Posterior Distribution Research Articles

Analysis of seepage failure probability for high core rockfill dams during rapid drawdown of reservoir water level

Rapid drawdown (RDD) is an abnormal reservoir flood control operation that may cause seepage failure of high core rockfill dams. In this paper, a novel framework combining Bayesian inference and transient unsaturated seepage simulation is proposed to evaluate the seepage failure probability of the core during RDD. The dynamic hydraulic boundaries utilized in the simulation are first determined by the performance of water release structures. The extreme learning machine (ELM) is then employed as the surrogate model to establish the mapping relationship between material parameters with simulated seepage pressure and hydraulic gradient. The Bayesian inference method is applied to characterize the uncertainty of material parameters based on seepage monitoring data. Finally, the posterior distribution samples of random parameters are utilized to calculate the seepage failure probability of the core. The seepage safety of the core under different drawdown rates is studied with a real 186 m high core rockfill dam. The results reveal that a faster RDD rate can lead to an increased probability of seepage failure. The maximum failure probability is 35.56 % for the four release structures working together (about 0.316 m/h), higher than 23.16 % for the three release structures (about 0.165 m/h). It is also found that the peak of seepage failure probability often occurs in the middle stage of RDD. Therefore, to ensure the safety of the project, it is important to control the RDD rate and carefully monitor the seepage pressure. From the point of view of seepage failure, the proposed method for evaluating the safety of core rockfill dams under RDD is of high guiding significance to the risk assessment of similar projects under extreme conditions.

Bayesian approach of elliptical loci and RAPID for damage localization in wind turbine blade

This investigation addresses the issue of damage detection and localization in wind turbine blade laminates. This paper proposes a novel approach that integrates the elliptical trajectory and probabilistic imaging method using the Bayesian framework. This method employs multiple damage-sensitive features to enhance the reliability and robustness of sensor arrays. The algorithm is optimized by analyzing the propagation characteristics of Lamb waves in composite blade laminates. A numerical simulation is conducted on a 1.5 MW wind turbine blade laminate model, incorporating the scattered wave signal, wave arrival time, and correlation coefficient as damage characteristic signals. Markov Chain Monte Carlo sampling method is adopted to obtain the posterior distribution of the damage location and achieve accurate localization of blade delamination damage. The experimental results indicate that the damage localization algorithm, which utilizes the Bayesian approach, achieves an accuracy of approximately 97.04% in localizing delamination damage in blade laminates.

Fine-resolution land cover mapping over large and mountainous areas for Lāna‘i, Hawaii using posterior probabilities, and expert knowledge

ABSTRACT The task of accurately mapping species-specific vegetation cover in remote and topographically complex regions like those found in Hawaiʻi presents unique challenges. This study leverages a machine learning approach to accurately classify vegetation into fine species-specific classes across the island of Lāna‘i, Hawaii, offering a novel methodology for tackling such challenges. Utilizing high-resolution WordView-2 satellite imagery, a neural network classifier and a custom lidar-based geometric correction, we introduced two new approaches to refine our high-resolution land cover classifications. This included the implementation of prior-based adjustments to class posterior probabilities to enhance land cover classification accuracy. Moreover, we developed mixed hierarchical classification maps that use class posterior probabilities to identify, at the pixel level, the finest land cover class that meets a user-defined confidence threshold. The resulting high-resolution land cover map for Lāna‘i captures the rich diversity and distribution of native and invasive plant species with high overall accuracy, generally exceeding 95%, based on independent ground control data. The capacity to produce wall-to-wall species-level vegetation maps provides a new window into monitoring vegetation dynamics on Lāna‘i and similarly remote and topographically complex regions, and contributes to our broader understanding of ecosystem responses to invasive species, climatic changes, and land management practices such as erosion and sediment control planning. Our approach offers a blueprint for similar efforts in other complex and remote ecosystems.

