Prior Specification Research Articles

Finding the needle by modeling the haystack: Pulmonary embolism in an emergency patient with cardiorespiratory manifestations

Background:Diagnoses from non-specific symptoms lead to reason over a huge space of hypotheses. Bayesian Networks (BN) can systematically address the task by integrating existing causal medical knowledge and clinical, possibly incomplete, case observations. Although prior specification of a causal BN structure can increase the generality of diagnostic predictions, particular care should be paid to avoid structural inconsistencies with the data at hand. Methods:The causal structure of a BN covering 129 cardiopulmonary disorders and 235 manifestations was elicited by experts. Markov chain Monte Carlo estimation of quantitative parameters was performed on the basis of 282 cases admitted to an Emergency Department and subsequently hospitalized. BN structural features underwent a refinement process to improve parameters sensitivity to data and diagnostic precision, until changes were consistent with published evidence. Discrimination/calibration measures of diagnostic performance (based on 284 test cases) were used to assess the predictive improvement after refinement. Results:Although one third of 1,417 parameter estimates remained anchored to prior settings, the BN updated by data returned adequately calibrated diagnostic predictions on the first six most observed disorders in the test sample, with an M-index of 0.91 (95% CI 0.87, 0.95) on 20 alternative diagnoses. Conclusions:Our large-sized BN effectively combines existing causal medical knowledge with sparse information contained in clinical records. It addresses a broad-spectrum differential diagnosis and informs physicians about possibly neglected cardiopulmonary conditions. The calibration analysis oriented the refinement of initial causal assumptions without loss of generality in diagnostic predictions.

Bayes Factor Model Comparisons Across Parameter Values for Mixed Models

Nested data structures, in which conditions include multiple trials and are fully crossed with participants, are often analyzed using repeated-measures analysis of variance or mixed-effects models. Typically, researchers are interested in determining whether there is an effect of the experimental manipulation. These kinds of analyses have different appropriate specifications for the null and alternative models, and a discussion on which is to be preferred and when is sorely lacking. van Doorn et al. (2021) performed three types of Bayes factor model comparisons on a simulated data set in order to examine which model comparison is most suitable for quantifying evidence for or against the presence of an effect of the experimental manipulation. Here, we extend their results by simulating multiple data sets for various scenarios and by using different prior specifications. We demonstrate how three different Bayes factor model comparison types behave under changes in different parameters, and we make concrete recommendations on which model comparison is most appropriate for different scenarios.

Applying Bayesian model selection to determine ecological covariates for recruitment and natural mortality in stock assessment

Abstract Incorporating ecological covariates into fishery stock assessments may improve estimates, but most covariates are estimated with error. Model selection criteria are often used to identify support for covariates, have some limitations and rely on assumptions that are often violated. For a more rigorous evaluation of ecological covariates, we used four popular selection criteria to identify covariates influencing natural mortality or recruitment in a Bayesian stock assessment of Pacific herring (Clupea pallasii) in Prince William Sound, Alaska. Within this framework, covariates were incorporated either as fixed effects or as latent variables (i.e. covariates have associated error). We found most support for pink salmon increasing natural mortality, which was selected by three of four criteria. There was ambiguous support for other fixed effects on natural mortality (walleye pollock and the North Pacific Gyre Oscillation) and recruitment (hatchery-released juvenile pink salmon and a 1989 regime shift). Generally, similar criteria values among covariates suggest no clear evidence for a consistent effect of any covariate. Models with covariates as latent variables were sensitive to prior specification and may provide potentially very different results. We recommend using multiple criteria and exploring different statistical assumptions about covariates for their use in stock assessment.

