Local functional connectivity abnormalities in mild cognitive impairment and Alzheimer's disease: A meta-analytic investigation using minimum Bayes factor activation likelihood estimation.

Functional magnetic resonance imaging research employing regional homogeneity (ReHo) analysis has uncovered aberrant local brain connectivity in individuals with mild cognitive impairment (MCI) and Alzheimer's disease (AD) in comparison with healthy controls. However, the precise localization, extent, and possible overlap of these aberrations are still not fully understood. To bridge this gap, we applied a novel meta-analytic and Bayesian method (minimum Bayes Factor Activation Likelihood Estimation, mBF-ALE) for a systematic exploration of local functional connectivity alterations in MCI and AD brains. We extracted ReHo data via a standardized MEDLINE database search, which included 35 peer-reviewed experiments, 1,256 individuals with AD or MCI, 1,118 healthy controls, and 205 x-y-z coordinates of ReHo variation. We then separated the data into two distinct datasets: one for MCI and the other for AD. Two mBF-ALE analyses were conducted, thresholded at "very strong evidence" (mBF ≥ 150), with a minimum cluster size of 200 mm³. We also assessed the spatial consistency and sensitivity of our Bayesian results using the canonical version of the ALE algorithm. For MCI, we observed two clusters of ReHo decrease and one of ReHo increase. Decreased local connectivity was notable in the left precuneus (Brodmann area - BA 7) and left inferior temporal gyrus (BA 20), while increased connectivity was evident in the right parahippocampal gyrus (BA 36). The canonical ALE confirmed these locations, except for the inferior temporal gyrus. In AD, one cluster each of ReHo decrease and increase were found, with decreased connectivity in the right posterior cingulate cortex (BA 30 extending to BA 23) and increased connectivity in the left posterior cingulate cortex (BA 31). These locations were confirmed by the canonical ALE. The identification of these distinct functional connectivity patterns sheds new light on the complex pathophysiology of MCI and AD, offering promising directions for future neuroimaging-based interventions. Additionally, the use of a Bayesian framework for statistical thresholding enhances the robustness of neuroimaging meta-analyses, broadening its applicability to small datasets.

Tip dating and Bayes factors provide insight into the divergences of crown bird clades across the end-Cretaceous mass extinction.

The origin of crown birds (Neornithes) remains contentious owing to conflicting divergence time hypotheses obtained from alternative sources of data. The fossil record suggests limited diversification of Neornithes in the Late Mesozoic and a substantial radiation in the aftermath of the Cretaceous-Palaeogene (K-Pg) mass extinction, approximately 66 Ma. Molecular clock studies, however, have yielded estimates for neornithine origins ranging from the Early Cretaceous (130 Ma) to less than 10 Myr before the K-Pg. We use Bayes factors to compare the fit of node ages from different molecular clock studies to an independent morphological dataset. Our results allow us to reject scenarios of crown bird origins deep in the Early Cretaceous, as well as an origin of crown birds within the last 10 Myr of the Cretaceous. The scenario best supported by our analyses is one where Neornithes originated between the Early and Late Cretaceous (ca 100 Ma), while numerous divergences within major neoavian clades either span or postdate the K-Pg. This study affirms the importance of the K-Pg on the diversification of modern birds, and the potential of combined-evidence tip-dating analyses to illuminate recalcitrant 'rocks versus clocks' debates.

Power priors for replication studies.

The ongoing replication crisis in science has increased interest in the methodology of replication studies. We propose a novel Bayesian analysis approach using power priors: The likelihood of the original study's data is raised to the power of α, and then used as the prior distribution in the analysis of the replication data. Posterior distribution and Bayes factor hypothesis tests related to the power parameter α quantify the degree of compatibility between the original and replication study. Inferences for other parameters, such as effect sizes, dynamically borrow information from the original study. The degree of borrowing depends on the conflict between the two studies. The practical value of the approach is illustrated on data from three replication studies, and the connection to hierarchical modeling approaches explored. We generalize the known connection between normal power priors and normal hierarchical models for fixed parameters and show that normal power prior inferences with a beta prior on the power parameter α align with normal hierarchical model inferences using a generalized beta prior on the relative heterogeneity variance I2. The connection illustrates that power prior modeling is unnatural from the perspective of hierarchical modeling since it corresponds to specifying priors on a relative rather than an absolute heterogeneity scale.