An optimal Bayesian intervention policy in response to unknown dynamic cell stimuli

Interventions in gene regulatory networks (GRNs) aim to restore normal functions of cells experiencing abnormal behavior, such as uncontrolled cell proliferation. The dynamic, uncertain, and complex nature of cellular processes poses significant challenges in determining the best interventions. Most existing intervention methods assume that cells are unresponsive to therapies, resulting in stationary and deterministic intervention solutions. However, cells in unhealthy conditions can dynamically respond to therapies through internal stimuli, leading to the recurrence of undesirable conditions. This paper proposes a Bayesian intervention policy that adaptively responds to cell dynamic responses according to the latest available information. The GRNs are modeled using a Boolean network with perturbation (BNp), and the fight between the cell and intervention is modeled as a two-player zero-sum game. Assuming an incomplete knowledge of cell stimuli, a recursive approach is developed to keep track of the posterior distribution of cell responses. The proposed Bayesian intervention policy takes action according to the posterior distribution and a set of Nash equilibrium policies associated with all possible cell responses. Analytical results demonstrate the superiority of the proposed intervention policy against several existing intervention techniques. Meanwhile, the performance of the proposed policy is investigated through comprehensive numerical experiments using the p53-MDM2 negative feedback loop regulatory network and melanoma network. The results demonstrate the empirical convergence of the proposed policy to the optimal Nash equilibrium policy.

Bayesian Linear Inverse Problems in Regularity Scales with Discrete Observations

AbstractWe obtain rates of contraction of posterior distributions in inverse problems with discrete observations. In a general setting of smoothness scales we derive abstract results for general priors, with contraction rates determined by discrete Galerkin approximation. The rate depends on the amount of prior concentration near the true function and the prior mass of functions with inferior Galerkin approximation. We apply the general result to non-conjugate series priors, showing that these priors give near optimal and adaptive recovery in some generality, Gaussian priors, and mixtures of Gaussian priors, where the latter are also shown to be near optimal and adaptive.

Postoperative pain trajectories in total hip arthroplasty.

Total hip arthroplasty (THA) is a common procedure to address pain and enhance function in hip disorders such as osteoarthritis. Despite its success, postoperative patient recovery exhibits considerable heterogeneity. This study aimed to investigate whether patients follow distinct pain trajectories following THA and identify the patient characteristics linked to suboptimal trajectories. This retrospective cohort study analyzed THA patients at a large academic centre (NYU Langone Orthopedic Hospital, New York, USA) from January 2018 to January 2023, who completed the Patient-Reported Outcomes Measurement Information System (PROMIS) pain intensity questionnaires, collected preoperatively at one-, three-, six-, 12-, and 24-month follow-up times. Growth mixture modelling (GMM) was used to model the trajectories. Optimal model fit was determined by Bayesian information criterion (BIC), Vuong-Lo-Mendell-Rubin likelihood ratio test (VLMR-LRT), posterior probabilities, and entropy values. Association between trajectory groups and patient characteristics were measured by multinomial logistic regression using the three-step approach. Among the 1,249 patients, a piecewise GMM model revealed three distinct pain trajectory groups: 56 patients (4.5%) in group 1; 1,144 patients (91.6%) in group 2; and 49 patients (3.9%) in group 3. Patients in group 2 experienced swift recovery post-THA and minimal preoperative pain. In contrast, groups 1 and 3 initiated with pronounced preoperative pain; however, only group 3 exhibited persistent long-term pain. Multinomial regression indicated African Americans were exceedingly likely to follow trajectory groups 1 (odds ratio (OR) 2.73) and 3 (OR 3.18). Additionally, odds of membership to group 3 increased by 12% for each BMI unit rise, by 19% for each added postoperative day, and by over four if discharged to rehabilitation services (OR 4.07). This study identified three distinct pain trajectories following THA, highlighting the role of individual patient factors in postoperative recovery. This emphasizes the importance of preoperatively addressing modifiable risk factors associated with suboptimal pain trajectories, particularly in at-risk patients.

Bayesian model updating of super high‐rise building for construction simulation

SummaryA finite element model was established using SAP2000 software for the C1 tower, a super high‐rise building in the second phase of the Nanjing Financial City project, and the construction process of the tower was simulated. The C1 tower adopts a frame core tube extension arm and ring truss structure system, with 87 floors above ground and five floors underground. The roof structure has an elevation of 416.6 m. Precise measurements of inter‐story compression deformation were conducted using advanced surveying equipment. Sensitivity analysis, based on the finite difference method, identified the shear wall elastic modulus, frame column elastic modulus, steel beam elastic modulus, and shear wall unit weight as four highly influential parameters. Employing the Bayesian principle, the Markov Chain Monte Carlo (MCMC) method was applied to determine the posterior density probability function of the parameters targeted for modification. Subsequently, the Metropolis–Hastings (MH) sampling algorithm was employed to refine the C1 Tower model. This refinement significantly reduced the root mean square error between the measured and simulated vertical displacements, achieving an error reduction of approximately 10% from 6.082 to around 2.160. The modified material parameters, for the most part, adhered to a normal distribution assumption and exhibited mean values in the posterior probability density functions for the elastic modulus of Q345 steel beams, C70 frame columns, and C60 shear walls. Compared to the initial finite element parameters, the variation range was approximately 13% to 17%. These results serve as a validation of the effectiveness of the proposed method.

