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As an alternative to the Frequentist p-value, the Bayes factor (or ratio of marginal likelihoods) has been regarded as one of the primary tools for Bayesian hypothesis testing. In recent years, several researchers have begun to re-analyze results from prominent medical journals, as well as from trials for FDA-approved drugs, to show that Bayes factors often give divergent conclusions from those of p-values. In this paper, we investigate the claim that Bayes factors are straightforward to interpret as directly quantifying the relative strength of evidence. In particular, we show that for nested hypotheses with consistent priors, the Bayes factor for the null over the alternative hypothesis is the posterior mean of the likelihood ratio. By re-analyzing 39 results previously published in the New England Journal of Medicine, we demonstrate how the posterior distribution of the likelihood ratio can be computed and visualized, providing useful information beyond the posterior mean alone.

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Molecular evolutionary rate variation is a key aspect of the evolution of many organisms that can be modeled using molecular clock models. For example, fixed local clocks revealed the role of episodic evolution in the emergence of SARS-CoV-2 variants of concern. Like all statistical models, however, the reliability of such inferences is contingent on an assessment of statistical evidence. We present a novel Bayesian phylogenetic approach for detecting episodic evolution. It consists of computing Bayes factors, as the ratio of posterior and prior odds of evolutionary rate increases, effectively quantifying support for the effect size. We conducted an extensive simulation study to illustrate the power of this method and benchmarked it to formal model comparison of a range of molecular clock models using (log) marginal likelihood estimation, and to inference under a random local clock model. Quantifying support for the effect size has higher sensitivity than formal model testing and is straight-forward to compute, because it only needs samples from the posterior and prior distribution. However, formal model testing has the advantage of accommodating a wide range molecular clock models. We also assessed the ability of an automated approach, known as the random local clock, where branches under episodic evolution may be detected without their a priori definition. In an empirical analysis of a data set of SARS-CoV-2 genomes, we find "very strong" evidence for episodic evolution. Our results provide guidelines and practical methods for Bayesian detection of episodic evolution, as well as avenues for further research into this phenomenon.

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Epidemiology has been transformed by the advent of Bayesian phylodynamic models that allow researchers to infer the geographic history of pathogen dispersal over a set of discrete geographic areas [1, 2]. These models provide powerful tools for understanding the spatial dynamics of disease outbreaks, but contain many parameters that are inferred from minimal geographic information (i.e., the single area in which each pathogen was sampled). Consequently, inferences under these models are inherently sensitive to our prior assumptions about the model parameters. Here, we demonstrate that the default priors used in empirical phylodynamic studies make strong and biologically unrealistic assumptions about the underlying geographic process. We provide empirical evidence that these unrealistic priors strongly (and adversely) impact commonly reported aspects of epidemiological studies, including: 1) the relative rates of dispersal between areas; 2) the importance of dispersal routes for the spread of pathogens among areas; 3) the number of dispersal events between areas, and; 4) the ancestral area in which a given outbreak originated. We offer strategies to avoid these problems, and develop tools to help researchers specify more biologically reasonable prior models that will realize the full potential of phylodynamic methods to elucidate pathogen biology and, ultimately, inform surveillance and monitoring policies to mitigate the impacts of disease outbreaks.

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We compare two approaches to using information about the signs of structural shocks at specific dates within a structural vector autoregression (SVAR): imposing "narrative restrictions" (NR) on the shock signs in an otherwise set-identified SVAR; and casting the information about the shock signs as a discrete-valued "narrative proxy" (NP) to point-identify the impulse responses. The NP is likely to be "weak" given that the sign of the shock is typically known in a small number of periods, in which case the weak-proxy robust confidence intervals in Montiel Olea, Stock, and Watson are the natural approach to conducting inference. However, we show both theoretically and via Monte Carlo simulations that these confidence intervals have distorted coverage-which may be higher or lower than the nominal level-unless the sign of the shock is known in a large number of periods. Regarding the NR approach, we show that the prior-robust Bayesian credible intervals from Giacomini, Kitagawa, and Read deliver coverage exceeding the nominal level, but which converges toward the nominal level as the number of NR increases.

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Nationwide opinions and international attitudes toward climate and environmental change are receiving increasing attention in both scientific and political communities. An often used way to measure these attitudes is by large-scale social surveys. However, the assumption for a valid country comparison, measurement invariance, is often not met, especially when a large number of countries are being compared. This makes a ranking of countries by the mean of a latent variable potentially unstable, and may lead to untrustworthy conclusions. Recently, more liberal approaches to assessing measurement invariance have been proposed, such as the alignment method in combination with Bayesian approximate measurement invariance. However, the effect of prior variances on the assessment procedure and substantive conclusions is often not well understood. In this article, we tested for measurement invariance of the latent variable "willingness to sacrifice for the environment" using Maximum Likelihood Multigroup Confirmatory Factor Analysis and Bayesian approximate measurement invariance, both with and without alignment optimization. For the Bayesian models, we used multiple priors to assess the impact on the rank order stability of countries. The results are visualized in such a way that the effect of different prior variances and models on group means and rankings becomes clear. We show that even when models appear to be a good fit to the data, there might still be an unwanted impact on the rank ordering of countries. From the results, we can conclude that people in Switzerland and South Korea are most motivated to sacrifice for the environment, while people in Latvia are less motivated to sacrifice for the environment.