Choosing priors in Bayesian ecological models by simulating from the prior predictive distribution

AbstractBayesian data analysis is increasingly used in ecology, but prior specification remains focused on choosing non‐informative priors (e.g., flat or vague priors). One barrier to choosing more informative priors is that priors must be specified on model parameters (e.g., intercepts, slopes, and sigmas), but prior knowledge often exists on the level of the response variable. This is particularly true for common models in ecology, like generalized linear mixed models that have a link function and potentially dozens of parameters, each of which needs a prior distribution. We suggest that this difficulty can be overcome by simulating from the prior predictive distribution and visualizing the results on the scale of the response variable. In doing so, some common choices for non‐informative priors on parameters can easily be seen to produce biologically impossible values of response variables. Such implications of prior choices are difficult to foresee without visualization. We demonstrate a workflow for prior selection using simulation and visualization with two ecological examples (predator–prey body sizes and spider responses to food competition). This approach is not new, but its adoption by ecologists will help to better incorporate prior information in ecological models, thereby maximizing one of the benefits of Bayesian data analysis.

A Comparison of Priors When Using Bayesian Regression to Estimate Oral Reading Fluency Slopes

Bayesian regression has emerged as a viable alternative for the estimation of curriculum-based measurement (CBM) growth slopes. Preliminary findings suggest such methods may yield improved efficiency relative to other linear estimators and can be embedded into data management programs for high-frequency use. However, additional research is needed, as Bayesian estimators require multiple specifications of the prior distributions. The current study evaluates the accuracy of several combinations of prior values, including three distributions of the residuals, two values of the expected growth rate, and three possible values for the precision of slope when using Bayesian simple linear regression to estimate fluency growth slopes for reading CBM. We also included traditional ordinary least squares (OLS) as a baseline contrast. Findings suggest that the prior specification for the residual distribution had, on average, a trivial effect on the accuracy of the slope. However, specifications for growth rate and precision of slope were influential, and virtually all variants of Bayesian regression evaluated were superior to OLS. Converging evidence from both simulated and observed data now suggests Bayesian methods outperform OLS for estimating CBM growth slopes and should be strongly considered in research and practice.

How many data clusters are in the Galaxy data set?

In model-based clustering, the Galaxy data set is often used as a benchmark data set to study the performance of different modeling approaches. Aitkin (Stat Model 1:287–304) compares maximum likelihood and Bayesian analyses of the Galaxy data set and expresses reservations about the Bayesian approach due to the fact that the prior assumptions imposed remain rather obscure while playing a major role in the results obtained and conclusions drawn. The aim of the paper is to address Aitkin’s concerns about the Bayesian approach by shedding light on how the specified priors influence the number of estimated clusters. We perform a sensitivity analysis of different prior specifications for the mixtures of finite mixture model, i.e., the mixture model where a prior on the number of components is included. We use an extensive set of different prior specifications in a full factorial design and assess their impact on the estimated number of clusters for the Galaxy data set. Results highlight the interaction effects of the prior specifications and provide insights into which prior specifications are recommended to obtain a sparse clustering solution. A simulation study with artificial data provides further empirical evidence to support the recommendations. A clear understanding of the impact of the prior specifications removes restraints preventing the use of Bayesian methods due to the complexity of selecting suitable priors. Also, the regularizing properties of the priors may be intentionally exploited to obtain a suitable clustering solution meeting prior expectations and needs of the application.

Alternative Tests of Independence in Two-Way Categorical Tables

The chi-squared test for independence in two-way categorical tables depends on the assumptions that the data follow the multinomial distribution. Thus, we suggest alternatives when the assumptions of multi nomial distribution do not hold. First, we consider the Bayes factor which is used for hypothesis testing in Bayesian statistics. Unfortunately, this has the problem that it is sensitive to the choice of prior distributions. We note here that the intrinsic Bayes factor is not appropriate because the prior distribu tions under consideration are all proper. Thus, we propose using Bayesian estimation which is generally not as sensitive to prior specifications as the Bayes factor. Our approach is to construct a 95% simultaneous credible re gion (i.e., a hyper-rectangle) for the interactions. A test that all interactions are zero is equivalent to a test of independence in two-way categorical tables. Thus, a 95% simultaneous credible region of the interactions provides a test of independence by inversion.

On the link between temperature and regional COVID-19 severity: Evidence from Italy.