Characterization of the complete mitochondrial genome of Paecilomyces variotii and comparative evolutionary mitochondriomics of 36 fungi.

Paecilomyces variotii has important economic value in stimulating crop growth, biodegradation, and other aspects. Up to now, there are no research reports on its mitochondrial genome. The mitochondrial genome of Paecilomyces variotii was determined with the next-generation sequencing method (Illumina, NovaSeq), and its characteristics were analyzed using various bioinformatics approaches. The length of complete mitochondrial genome sequence of P. variotii is 40,965bp and consists of 14 protein-coding genes, 2 ribosomal RNA genes, 1 ribosomal protein S3 gene, 26 transport RNA genes. The results of phylogenetics analysis using Bayesian inference and Maximum likelihood methods showed that P. variotii belongs to the Eurotiales order in the Thermoascaceae family, and 9 genera within the Eurotiomycetes class were effectively distinguished with high support rates (bootstrap value > 92% and posterior probabilities > 99%). The analysis of synonymous substitution rates and nonsynonymous substitution rates indicated that the Ka/Ks values of the 14 PCGs in the mitochondrial genomes of the two orders in the Eurotiomycetes class ranged from 0 to 0.4333. This study revealed the structural and sequence information characteristics of the mitochondrial genome of P. variotii, and the phylogenetic results strongly support its classification within the family Thermoascaceae, consistent with traditional morphological taxonomy studies. The 14 PCGs in the mitochondrial genomes of the two orders in the Eurotiomycetes class are subject to strong purifying (negative) selection. The results of this research provides an important molecular basis for the development of genomics, evolutionary genetics and molecular markers of P. variotii in the future.

Methylphenidate and Short-Term Cardiovascular Risk

There are concerns about the safety of medications for treatment of attention-deficit/hyperactivity disorder (ADHD), with mixed evidence on possible cardiovascular risk. To assess whether short-term methylphenidate use is associated with risk of cardiovascular events. This retrospective, population-based cohort study was based on national Swedish registry data. Participants were individuals with ADHD aged 12 to 60 years with dispensed prescriptions of methylphenidate between January 1, 2007, and June 30, 2012. Each person receiving methylphenidate (n = 26 710) was matched on birth date, sex, and county to up to 10 nonusers without ADHD (n = 225 672). Statistical analyses were performed from September 13, 2022, to May 16, 2023. Rates of cardiovascular events, including ischemic heart disease, venous thromboembolism, heart failure, or tachyarrhythmias, 1 year before methylphenidate treatment and 6 months after treatment initiation were compared between individuals receiving methylphenidate and matched controls using a bayesian within-individual design. Analyses were stratified by history of cardiovascular events. The cohort included 252 382 individuals (15 442 [57.8% men]; median age, 20 (IQR, 15-31) years). The overall incidence of cardiovascular events was 1.51 per 10 000 person-weeks (95% highest density interval [HDI], 1.35-1.69) for individuals receiving methylphenidate and 0.77 (95% HDI, 0.73-0.82) for the matched controls. Individuals treated with methylphenidate had an 87% posterior probability of having a higher rate of cardiovascular events after treatment initiation (incidence rate ratio [IRR], 1.41; 95% HDI, 1.09-1.88) compared with matched controls (IRR, 1.18; 95% HDI, 1.02-1.37). The posterior probabilities were 70% for at least a 10% increased risk of cardiovascular events in individuals receiving methylphenidate vs 49% in matched controls. No difference was found in this risk between individuals with and without a history of cardiovascular disease (IRR, 1.11; 95% HDI, 0.58-2.13). In this cohort study, individuals receiving methylphenidate had a small increased cardiovascular risk vs matched controls in the 6 months after treatment initiation. However, there was little evidence for an increased risk of 20% or higher and for differences in risk increase between people with and without a history of cardiovascular disease. Therefore, before treatment initiation, careful consideration of the risk-benefit trade-off of methylphenidate would be useful, regardless of cardiovascular history.

Bayesian framework for multi-source data integration-Application to human extrapolation from preclinical studies.

