Bayesian revision of a prior given prior-data conflict,expert opinion,or a similar insight: a large-deviation approach

Learning from model diagnostics that a prior distribution must be replaced by one that conflicts less with the data raises the question of which prior should instead be used for inference and decision. The same problem arises when a decision maker learns that one or more reliable experts express unexpected beliefs. In both cases, coherence of the solution would be guaranteed by applying Bayes's theorem to a distribution of prior distributions that effectively assigns the initial prior distribution a probability arbitrarily close to 1. The new distribution for inference would then be the distribution of priors conditional on the insight that the prior distribution lies in a closed convex set that does not contain the initial prior. A readily available distribution of priors needed for such conditioning is the law of the empirical distribution of sufficiently large number of independent parameter values drawn from the initial prior. According to the Gibbs conditioning principle from the theory of large deviations, the resulting new prior distribution minimizes the entropy relative to the initial prior. While minimizing relative entropy accommodates the necessity of going beyond the initial prior without departing from it any more than the insight demands, the large-deviation derivation also ensures the advantages of Bayesian coherence. This approach is generalized to uncertain insights by allowing the closed convex set of priors to be random.     

A quasi-Bayesian model averaging approach for conditional quantile models

The value at risk (VaR) is a risk measure that is widely used by financial institutions to allocate risk. VaR forecast estimation involves the evaluation of conditional quantiles based on the currently available information. Recent advances in VaR evaluation incorporate conditional variance into the quantile estimation, which yields the conditional autoregressive VaR (CAViaR) models. However, uncertainty with regard to model selection in CAViaR model estimators raises the issue of identifying the better quantile predictor via averaging. In this study, we propose a quasi-Bayesian model averaging method that generates combinations of conditional VaR estimators based on single CAViaR models. This approach provides us a basis for comparing single CAViaR models against averaged ones for their ability to forecast VaR. We illustrate this method using simulated and financial daily return data series. The results demonstrate significant findings with regard to the use of averaged conditional VaR estimates when forecasting quantile risk.     

Bayesian mixture model averaging for identifying the different gene expressions of chickpea (Cicer arietinum) plant tissue

Identification of different gene expressions of chickpea (Cicer arietinum) plant tissue is needed in order to develop new varieties of chickpea plant which is resistant to disease through the insertion of genes. This plant is the third legume plant of the Leguminosae (Fabaceae) family and is much needed in the world due to its high-protein seeds and roots that contain symbiotic nitrogen-fixing bacteria. This paper has succeeded to demonstrate the work of Bayesian mixture model averaging (BMMA) approach to identify the different gene expressions of chickpea plant tissue in Indonesia. The results show that the best BMMA normal models contain from 727 (73%) up to 939 (94%) models from 1,000 generated mixture normal models. The fitted BMMA models to gene expression differences data on average is 0.2878511 for Kolmogorov–Smirnov (KS) and 0.1278080 for continuous rank probability score (CRPS). Based on these BMMA models, there are three groups of gene IDs: downregulated, regulated, and upregulated. The results of this grouping can be useful to find new varieties of chickpea plants that are more resistant to disease. The BMMA normal models coupled with Occam's window as a data-driven modeling have succeed to demonstrate the work of building the gene expression differences microarray experiments data.     

Bayesian credit ratings: A random forest alternative approach

Cerciello and Giudici (2014 Cerciello, P., Giudici, P. (2014). Bayesian credit ratings. Commun. Stat. Theory Methods. 43:867–878.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) proposed a Bayesian approach to improve the ordinal variable selection in credit rating assessment. However, no comparison has been made with other methods and the predictive power was not tested. This study proposes an integrated framework of random forest (RF)-based methods and Bayesian model averaging (BMA) to validate and investigate the ordinal variable importance in evaluating credit risk and predicting default in greater depth. The proposed approach was superior to the Cerciello and Giudici method in terms of predictive accuracy and interpretability when applied to a European credit risk database.     

Credit risk assessment with Bayesian model averaging

Many credit risk models are based on the selection of a single logistic regression model, on which to base parameter estimation. When many competing models are available, and without enough guidance from economical theory, model averaging represents an appealing alternative to the selection of single models. Despite model averaging approaches have been present in statistics for many years, only recently they are starting to receive attention in economics and finance applications. This contribution shows how Bayesian model averaging can be applied to credit risk estimation, a research area that has received a great deal of attention recently, especially in the light of the global financial crisis of the last few years and the correlated attempts to regulate international finance. The paper considers the use of logistic regression models under the Bayesian Model Averaging paradigm. We argue that Bayesian model averaging is not only more correct from a theoretical viewpoint, but also slightly superior, in terms of predictive performance, with respect to single selected models.     