This study analyzes the link between temperature and COVID‐19 incidence in a sample of Italian regions during the period that covers the first epidemic wave of 2020. To that end, Bayesian model averaging techniques are used to analyze the relevance of temperature together with a set of additional climatic, demographic, social, and health policy factors. The robustness of individual predictors is measured through posterior inclusion probabilities. The empirical analysis provides conclusive evidence on the role played by temperature given that it appears as one of the most relevant determinants reducing regional coronavirus disease 2019 (COVID‐19) severity. The strong negative link observed in our baseline analysis is robust to the specification of priors, the scale of analysis, the correction of measurement errors in the data due to under‐reporting, the time window considered, and the inclusion of spatial effects in the model. In a second step, we compute relative importance metrics that decompose the variability explained by the model. We find that cross‐regional temperature differentials explain a large share of the observed variation on the number of infections.

Identifying groups of determinants in Bayesian model averaging using Dirichlet process clustering

AbstractModel uncertainty is a pervasive problem in regression applications. Bayesian model averaging (BMA) takes model uncertainty into account and identifies robust determinants. However, it requires the specification of suitable model priors. Mixture model priors are appealing because they explicitly account for different groups of covariates as robust determinants. Specific Dirichlet process clustering (DPC) model priors are proposed; their correspondence to the binomial model prior derived and methods to perform the BMA analysis including a DPC postprocessing procedure to identify groups of determinants are outlined. The application of these model priors is demonstrated in a simulation exercise and in an empirical analysis of cross‐country economic growth data. The BMA analysis is performed using the Markov chain Monte Carlo model composition sampler to obtain samples from the posterior of the model specifications. Results are compared with those obtained under a beta‐binomial and a collinearity‐adjusted dilution model prior.

Optimal Bayesian hierarchical model to accelerate the development of tissue-agnostic drugs and basket trials

Tissue-agnostic trials and basket trials enroll patients based on their genetic biomarkers, not tumor type, in an attempt to determine if a new drug can successfully treat disease conditions based on biomarkers. The Bayesian hierarchical model (BHM) provides an attractive approach to design phase II tissue-agnostic trials by allowing information borrowing across multiple disease types. In this article, we elucidate two intrinsic and inevitable issues that may limit the use of BHM to tissue-agnostic trials: sensitivity to the prior specification of the shrinkage parameter and the competing “interest” among disease types in increasing power and controlling type I error. To address these issues, we propose the optimal BHM (OBHM) and clustered OBHM (COBHM) approaches. In these approach, we first specify a flexible utility function to quantify the tradeoff between type I error and power across disease types based on the study objectives, and then we select the prior of the shrinkage parameter to optimize the utility function of clinical and regulatory interest. COBHM further utilizes a simple Bayesian rule to cluster tumor types into sensitive and insensitive subgroups to achieve more accurate information borrowing. Simulation study shows that the OBHM and especially COBHM have desirable operating characteristics, outperforming some existing methods. COBHM effectively balances power and type I error, addresses the sensitivity of the prior selection, and reduces the “unwarranted” subjectivity in the prior selection. It provides a systematic, rigorous way to apply BHM and solve the common problem of blindingly using a non-informative inverse-gamma prior (with a large variance) or priors arbitrarily chosen that may lead to problematic statistical properties.

Coopetition for innovation - the more, the better? An empirical study based on preference disaggregation analysis

Companies rely widely on coopetition, that is, cooperation with competitors, to foster their innovation. However, despite its increasing popularity, the impact of coopetition on innovation remains unclear. Different theories predict different results, and empirical evidence suggests that coopetition can either improve or reduce innovation performance, or even make no impact. The lack of consensus makes further exploration of this important relationship essential. Our study attempts to shed new light on the impact of sourcing from competitors on innovation, using a data-driven exploratory model that needs no prior specification of the direction and form of this relationship. We use a recently developed analytical method based on the preference disaggregation approach to analyze data from a sample of 112 firms operating in the petroleum, chemical, and pharmaceutical industries. We find that sourcing from competitors fosters the innovation performance of firms with financial constraints (small firms receiving no financial support from the government for innovation). In contrast, our results show that sourcing from competitors has a non-linear negative impact on large and small firms receiving such financial support, and that sourcing from competitors is much riskier for small firms in this category. Our findings contribute to the coopetition for innovation literature by highlighting that the relationship between sourcing from competitors and innovation performance depends on firm characteristics, i.e., size and financial capability. Our use of a preference disaggregation method confirms its value when studying relationships between variables where the conceptual and empirical evidence leaves this relationship unclear.