In preclinical investigations, for example, in in vitro, in vivo, and in silico studies, the pharmacokinetic, pharmacodynamic, and toxicological characteristics of a drug are evaluated before advancing to first-in-man trial. Usually, each study is analyzed independently and the human dose range does not leverage the knowledge gained from all studies. Taking into account all preclinical data through inferential procedures can be particularly interesting in obtaining a more precise and reliable starting dose and dose range. Our objective is to propose a Bayesian framework for multi-source data integration, customizable, and tailored to the specific research question. We focused on preclinical results extrapolated to humans, which allowed us to predict the quantities of interest (e.g. maximum tolerated dose, etc.) in humans. We build an approach, divided into four steps, based on a sequential parameter estimation for each study, extrapolation to human, commensurability checking between posterior distributions and final information merging to increase the precision of estimation. The new framework is evaluated via an extensive simulation study, based on a real-life example in oncology. Our approach allows us to better use all the information compared to a standard framework, reducing uncertainty in the predictions and potentially leading to a more efficient dose selection.

Robust Gaussian process regression based on bias trimming

This paper presents a new robust Gaussian process regression (GPR) algorithm based on identifying and trimming outliers, and it can reduce the computation time and improve the accuracy compared with other robust algorithms. Currently, most existing robust GPR approaches model noise with heavy-tailed distributions that are less sensitive to outliers, such as the Laplace and Student-t distributions. However, a model that is designed this way will be analytically intractable, and the computationally expensive Bayesian approximation method needs to be adopted to obtain the posterior distribution. In our method, the deviation between normal points and outliers is modeled as a bias value, so the non-Gaussian likelihood approach, which consumes substantial time, is not necessary. Nevertheless, the decrease in calculation time is accompanied by a decrease in accuracy. To solve this problem, the trimming algorithm with an adaptive threshold is adopted to improve the prediction precision. Therefore, the proposed approach is simple but very computationally attractive. We conducted various simulation experiments to numerically evaluate the model with different outlier ratios and noise levels. We compared the approach with existing robust Gaussian process methods and other kinds of anomaly detection algorithms to demonstrate the excellent performance of the proposed method. Finally, industrial datasets are used to verify the method, and the results illustrate that it can significantly improve the practicality of the measurement system.

Evidence based on Mendelian randomization and colocalization analysis strengthens causal relationships between structural changes in specific brain regions and risk of amyotrophic lateral sclerosis.

Amyotrophic lateral sclerosis (ALS) is a neurodegenerative disease characterized by the degeneration of motor neurons in the brain and spinal cord with a poor prognosis. Previous studies have observed cognitive decline and changes in brain morphometry in ALS patients. However, it remains unclear whether the brain structural alterations contribute to the risk of ALS. In this study, we conducted a bidirectional two-sample Mendelian randomization (MR) and colocalization analysis to investigate this causal relationship. Summary data of genome-wide association study were obtained for ALS and the brain structures, including surface area (SA), thickness and volume of subcortical structures. Inverse-variance weighted (IVW) method was used as the main estimate approach. Sensitivity analysis was conducted detect heterogeneity and pleiotropy. Colocalization analysis was performed to calculate the posterior probability of causal variation and identify the common genes. In the forward MR analysis, we found positive associations between the SA in four cortical regions (lingual, parahippocampal, pericalcarine, and middle temporal) and the risk of ALS. Additionally, decreased thickness in nine cortical regions (caudal anterior cingulate, frontal pole, fusiform, inferior temporal, lateral occipital, lateral orbitofrontal, pars orbitalis, pars triangularis, and pericalcarine) was significantly associated with a higher risk of ALS. In the reverse MR analysis, genetically predicted ALS was associated with reduced thickness in the bankssts and increased thickness in the caudal middle frontal, inferior parietal, medial orbitofrontal, and superior temporal regions. Colocalization analysis revealed the presence of shared causal variants between the two traits. Our results suggest that altered brain morphometry in individuals with high ALS risk may be genetically mediated. The causal associations of widespread multifocal extra-motor atrophy in frontal and temporal lobes with ALS risk support the notion of a continuum between ALS and frontotemporal dementia. These findings enhance our understanding of the cortical structural patterns in ALS and shed light on potentially viable therapeutic targets.