Generalized Least Squares Model Averaging

In this article, we propose a method of averaging generalized least squares estimators for linear regression models with heteroskedastic errors. The averaging weights are chosen to minimize Mallows’ C_p-like criterion. We show that the weight vector selected by our method is optimal. It is also shown that this optimality holds even when the variances of the error terms are estimated and the feasible generalized least squares estimators are averaged. The variances can be estimated parametrically or nonparametrically. Monte Carlo simulation results are encouraging. An empirical example illustrates that the proposed method is useful for predicting a measure of firms’ performance.     

Extended Bayesian model averaging for heritability in twin studies

Family studies are often conducted to examine the existence of familial aggregation. Particularly, twin studies can model separately the genetic and environmental contribution. Here we estimate the heritability of quantitative traits via variance components of random-effects in linear mixed models (LMMs). The motivating example was a myopia twin study containing complex nesting data structures: twins and siblings in the same family and observations on both eyes for each individual. Three models are considered for this nesting structure. Our proposal takes into account the model uncertainty in both covariates and model structures via an extended Bayesian model averaging (EBMA) procedure. We estimate the heritability using EBMA under three suggested model structures. When compared with the results under the model with the highest posterior model probability, the EBMA estimate has smaller variation and is slightly conservative. Simulation studies are conducted to evaluate the performance of variance-components estimates, as well as the selections of risk factors, under the correct or incorrect structure. The results indicate that EBMA, with consideration of uncertainties in both covariates and model structures, is robust in model misspecification than the usual Bayesian model averaging (BMA) that considers only uncertainty in covariates selection.     

Forecasting Equity Premium: Global Historical Average Versus Local Historical Average and Constraints

The equity premium, return on equity minus return on risk-free asset, is expected to be positive. We consider imposing such positivity constraint in local historical average (LHA) in nonparametric kernel regression framework. It is also extended to the semiparametric single index model when multiple predictors are used. We construct the constrained LHA estimator via an indicator function which operates as “model-selection” between the unconstrained LHA and the bound of the constraint (zero for the positivity constraint). We smooth the indicator function by bagging, which operates as “model-averaging” and yields a combined forecast of unconstrained LHA forecasts and the bound of the constraint. The local combining weights are determined by the probability that the constraint is binding. Asymptotic properties of the constrained LHA estimators without and with bagging are established, which show how the positive constraint and bagging can help reduce the asymptotic variance and mean squared errors. Monte Carlo simulations are conducted to show the finite sample behavior of the asymptotic properties. In predicting U.S. equity premium, we show that substantial nonlinearity can be captured by LHA and that the local positivity constraint can improve out-of-sample prediction of the equity premium.     

Multiple Regression Model Averaging and the Focused Information Criterion With an Application to Portfolio Choice

ABSTRACT

We consider multiple regression (MR) model averaging using the focused information criterion (FIC). Our approach is motivated by the problem of implementing a mean-variance portfolio choice rule. The usual approach is to estimate parameters ignoring the intention to use them in portfolio choice. We develop an estimation method that focuses on the trading rule of interest. Asymptotic distributions of submodel estimators in the MR case are derived using a localization framework. The localization is of both regression coefficients and error covariances. Distributions of submodel estimators are used for model selection with the FIC. This allows comparison of submodels using the risk of portfolio rule estimators. FIC model averaging estimators are then characterized. This extension further improves risk properties. We show in simulations that applying these methods in the portfolio choice case results in improved estimates compared with several competitors. An application to futures data shows superior performance as well.     

Potential Uncertainty Reduction in Model‐Averaged Benchmark Dose Estimates Informed by an Additional Dose Study

A methodology is presented for assessing the information value of an additional dosage experiment in existing bioassay studies. The analysis demonstrates the potential reduction in the uncertainty of toxicity metrics derived from expanded studies, providing insights for future studies. Bayesian methods are used to fit alternative dose‐response models using Markov chain Monte Carlo (MCMC) simulation for parameter estimation and Bayesian model averaging (BMA) is used to compare and combine the alternative models. BMA predictions for benchmark dose (BMD) are developed, with uncertainty in these predictions used to derive the lower bound BMDL. The MCMC and BMA results provide a basis for a subsequent Monte Carlo analysis that backcasts the dosage where an additional test group would have been most beneficial in reducing the uncertainty in the BMD prediction, along with the magnitude of the expected uncertainty reduction. Uncertainty reductions are measured in terms of reduced interval widths of predicted BMD values and increases in BMDL values that occur as a result of this reduced uncertainty. The methodology is illustrated using two existing data sets for TCDD carcinogenicity, fitted with two alternative dose‐response models (logistic and quantal‐linear). The example shows that an additional dose at a relatively high value would have been most effective for reducing the uncertainty in BMA BMD estimates, with predicted reductions in the widths of uncertainty intervals of approximately 30%, and expected increases in BMDL values of 5–10%. The results demonstrate that dose selection for studies that subsequently inform dose‐response models can benefit from consideration of how these models will be fit, combined, and interpreted.