Bayesian nonparametrics for directional statistics

We introduce a density basis of the trigonometric polynomials that is suitable to mixture modelling. Statistical and geometric properties are derived, suggesting it as a circular analogue to the Bernstein polynomial densities. Nonparametric priors are constructed using this basis and a simulation study shows that the use of the resulting Bayes estimator may provide gains over comparable circular density estimators previously suggested in the literature.From a theoretical point of view, we propose a general prior specification framework for density estimation on compact metric space using sieve priors. This is tailored to density bases such as the one considered herein and may also be used to exploit their particular shape-preserving properties. Furthermore, strong posterior consistency is shown to hold under notably weak regularity assumptions and adaptive convergence rates are obtained in terms of the approximation properties of positive linear operators generating our models.

Chemical Kinetics Bayesian Inference Toolbox (CKBIT)

The robust estimation of chemical kinetic parameters and their associated uncertainty is essential in the field of chemistry and catalysis. The Chemical Kinetics Bayesian Inference Toolbox (CKBIT) is a Python software library introduced to enable users to implement advanced Bayesian inference techniques for kinetic parameter estimation and uncertainty quantification. Leveraging functionalities of other open source Python packages and offering simplified implementation through minimal user-required coding and straightforward Excel input files, CKBIT aspires to make the inference method easily accessible for chemical kinetics. CKBIT provides maximum a posteriori, Markov chain Monte Carlo, and variational inference estimation options. Users may apply these functionalities to estimate activation energies, reaction orders, and pre-exponential terms from chemical reaction data from batch reactors, continuous stirred-tank reactors, and plug flow reactors. The availability of prior distribution specification and the implementation of hierarchical modeling in CKBIT provide a heightened level of accuracy in estimates of kinetic parameters and their uncertainties. Program summaryProgram title: CKBITCPC Library link to program files:https://doi.org/10.17632/tnzk2jvffs.1Developer's repository link:https://github.com/VlachosGroup/ckbitCode Ocean capsule:https://codeocean.com/capsule/8389927Licensing provisions: MIT licenseProgramming language: PythonNature of problem: Rigorous estimation and uncertainty quantification of kinetic rate parameters are necessary for chemical researchers to create physical models with explicit levels of certainty. Advanced statistical treatments of Bayesian inference meet this need with a high level of rigor. However, current software available for Bayesian inference is complex and nuanced in its implementation, preventing widespread adoption among researchers.Solution method: We present a Python package with a modular approach to applying different Bayesian inference techniques for kinetic rate parameter estimation and uncertainty quantification. Optimal point estimates can be obtained through maximum a posteriori estimation, while full probability distributions of parameters can be generated through Markov chain Monte Carlo estimation or variational inference. The code is straightforward to use in contrast to current Bayesian inference software, and it interfaces with Excel for ease of data entry.

Imprecise credibility theory

AbstractThe classical credibility theory is a cornerstone of experience rating, especially in the field of property and casualty insurance. An obstacle to putting the credibility theory into practice is the conversion of available prior information into a precise choice of crucial hyperparameters. In most real-world applications, the information necessary to justify a precise choice is lacking, so we propose animprecise credibility estimatorthat honestly acknowledges the imprecision in the hyperparameter specification. This results in an interval estimator that is doubly robust in the sense that it retains the credibility estimator’s freedom from model specification and fast asymptotic concentration, while simultaneously being insensitive to prior hyperparameter specification.