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Bayesian inference has recently been identified as an approach for estimating a subjects' pose from noisy marker position data. Previous research suggests that Bayesian inference markedly reduces error for inverse kinematic problems relative to traditional least-squares approaches with estimators having reduced variance despite both least-squares and Bayesian estimators being unbiased. This result is surprising as Bayesian estimators are typically similar to least-squares approaches unless highly informative prior distributions are used. As a result the purpose of this work was to examine the sensitivity of Bayesian inverse kinematics solutions to the prior distribution. Our results highlight that Bayesian solutions to inverse kinematics are sensitive to the choice of prior and that the previously reported superior performance of Bayesian inference is likely due to an overly informative prior distribution which unrealistically uses knowledge of the true kinematic pose. When more realistic, 'weakly-informative' priors, which do not use the known kinematic pose are used then any improvements in estimator accuracy are minimal when compared to the traditional least squares approach. However, with appropriate priors, Bayesian inference can propagate uncertainties related to marker position to uncertainty in joint angles, a valuable contribution for kinematic analyses. When using Bayesian methods, we recommend researchers use weakly-informative priors and conduct a sensitivity analysis to highlight the effects of prior choice on analysis outcomes.

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We examine issues of prior sensitivity in a semi-parametric hierarchical extension of the INAR(p) model with innovation rates clustered according to a Pitman-Yor process placed at the top of the model hierarchy. Our main finding is a graphical criterion that guides the specification of the hyperparameters of the Pitman-Yor process base measure. We show how the discount and concentration parameters interact with the chosen base measure to yield a gain in terms of the robustness of the inferential results. The forecasting performance of the model is exemplified in the analysis of a time series of worldwide earthquake events, for which the new model outperforms the original INAR(p) model.

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Bayesian structural equation modeling (BSEM) is a flexible tool for the exploration and estimation of sparse factor loading structures; that is, most cross-loading entries are zero and only a few important cross-loadings are nonzero. The current investigation was focused on the BSEM with small-variance normal distribution priors (BSEM-N) for both variable selection and model estimation. The prior sensitivity in BSEM-N was explored in factor analysis models with sparse loading structures through a simulation study (Study 1) and an empirical example (Study 2). Study 1 examined the prior sensitivity in BSEM-N based on the model fit, population model recovery, true and false positive rates, and parameter estimation. Seven shrinkage priors on cross-loadings and five noninformative/vague priors on other model parameters were examined. Study 2 provided a real data example to illustrate the impact of various priors on model fit and parameter selection and estimation. Results indicated that when the 95% credible intervals of shrinkage priors barely covered the population cross-loading values, it resulted in the best balance between true and false positives. If the goal is to perform variable selection, a sparse cross-loading structure is required, preferably with a minimal number of nontrivial cross-loadings and relatively high primary loading values. To improve parameter estimates, a relatively large prior variance is preferred. When cross-loadings are relatively large, BSEM-N with zero-mean priors is not recommended for the estimation of cross-loadings and factor correlations.

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Informative hypotheses are increasingly being used in psychological sciences because they adequately capture researchers' theories and expectations. In the Bayesian framework, the evaluation of informative hypotheses often makes use of default Bayes factors such as the fractional Bayes factor. This paper approximates and adjusts the fractional Bayes factor such that it can be used to evaluate informative hypotheses in general statistical models. In the fractional Bayes factor a fraction parameter must be specified which controls the amount of information in the data used for specifying an implicit prior. The remaining fraction is used for testing the informative hypotheses. We discuss different choices of this parameter and present a scheme for setting it. Furthermore, a software package is described which computes the approximated adjusted fractional Bayes factor. Using this software package, psychological researchers can evaluate informative hypotheses by means of Bayes factors in an easy manner. Two empirical examples are used to illustrate the procedure.

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The Drug Information Association Bayesian Scientific Working Group (BSWG) was formed in 2011 with a vision to ensure that Bayesian methods are well understood and broadly utilized for design and analysis and throughout the medical product development process, and to improve industrial, regulatory, and economic decision making. The group, composed of individuals from academia, industry, and regulatory, has as its mission to facilitate the appropriate use and contribute to the progress of Bayesian methodology. In this paper, the safety sub-team of the BSWG explores the use of Bayesian methods when applied to drug safety meta-analysis and network meta-analysis. Guidance is presented on the conduct and reporting of such analyses. We also discuss different structural model assumptions and provide discussion on prior specification. The work is illustrated through a case study involving a network meta-analysis related to the cardiovascular safety of non-steroidal anti-inflammatory drugs.

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