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1. Understanding and mitigating difficulties in posterior predictive evaluation

   Agrawal, A; Domke, J (July 2025, PMLR)
2. Assessing the Adequacy of Morphological Models Using Posterior Predictive Simulations

   https://doi.org/10.1093/sysbio/syae055  

   Mulvey, Laura P_A; May, Michael R; Brown, Jeremy M; Höhna, Sebastian; Wright, April M; Warnock, Rachel C_M (October 2024, Systematic Biology)

   Klopfstein, Seraina (Ed.)

   Abstract Reconstructing the evolutionary history of different groups of organisms provides insight into how life originated and diversified on Earth. Phylogenetic trees are commonly used to estimate this evolutionary history. Within Bayesian phylogenetics a major step in estimating a tree is in choosing an appropriate model of character evolution. While the most common character data used is molecular sequence data, morphological data remains a vital source of information. The use of morphological characters allows for the incorporation fossil taxa, and despite advances in molecular sequencing, continues to play a significant role in neontology. Moreover, it is the main data source that allows us to unite extinct and extant taxa directly under the same generating process. We therefore require suitable models of morphological character evolution, the most common being the Mk Lewis model. While it is frequently used in both palaeobiology and neontology, it is not known whether the simple Mk substitution model, or any extensions to it, provide a sufficiently good description of the process of morphological evolution. In this study we investigate the impact of different morphological models on empirical tetrapod datasets. Specifically, we compare unpartitioned Mk models with those where characters are partitioned by the number of observed states, both with and without allowing for rate variation across sites and accounting for ascertainment bias. We show that the choice of substitution model has an impact on both topology and branch lengths, highlighting the importance of model choice. Through simulations, we validate the use of the model adequacy approach, posterior predictive simulations, for choosing an appropriate model. Additionally, we compare the performance of model adequacy with Bayesian model selection. We demonstrate how model selection approaches based on marginal likelihoods are not appropriate for choosing between models with partition schemes that vary in character state space (i.e., that vary in Q-matrix state size). Using posterior predictive simulations, we found that current variations of the Mk model are often performing adequately in capturing the evolutionary dynamics that generated our data. We do not find any preference for a particular model extension across multiple datasets, indicating that there is no “one size fits all” when it comes to morphological data and that careful consideration should be given to choosing models of discrete character evolution. By using suitable models of character evolution, we can increase our confidence in our phylogenetic estimates, which should in turn allow us to gain more accurate insights into the evolutionary history of both extinct and extant taxa. 

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

   https://doi.org/10.1093/molbev/msae051  

   Fabreti, Luiza Guimarães; Coghill, Lyndon M; Thomson, Robert C; Höhna, Sebastian; Brown, Jeremy M (March 2024, Molecular Biology and Evolution)

   Pupko, Tal (Ed.)

   Abstract 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. 

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4. A Bayesian Saturated Model Approach to Posterior Predictive Model Checks in Confirmatory Factor Analysis

   https://doi.org/10.1080/00273171.2019.1700773  

   Zhang, Jihong; Templin, Jonathan; Mintz, Catherine E. (January 2020, Multivariate Behavioral Research)
5. A model comparison approach to posterior predictive model checks in Bayesian confirmatory factor analysis.

   https://doi.org/10.1080/10705511.2021.2012682  

   Zhang, J. (January 2022, Structural equation modeling)