Testing Conditional Independence in Psychometric Networks: An Analysis of Three Bayesian Methods.

Network psychometrics uses graphical models to assess the network structure of psychological variables. An important task in their analysis is determining which variables are unrelated in the network, i.e., are independent given the rest of the network variables. This conditional independence structure is a gateway to understanding the causal structure underlying psychological processes. Thus, it is crucial to have an appropriate method for evaluating conditional independence and dependence hypotheses. Bayesian approaches to testing such hypotheses allow researchers to differentiate between absence of evidence and evidence of absence of connections (edges) between pairs of variables in a network. Three Bayesian approaches to assessing conditional independence have been proposed in the network psychometrics literature. We believe that their theoretical foundations are not widely known, and therefore we provide a conceptual review of the proposed methods and highlight their strengths and limitations through a simulation study. We also illustrate the methods using an empirical example with data on Dark Triad Personality. Finally, we provide recommendations on how to choose the optimal method and discuss the current gaps in the literature on this important topic.

The flaw of averages: Bayes factors as posterior means of the likelihood ratio.

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.

Vertical Center-of-Mass Braking and Motor Performance during Gait Initiation in Young Healthy Adults, Elderly Healthy Adults, and Patients with Parkinson's Disease: A Comparison of Force-Plate and Markerless Motion Capture Systems.

BACKGROUND: This study tested the agreement between a markerless motion capture system and force-plate system ("gold standard") to quantify stability control and motor performance during gait initiation. METHODS: Healthy adults (young and elderly) and patients with Parkinson's disease performed gait initiation series at spontaneous and maximal velocity on a system of two force-plates placed in series while being filmed by a markerless motion capture system. Signals from both systems were used to compute the peak of forward center-of-mass velocity (indicator of motor performance) and the braking index (indicator of stability control). RESULTS: Descriptive statistics indicated that both systems detected between-group differences and velocity effects similarly, while a Bland-Altman plot analysis showed that mean biases of both biomechanical indicators were virtually zero in all groups and conditions. Bayes factor 01 indicated strong (braking index) and moderate (motor performance) evidence that both systems provided equivalent values. However, a trial-by-trial analysis of Bland-Altman plots revealed the possibility of differences >10% between the two systems. CONCLUSION: Although non-negligible differences do occur, a markerless motion capture system appears to be as efficient as a force-plate system in detecting Parkinson's disease and velocity condition effects on the braking index and motor performance.

Evaluation of the Margin of Stability during Gait Initiation in Young Healthy Adults, Elderly Healthy Adults and Patients with Parkinson's Disease: A Comparison of Force Plate and Markerless Motion Capture Systems.

Gait initiation (GI) is a functional task classically used in the literature to evaluate the capacity of individuals to maintain postural stability. Postural stability during GI can be evaluated through the "margin of stability" (MoS), a variable that is often computed from force plate recordings. The markerless motion capture system (MLS) is a recent innovative technology based on deep learning that has the potential to compute the MoS. This study tested the agreement between a force plate measurement system (FPS, gold standard) and an MLS to compute the MoS during GI. Healthy adults (young [YH] and elderly [EH]) and Parkinson's disease patients (PD) performed GI series at spontaneous (SVC) and maximum velocity (MVC) on an FPS while being filmed by a MLS. Descriptive statistics revealed a significant effect of the group (YH vs. EH vs. PD) and velocity condition (SVC vs. MVC) on the MoS but failed to reveal any significant effect of the system (MLS vs. PFS) or interaction between factors. Bland-Altman plot analysis further showed that mean MoS biases were zero in all groups and velocity conditions, while the Bayes factor 01 indicated "moderate evidence" that both systems provided equivalent MoS. Trial-by-trial analysis of Bland-Altman plots, however, revealed that differences of >20% between the two systems did occur. Globally taken, these findings suggest that the two systems are similarly effective in detecting an effect of the group and velocity on the MoS. These findings may have important implications in both clinical and laboratory settings due to the ease of use of the MLS compared to the FPS.