Kinetically coupled scalar fields model and cosmological tensions

ABSTRACT In this paper, we investigate the kinetically coupled early dark energy (EDE) and scalar field dark matter to address cosmological tensions. The EDE model presents an intriguing theoretical approach to resolving the Hubble tension, but it exacerbates the large-scale structure tension. We consider the interaction between dark matter and EDE, such that the drag of dark energy on dark matter suppresses structure growth, which can alleviate large-scale structure tension. We replace cold dark matter with scalar field dark matter, which has the property of suppressing structure growth on small scales. We employed the Markov Chain Monte Carlo method to constrain the model parameters, our new model reveals a non-zero coupling constant of 0.030 ± 0.026 at a 68 per cent confidence level. The coupled model yields the Hubble constant value of $72.38^{+0.71}_{-0.82}$ km s−1 Mpc−1, which resolves the Hubble tension. However, similar to the EDE model, it also obtains a larger S8 value compared to the ΛCDM model, further exacerbating the large-scale structure tension. The EDE model and the new model yield the best-fitting values of 0.8316 and 0.8146 for S8, respectively, indicating that the new model partially alleviates the negative effect of the EDE model. However, this signature disappears when comparing marginalized posterior probabilities, and both models produce similar results. The values obtained from the EDE model and the new model are $0.822^{+0.011}_{-0.0093}$ and $0.819^{+0.013}_{-0.0092}$, respectively, at a 68 per cent confidence level.

The credit-augmented Divisia aggregates and the monetary business cycle

Abstract We follow Belongia and Ireland (2021) and investigate the role that the Center for Financial Stability credit card-augmented Divisia monetary aggregates could play in monetary policy and business cycle analysis. We use Bayesian methods to estimate a structural VAR under priors that reflect Keynesian channels of monetary transmission, but produce posterior distributions for the structural parameters consistent with classical channels. We also find that valuable information is contained in the credit-augmented Divisia monetary aggregates and that they perform even better than the conventional Divisia aggregates, in terms of highlighting the role of the money supply in aggregate demand.

CUQIpy: I. Computational uncertainty quantification for inverse problems in Python

This paper introduces CUQIpy, a versatile open-source Python package for computational uncertainty quantification (UQ) in inverse problems, presented as Part I of a two-part series. CUQIpy employs a Bayesian framework, integrating prior knowledge with observed data to produce posterior probability distributions that characterize the uncertainty in computed solutions to inverse problems. The package offers a high-level modeling framework with concise syntax, allowing users to easily specify their inverse problems, prior information, and statistical assumptions. CUQIpy supports a range of efficient sampling strategies and is designed to handle large-scale problems. Notably, the automatic sampler selection feature analyzes the problem structure and chooses a suitable sampler without user intervention, streamlining the process. With a selection of probability distributions, test problems, computational methods, and visualization tools, CUQIpy serves as a powerful, flexible, and adaptable tool for UQ in a wide selection of inverse problems. Part II of the series focuses on the use of CUQIpy for UQ in inverse problems with partial differential equations.

Bayesian estimation in generalized linear models for longitudinal data with hyperspherical coordinates

Under the framework of generalized linear models (GLM), the generalized estimating equation (GEE) method is typically applied for longitudinal data analysis. However, there are a series of problems due to the misspecification of the within-subject correlation structure, especially in Bayesian estimation. To handle these difficulties, in this paper, we construct a class of generalized estimating equations for longitudinal data with hyperspherical coordinates (HPC) and propose a Bayesian approach established through empirical likelihood (EL). Additionally, an efficient Markov chain Monte Carlo (MCMC) procedure is developed for the required computation of the posterior distribution. As proved by the simulation studies and an application to a real longitudinal data set, our method not only performs better than traditional empirical likelihood estimation and Bayesian estimation with partial autocorrelations (PAC) but also is suitable for non-Gaussian data.

Bayesian Estimation of Past Astronomical Frequencies, Lunar Distance, and Length of Day From Sediment Cycles

