, behavioral, and heterogeneity effects. Bayesian versions of these models are presented and use of Akaike's Information Criterion (AIC) and the Deviance Information Criterion (DIC) are explored as model selection tools, through simulation and real dataset analysis.

Intrinsic priors for testing exponential means

Hybrid Bayes factors for genome-wide association studies when a robust test is used

This dissertation consists of five chapters with three distinct but related research projects. In Chapter 1, we introduce some necessary definitions related to the research work. In Chapter 2, we develop Bayes factor based testing procedures for a general linear hypothesis of the regression coefficients in the context of the normal linear models. We propose two calibration schemes to deal with asymmetry in information of Bayes factor: (i) by controlling Type I error probability of the Bayes factor and (ii) by balancing Type I and Type II error probabilities of the Bayes factor. We evaluate the finite sample performance of the proposed Bayes factors via simulation studies and a real-data application. Experimental results have shown than the proposed Bayes factors perform well in testing the general linear hypothesis of the regression coefficient. In Chapter 3, we consider Bayesian quantile analysis for testing constrained hypotheses in linear models, in which the quantiles the parameters satisfy a simple order restriction. We develop a Bayesian hierarchical model based on the specification the asymmetric Laplace distribution for the error component. We propose a non-iterative sampling algorithm in the Expectation-Maximization (EM) structure to generate independently and identically distributed posterior samples from their posterior distributions of the parameters. Then we adopt the Savage-Dickey density ratios to conduct the multiple comparison with simply order constraints. Simulation studies were conducted to compare the finite sample performance of the proposed non-iterative sampling algorithm with the Gibbs sampling algorithm. In Chapter 4, we consider objective Bayesian analysis for the concordance correlation coefficient (CCC), which is one of the most commonly used metrics to assess agreement of different methods in many practical applications. We develop an objective Bayesian framework for estimating the CCC based a combined use of the multivariate student's t-distribution with noninformative Independence Jeffreys prior for the unknown parameters. Extensive simulation studies are conducted to compare the performance of the proposed Bayesian estimates with the ones under the subjective priors in the literature. In Chapter 5, we discuss some ongoing projects related to our research work mentioned above and some interesting problems for future work.

Selecting the best linear regression model

Model Selection Techniques and Merging Rules for Range Data Segmentation Algorithms

A better estimation of surface reaction efficiency of semiconductor-grade graphite with atomic nitrogen, as well as the calibration error are calculated using Bayesian updating based on experimental data. Compared with a conventional deterministic model, the stochastic model approach is a powerful tool in the sense that the model is capable of taking into account underlying error correlations among the data quantities. In this paper, we investigate four different stochastic models (called “stochastic system model classes” herein) corresponding to different descriptions of modeling and measurement error structures, given one deterministic physical model. These stochastic system model classes differ in the covariance matrix structure that is used in the uncertainty model to represent uncertainties associated with the physical model and experimental measurements. For each model class, Bayesian inference is used to estimate the posterior probabilities of the physical model parameters as well as of the stochastic model parameters. Model comparison and selection are then applied based on two measures including Bayesian evidence and Bayesian information criterion, as well as the deviance information criterion. Both measures suggest the stochastic model class, which considers that a correlation between errors in two data quantities among different data points is the most plausible. With the stochastic model class, the range of uncertainty in surface reaction efficiency is estimated to be about two orders of magnitude at .