Bayesian sequential designs in studies with multilevel data.

In many studies in the social and behavioral sciences, the data have a multilevel structure, with subjects nested within clusters. In the design phase of such a study, the number of clusters to achieve a desired power level has to be calculated. This requires a priori estimates of the effect size and intraclass correlation coefficient. If these estimates are incorrect, the study may be under- or overpowered. This may be overcome by using a group-sequential design, where interim tests are done at various points in time of the study. Based on interim test results, a decision is made to either include additional clusters or to reject the null hypothesis and conclude the study. This contribution introduces Bayesian sequential designs as an alternative to group-sequential designs. This approach compares various hypotheses based on the support in the data for each of them. If neither hypothesis receives a sufficient degree of support, additional clusters are included in the study and the Bayes factor is recalculated. This procedure continues until one of the hypotheses receives sufficient support. This paper explains how the Bayes factor is used as a measure of support for a hypothesis and how a Bayesian sequential design is conducted. A simulation study in the setting of a two-group comparison was conducted to study the effects of the minimum and maximum number of clusters per group and the desired degree of support. It is concluded that Bayesian sequential designs are a flexible alternative to the group sequential design.

Bayesian evidence synthesis as a flexible alternative to meta-analysis: A simulation study and empirical demonstration.

Synthesizing results across multiple studies is a popular way to increase the robustness of scientific findings. The most well-known method for doing this is meta-analysis. However, because meta-analysis requires conceptually comparable effect sizes with the same statistical form, meta-analysis may not be possible when studies are highly diverse in terms of their research design, participant characteristics, or operationalization of key variables. In these situations, Bayesian evidence synthesis may constitute a flexible and feasible alternative, as this method combines studies at the hypothesis level rather than at the level of the effect size. This method therefore poses less constraints on the studies to be combined. In this study, we introduce Bayesian evidence synthesis and show through simulations when this method diverges from what would be expected in a meta-analysis to help researchers correctly interpret the synthesis results. As an empirical demonstration, we also apply Bayesian evidence synthesis to a published meta-analysis on statistical learning in people with and without developmental language disorder. We highlight the strengths and weaknesses of the proposed method and offer suggestions for future research.

Using the 'Leapfrog' Design as a Simple Form of Adaptive Platform Trial to Develop, Test, and Implement Treatment Personalization Methods in Routine Practice.

The route for the development, evaluation and dissemination of personalized psychological therapies is complex and challenging. In particular, the large sample sizes needed to provide adequately powered trials of newly-developed personalization approaches means that the traditional treatment development route is extremely inefficient. This paper outlines the promise of adaptive platform trials (APT) embedded within routine practice as a method to streamline development and testing of personalized psychological therapies, and close the gap to implementation in real-world settings. It focuses in particular on a recently-developed simplified APT design, the 'leapfrog' trial, illustrating via simulation how such a trial may proceed and the advantages it can bring, for example in terms of reduced sample sizes. Finally it discusses models of how such trials could be implemented in routine practice, including potential challenges and caveats, alongside a longer-term perspective on the development of personalized psychological treatments.

On the Nuisance Parameter Elimination Principle in Hypothesis Testing.

The Non-Informative Nuisance Parameter Principle concerns the problem of how inferences about a parameter of interest should be made in the presence of nuisance parameters. The principle is examined in the context of the hypothesis testing problem. We prove that the mixed test obeys the principle for discrete sample spaces. We also show how adherence of the mixed test to the principle can make performance of the test much easier. These findings are illustrated with new solutions to well-known problems of testing hypotheses for count data.

The Relationships Between External and Internal Training Loads in Mixed Martial Arts.