Robust Bayesian approach to logistic regression modeling in small sample size utilizing a weakly informative student’s t prior distribution

This work discourages using the logistic regression (LR) model for estimative purposes when the sample size is small. We propose a new model called the “Robust Bayesian Logistic (RBL) model” that minimizes bias in the estimated logistic regression coefficients when the sample size is small, and covariate corruption is suspected. For the prior specification in the proposed RBL model, all the logistic regression coefficients are assigned independent Student’s t-distribution with the location parameter 0, scale parameter 1, and degree of freedom 7 for the constant term and degree of freedom of 1 for all other regression coefficients. In our experimental study, the proposed RBL model outperforms the Logistic Regression (LR) model by having a lower mean squared error (MSE) in the regression coefficients estimated for all the sample sizes considered. The proposed RBL model has a lower standard deviation than the LR model for all the estimated regression coefficients on the real-life dataset. We suggest that the proposed RBL model be considered for logistic modeling since it generates stable, consistent, and reliable estimates, especially when the sample size is small. The proposed RBL model is a fully Bayesian method implemented in the R environment using the RJAGS package.

EcoQBNs: First Application of Ecological Modeling with Quantum Bayesian Networks.

A recent advancement in modeling was the development of quantum Bayesian networks (QBNs). QBNs generally differ from BNs by substituting traditional Bayes calculus in probability tables with the quantum amplification wave functions. QBNs can solve a variety of problems which are unsolvable by, or are too complex for, traditional BNs. These include problems with feedback loops and temporal expansions; problems with non-commutative dependencies in which the order of the specification of priors affects the posterior outcomes; problems with intransitive dependencies constituting the circular dominance of the outcomes; problems in which the input variables can affect each other, even if they are not causally linked (entanglement); problems in which there may be >1 dominant probability outcome dependent on small variations in inputs (superpositioning); and problems in which the outcomes are nonintuitive and defy traditional probability calculus (Parrondo’s paradox and the violation of the Sure Thing Principle). I present simple examples of these situations illustrating problems in prediction and diagnosis, and I demonstrate how BN solutions are infeasible, or at best require overly-complex latent variable structures. I then argue that many problems in ecology and evolution can be better depicted with ecological QBN (EcoQBN) modeling. The situations that fit these kinds of problems include noncommutative and intransitive ecosystems responding to suites of disturbance regimes with no specific or single climax condition, or that respond differently depending on the specific sequence of the disturbances (priors). Case examples are presented on the evaluation of habitat conditions for a bat species, representing state-transition models of a boreal forest under disturbance, and the entrainment of auditory signals among organisms. I argue that many current ecological analysis structures—such as state-and-transition models, predator–prey dynamics, the evolution of symbiotic relationships, ecological disturbance models, and much more—could greatly benefit from a QBN approach. I conclude by presenting EcoQBNs as a nascent field needing the further development of the quantum mathematical structures and, eventually, adjuncts to existing BN modeling shells or entirely new software programs to facilitate model development and application.

Parameter Specification in Bayesian CFA: An Exploration of Multivariate and Separation Strategy Priors

ABSTRACT The impact of parameter and prior specifications on Bayesian SEM estimates is examined through two simulation studies. The model of focus was a CFA. Simulation conditions for Study 1 included varying sample size, the strength of the factor loadings (also tied to issues of reliability), factor correlation strength, and estimation conditions tied to different parameter specifications. Study 2 extended these factors and included non-zero cross-loadings to highlight the flexibility that Bayesian methods afford CFAs. The main goal of these studies was to examine the impact of different parameter specifications, as crossed with different forms of prior distributions, on the accuracy of parameter estimates–examined via relative bias. We examined several parameter specification conditions focused on the latent factor covariance specification, and then crossed these conditions with different prior forms (multivariate and separation strategy priors). Findings highlight where parameter specification implemented had an overall larger impact on the accuracy of results obtained.