AbstractAstronomical cycles recorded in stratigraphic sequences offer a powerful data source to estimate Earth's axial precession frequency k, as well as the frequency of rotation of the planetary perihelia (gi) and of the ascending nodes of their orbital planes (si). Together, these frequencies control the insolation cycles (eccentricity, obliquity and climatic precession) that affect climate and sedimentation, providing a geologic record of ancient Solar system behavior spanning billions of years. Here, we introduce two Bayesian methods that harness stratigraphic data to quantitatively estimate ancient astronomical frequencies and their uncertainties. The first method (TimeOptB) calculates the posterior probability density function (PDF) of the axial precession frequency k and of the sedimentation rate u for a given cyclostratigraphic data set, while setting the Solar system frequencies gi and si to fixed values. The second method (TimeOptBMCMC) applies an adaptive Markov chain Monte Carlo algorithm to efficiently sample the posterior PDF of all the parameters that affect astronomical cycles recorded in stratigraphy: five gi, five si, k, and u. We also include an approach to assess the significance of detecting astronomical cycles in cyclostratigraphic records. The methods provide an extension of current approaches that is computationally efficient and well suited to recover the history of astronomical cycles, Earth‐Moon history, and the evolution of the Solar system from geological records. As case studies, data from the Xiamaling Formation (N. China, 1.4 Ga) and ODP Site 1262 (S. Atlantic, 55 Ma) are evaluated, providing updated estimates of astronomical frequencies, Earth‐Moon history, and secular resonance terms.

First molecular observation on Mylonchulus hawaiiensis from South Africa.

During a survey of soil nematodes in South Africa, a species of predatory nematodes, namely Mylonchulus hawaiiensis was recovered from soil in the Magoebaskloof mountain of Limpopo Province, South Africa. The morphology of the material studied fit well with the previous populations of the same species. A molecular study of 18S rDNA region of M. hawaiiensis indicated a 100% similarity between the South African population and the Japanese population of M. hawaiiensis (AB361438; AB361439; AB361440; AB361442). In addition, phylogenetic analysis placed all M. hawaiiensis in a group with 0.97 posterior probability. Additionally, ITS rDNA of M. hawaiiensis amplified for the first time. However, Principal component analysis (PCA) showed a morphological variation among the different populations of M. hawaiiensis. In addition, haplotype analysis also revealed that the South African population is close to the Japanese population. In conclusion, 18S rDNA was a good marker for detecting M. hawaiiensis. Measurement, photographs, and phylogenetic position of South African M. hawaiinesis are given.

The autocorrelated Bayesian sampler: A rational process for probability judgments, estimates, confidence intervals, choices, confidence judgments, and response times.

Normative models of decision-making that optimally transform noisy (sensory) information into categorical decisions qualitatively mismatch human behavior. Indeed, leading computational models have only achieved high empirical corroboration by adding task-specific assumptions that deviate from normative principles. In response, we offer a Bayesian approach that implicitly produces a posterior distribution of possible answers (hypotheses) in response to sensory information. But we assume that the brain has no direct access to this posterior, but can only sample hypotheses according to their posterior probabilities. Accordingly, we argue that the primary problem of normative concern in decision-making is integrating stochastic hypotheses, rather than stochastic sensory information, to make categorical decisions. This implies that human response variability arises mainly from posterior sampling rather than sensory noise. Because human hypothesis generation is serially correlated, hypothesis samples will be autocorrelated. Guided by this new problem formulation, we develop a new process, the Autocorrelated Bayesian Sampler (ABS), which grounds autocorrelated hypothesis generation in a sophisticated sampling algorithm. The ABS provides a single mechanism that qualitatively explains many empirical effects of probability judgments, estimates, confidence intervals, choice, confidence judgments, response times, and their relationships. Our analysis demonstrates the unifying power of a perspective shift in the exploration of normative models. It also exemplifies the proposal that the "Bayesian brain" operates using samples not probabilities, and that variability in human behavior may primarily reflect computational rather than sensory noise. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

The Expected Behaviors of Posterior Predictive Tests and Their Unexpected Interpretation.

Poor fit between models of sequence or trait evolution and empirical data is known to cause biases and lead to spurious conclusions about evolutionary patterns and processes. Bayesian posterior prediction is a flexible and intuitive approach for detecting such cases of poor fit. However, the expected behavior of posterior predictive tests has never been characterized for evolutionary models, which is critical for their proper interpretation. Here, we show that the expected distribution of posterior predictive P-values is generally not uniform, in contrast to frequentist P-values used for hypothesis testing, and extreme posterior predictive P-values often provide more evidence of poor fit than typically appreciated. Posterior prediction assesses model adequacy under highly favorable circumstances, because the model is fitted to the data, which leads to expected distributions that are often concentrated around intermediate values. Nonuniform expected distributions of P-values do not pose a problem for the application of these tests, however, and posterior predictive P-values can be interpreted as the posterior probability that the fitted model would predict a dataset with a test statistic value as extreme as the value calculated from the observed data.