PURPOSE: As a multidisciplined combat sport, relationships between external and internal training loads and intensities of mixed martial arts (MMA) have not been described. The aim of this study was to determine the external loads and intensities of MMA training categories and their relationship to internal loads and intensities. METHODS: Twenty MMA athletes (age = 23.3 [5.3] y, mass = 72.1 [7.2] kg, stature = 171.5 [8.4] cm) were observed for 2 consecutive weeks. Internal load and intensity (session rating of perceived exertion [sRPE]) were calculated using the Foster RPE for the session overall (sRPE-training load [TL]) and segmented RPE (segRPE-TL) for each training category: warm-up, striking drills, wrestling drills, Brazilian jiujitsu (BJJ) drills, striking sparring, wrestling sparring, BJJ sparring, and MMA sparring. External load and intensity were measured via Catapult OptimEye S5 for the full duration of each session using accumulated Playerload (PLdACC) and PLdACC per minute (PLdACC·min-1). Differences in loads between categories and days were assessed via Bayesian analysis of variance (BF10 ≥ 3). Predictive relationships between internal and external variables were calculated using Bayesian regression. RESULTS: Session overall sRPE-TL = 448.6 (191.1) arbitrary units (AU); PLdACC = 310.6 (112) AU. Category segRPE-TL range = 33.8 (22.6) AU (warm-up) to 122.8 (54.6) AU (BJJ drills). Category PLdACC range = 44 (36.3) AU (warm-up) to 125 (58.8) AU (MMA sparring). Neither sRPE-TL nor PLdACC changed between days. PLdACC was different between categories. Evidence for regressions was strong-decisive except for BJJ drills (BF10 = 7, moderate). R2 range = .50 to .77, except for warm-up (R2 = .17), BJJ drills (R2 = .27), BJJ sparring (R2 = .49), and session overall (R2 = .13). CONCLUSIONS: While MMA training categories may be differentiated in terms of external load, overall session external load does not change within or between weeks. Resultant regression equations may be used to appropriately plan MMA technical/tactical training loads.

Strength of Statistical Evidence for the Efficacy of Cancer Drugs: A Bayesian Re-Analysis of Randomized Trials Supporting FDA Approval.

OBJECTIVE: To quantify the strength of statistical evidence of randomised controlled trials (RCTs) for novel cancer drugs approved by the Food and Drug Administration (FDA) in the last two decades. STUDY DESIGN AND SETTING: We used data on overall survival (OS), progression-free survival (PFS), and tumour response (TR) for novel cancer drugs approved for the first time by the FDA between January 2000 and December 2020. We assessed strength of statistical evidence by calculating Bayes Factors (BFs) for all available endpoints, and we pooled evidence using Bayesian fixed-effect meta-analysis for indications approved based on two RCTs. Strength of statistical evidence was compared between endpoints, approval pathways, lines of treatment, and types of cancer. RESULTS: We analysed the available data from 82 RCTs corresponding to 68 indications supported by a single RCT and seven indications supported by two RCTs. Median strength of statistical evidence was ambiguous for OS (BF = 1.9; IQR 0.5-14.5), and strong for PFS (BF = 24,767.8; IQR 109.0-7.3*106) and TR (BF = 113.9; IQR 3.0-547,100). Overall, 44 indications (58.7%) were approved without clear statistical evidence for OS improvements and seven indications (9.3%) were approved without statistical evidence for improvements on any endpoint. Strength of statistical evidence was lower for accelerated approval compared to non-accelerated approval across all three endpoints. No meaningful differences were observed for line of treatment and cancer type. LIMITATIONS: This analysis is limited to statistical evidence. We did not consider non-statistical factors (e.g., risk of bias, quality of the evidence). CONCLUSION: BFs offer novel insights into the strength of statistical evidence underlying cancer drug approvals. Most novel cancer drugs lack strong statistical evidence that they improve OS, and a few lack statistical evidence for efficacy altogether. These cases require a transparent and clear explanation. When evidence is ambiguous, additional post-marketing trials could reduce uncertainty.

Beyond 'statistical significance': A nontechnical primer of Bayesian statistics and Bayes factors for health researchers.

RATIONALE: Hypothesis testing is integral to health research and is commonly completed through frequentist statistics focused on computing p values. p Values have been long criticized for offering limited information about the relationship of variables and strength of evidence concerning the plausibility, presence and certainty of associations among variables. Bayesian statistics is a potential alternative for inference-making. Despite emerging discussion on Bayesian statistics across various disciplines, the uptake of Bayesian statistics in health research is still limited. AIM: To offer a primer on Bayesian statistics and Bayes factors for health researchers to gain preliminary knowledge of its use, application and interpretation in health research. METHODS: Theoretical and empirical literature on Bayesian statistics and methods were used to develop this methodological primer. CONCLUSIONS: Using Bayesian statistics in health research without a careful and complete understanding of its underlying philosophy and differences from frequentist testing, estimation and interpretation methods can result in similar ritualistic use as done for p values. IMPLICATIONS: Health researchers should supplement frequentists statistics with Bayesian statistics when analysing research data. The overreliance on p values for clinical decisions making should be avoided. Bayes factors offer a more intuitive measure of assessing the strength of evidence for null and alternative hypothesis.