Evaluating the informativeness of the Besag-York-Mollié CAR model

The use of the conditional autoregressive framework proposed by Besag, York, and Mollié (1991; BYM) is ubiquitous in Bayesian disease mapping and spatial epidemiology. While it is understood that Bayesian inference is based on a combination of the information contained in the data and the information contributed by the model, quantifying the contribution of the model relative to the information in the data is often non-trivial. Here, we provide a measure of the contribution of the BYM framework by first considering the simple Poisson-gamma setting in which quantifying the prior’s contribution is quite clear. We then propose a relationship between gamma and lognormal priors that we then extend to cover the framework proposed by BYM. Following a brief simulation study in which we illustrate the accuracy of our lognormal approximation of the gamma prior, we analyze a dataset comprised of county-level heart disease-related death data across the United States. In addition to demonstrating the potential for the BYM framework to correspond to a highly informative prior specification, we also illustrate the sensitivity of death rate estimates to changes in the informativeness of the BYM framework.

Systematically Defined Informative Priors in Bayesian Estimation: An Empirical Application on the Transmission of Internalizing Symptoms Through Mother-Adolescent Interaction Behavior.

BackgroundBayesian estimation with informative priors permits updating previous findings with new data, thus generating cumulative knowledge. To reduce subjectivity in the process, the present study emphasizes how to systematically weigh and specify informative priors and highlights the use of different aggregation methods using an empirical example that examined whether observed mother-adolescent positive and negative interaction behavior mediate the associations between maternal and adolescent internalizing symptoms across early to mid-adolescence in a 3-year longitudinal multi-method design.MethodsThe sample consisted of 102 mother-adolescent dyads (39.2% girls, Mage T1 = 13.0). Mothers and adolescents reported on their internalizing symptoms and their interaction behaviors were observed during a conflict task. We systematically searched for previous studies and used an expert-informed weighting system to account for their relevance. Subsequently, we aggregated the (power) priors using three methods: linear pooling, logarithmic pooling, and fitting a normal distribution to the linear pool by means of maximum likelihood estimation. We compared the impact of the three differently specified informative priors and default priors on the prior predictive distribution, shrinkage, and the posterior estimates.ResultsThe prior predictive distributions for the three informative priors were quite similar and centered around the observed data mean. The shrinkage results showed that the logarithmic pooled priors were least affected by the data. Most posterior estimates were similar across the different priors. Some previous studies contained extremely specific information, resulting in bimodal posterior distributions for the analyses with linear pooled prior distributions. The posteriors following the fitted normal priors and default priors were very similar. Overall, we found that maternal, but not adolescent, internalizing symptoms predicted subsequent mother-adolescent interaction behavior, whereas negative interaction behavior seemed to predict subsequent internalizing symptoms. Evidence regarding mediation effects remained limited.ConclusionA systematic search for previous information and an expert-built weighting system contribute to a clear specification of power priors. How information from multiple previous studies should be included in the prior depends on theoretical considerations (e.g., the prior is an updated Bayesian distribution), and may also be affected by pragmatic considerations regarding the impact of the previous results at hand (e.g., extremely specific previous results).

Bayes linear analysis for ordinary differential equations

Differential equation models are used in a wide variety of scientific fields to describe the behaviour of physical systems. Commonly, solutions to given systems of differential equations are not available in closed-form; in such situations, the solution to the system is generally approximated numerically. The numerical solution obtained will be systematically different from the (unknown) true solution implicitly defined by the differential equations. Even if it were known, this true solution would be an imperfect representation of the behaviour of the real physical system that it was designed to represent. A Bayesian framework is proposed which handles all sources of numerical and structural uncertainty encountered when using ordinary differential equation (ODE) models to represent real-world processes. The model is represented graphically, and the graph proves to be useful tool, both for deriving a full prior belief specification and for inferring model components given observations of the real system. A general strategy for modelling the numerical discrepancy induced through choice of a particular solver is outlined, in which the variability of the numerical discrepancy is fixed to be proportional to the length of the solver time-step and a grid-refinement strategy is used to study its structure in detail. A Bayes linear adjustment procedure is presented, which uses a junction tree derived from the originally specified directed graphical model to propagate information efficiently between model components, lessening the computational demands associated with the inference. The proposed framework is illustrated through application to two examples: a model for the trajectory of an airborne projectile moving subject to gravity and air resistance, and a model for the coupled motion of a set of ringing bells and the tower which houses them.