Bayesian Reconciliation of Return Predictability.

This article considers a stable vector autoregressive (VAR) model and investigates return predictability in a Bayesian context. The bivariate VAR system comprises asset returns and a further prediction variable, such as the dividend-price ratio, and allows pinning down the question of return predictability to the value of one particular model parameter. We develop a new shrinkage type prior for this parameter and compare our Bayesian approach to ordinary least squares estimation and to the reduced-bias estimator proposed in Amihud and Hurvich (2004. "Predictive Regressions: A Reduced-Bias Estimation Method." Journal of Financial and Quantitative Analysis 39: 813-41). A simulation study shows that the Bayesian approach dominates the reduced-bias estimator in terms of observed size (false positive) and power (false negative). We apply our methodology to a system comprising annual CRSP value-weighted returns running, respectively, from 1926 to 2004 and from 1953 to 2021, and the logarithmic dividend-price ratio. For the first sample, the Bayesian approach supports the hypothesis of no return predictability, while for the second data set weak evidence for predictability is observed. Then, instead of the dividend-price ratio, some prediction variables proposed in Welch and Goyal (2008. "A Comprehensive Look at the Empirical Performance of Equity Premium Prediction." Review of Financial Studies 21: 1455-508) are used. Also with these prediction variables, only weak evidence for return predictability is supported by Bayesian testing. These results are corroborated with an out-of-sample forecasting analysis.

Bayesian tensor-on-tensor regression with efficient computation.

We propose a Bayesian tensor-on-tensor regression approach to predict a multidimensional array (tensor) of arbitrary dimensions from another tensor of arbitrary dimensions, building upon the Tucker decomposition of the regression coefficient tensor. Traditional tensor regression methods making use of the Tucker decomposition either assume the dimension of the core tensor to be known or estimate it via cross-validation or some model selection criteria. However, no existing method can simultaneously estimate the model dimension (the dimension of the core tensor) and other model parameters. To fill this gap, we develop an efficient Markov Chain Monte Carlo (MCMC) algorithm to estimate both the model dimension and parameters for posterior inference. Besides the MCMC sampler, we also develop an ultra-fast optimization-based computing algorithm wherein the maximum a posteriori estimators for parameters are computed, and the model dimension is optimized via a simulated annealing algorithm. The proposed Bayesian framework provides a natural way for uncertainty quantification. Through extensive simulation studies, we evaluate the proposed Bayesian tensor-on-tensor regression model and show its superior performance compared to alternative methods. We also demonstrate its practical effectiveness by applying it to two real-world datasets, including facial imaging data and 3D motion data.

Bayesian analysis of psychological capital in peruvian university students: Differences by sex and age.

Psychological capital (PsyCap) constitutes a positive personal resource that enhances better well-being and academic performance in university students. Initially addressed in the organizational realm and recently in the academic one. This study aimed to establish the differences in PsyCap according to gender and age in Peruvian university students. A quantitative, comparative, non-experimental, and cross-sectional study was conducted with 708 students (77.4 % women and 22.6 % men), aged between 18 and 61 years (M = 22.1; SD = 5.95), selected in a non-probabilistic manner, who completed the Psychological Capital Questionnaire (PCQ-12). The results indicate very strong evidence supporting the existence of significant differences between different age groups, suggesting that the observed variations are not due to chance but reflect real differences between ages. Regarding gender, the data do not provide enough information to confidently assert whether there are significant differences between men and women in relation to psychological capital (PsyCap) and its dimensions. This implies that we cannot confirm whether gender influences these variables. These findings highlight the need to consider age when assessing and intervening in PsyCap in university students.

Bayesian meta-analysis of phase 3 results of aducanumab, lecanemab, donanemab, and high-dose gantenerumab in prodromal and mild Alzheimer's disease.

INTRODUCTION: Phase 3 trials using the anti-amyloid antibodies aducanumab, lecanemab, donanemab, and high-dose gantenerumab in prodromal and mild Alzheimer's disease dementia were heterogeneous in respect to statistical significance of effects. However, heterogeneity of results has not yet directly be quantified. METHODS: We used Bayesian random effects meta-analysis to quantify evidence for or against a treatment effect, and assessed the size of the effect and its heterogeneity. Data were extracted from published studies where available and Web based data reports, assuming a Gaussian data generation process. RESULTS: We found moderate evidence in favor of a treatment effect (Bayes factor = 13.2). The effect was moderate to small with -0.33 (95% credible interval -0.54 to -0.10) points on the Clinical Dementia Rating - Sum of Boxes (CDR-SB) scale. The heterogeneity parameter was low to moderate with 0.21 (0.04 to 0.45) CDR-SB points. DISCUSSION: Heterogeneity across studies was moderate despite some trials reaching statistical significance, while others did not. This suggests that the negative aducanumab and gantenerumab trials are in full agreement with the expected effect sizes.

A Bayesian competing risk analysis of renal cancer patients based on SEER database.

BACKGROUND: Renal cell carcinoma (RCC) remains a global health concern due to its poor survival rate. This study aimed to investigate the influence of medical determinants and socioeconomic status on survival outcomes of RCC patients. We analyzed the survival data of 41,563 RCC patients recorded under the Surveillance, Epidemiology, and End Results (SEER) program from 2012 to 2020. METHODS: We employed a competing risk model, assuming lifetime of RCC patients under various risks follows Chen distribution. This model accounts for uncertainty related to survival time as well as causes of death, including missing cause of death. For model analysis, we utilized Bayesian inference and obtained the estimate of various key parameters such as cumulative incidence function (CIF) and cause-specific hazard. Additionally, we performed Bayesian hypothesis testing to assess the impact of multiple factors on the survival time of RCC patients. RESULTS: Our findings revealed that the survival time of RCC patients is significantly influenced by gender, income, marital status, chemotherapy, tumor size, and laterality. However, we observed no significant effect of race and origin on patient's survival time. The CIF plots indicated a number of important distinctions in incidence of causes of death corresponding to factors income, marital status, race, chemotherapy, and tumor size. CONCLUSIONS: The study highlights the impact of various medical and socioeconomic factors on survival time of RCC patients. Moreover, it also demonstrates the utility of competing risk model for survival analysis of RCC patients under Bayesian paradigm. This model provides a robust and flexible framework to deal with missing data, which can be particularly useful in real-life situations where patients information might be incomplete.

A Modern Approach to Stability Studies via Bayesian Linear Mixed Models Incorporating Auxiliary Effects.

In preparation to the launch of a pharmaceutical product, an estimate of its shelf life via stability testing is required by regulatory agencies. The ICH-Q1E guidance has been the worldwide reference to reach this objective, but in recent years several authors have criticized many of its aspects. To that end we discuss a complete Bayesian transcript of the ICH-Q1E, treating all the apparent shortcomings, while also addressing the presence of multiple batches using a linear mixed model (LMM) for proper shelf life prediction by explicitly modelling the batch-to-batch variability. This comprises a redefinition of the linear models proposed in the ICH-Q1E by suitable LMM counterparts, and a Bayesian analogue for model selection, which is more intuitive and remedies detrimental features of the ICH approach. In that context, a proper mathematical foundation of shelf life is provided that we use to investigate and mathematically compare the two available approaches to shelf life determination via shelf life distribution and batch distribution. The discussed method is then tested and evaluated using real data in comparison with the ICH-Q1E approach demonstrating their approximate equivalency for 6 batches. As a major objective, we extended the LMM with auxiliary fixed effects, here the concentration, which interconnect data sets allowing a prediction of shelf lives for concentrations lacking a sufficient number of batches. This establishes a novel approach to accelerate the speed to submission while retaining the patients' safety. Both case studies underline the inherent superiority of LMMs within a Bayesian framework regarding predictability and interpretability, and we hope that the relevant authorities will accept this approach in the future.