Population Predictive Checks

Bayesian modeling has become a staple for researchers analyzing data. Thanks to recent developments in approximate posterior inference, modern researchers can easily build, use, and revise complicated Bayesian models for large and rich data. These new abilities, however, bring into focus the problem of model assessment. Researchers need tools to diagnose the fitness of their models, to understand where a model falls short, and to guide its revision. In this paper we develop a new method for Bayesian model checking, the population predictive check (Pop-PC). Pop-PCs are built on posterior predictive checks (PPC), a seminal method that checks a model by assessing the posterior predictive distribution on the observed data. Though powerful, PPCs use the data twice---both to calculate the posterior predictive and to evaluate it---which can lead to overconfident assessments. Pop-PCs, in contrast, compare the posterior predictive distribution to the population distribution of the data. This strategy blends Bayesian modeling with frequentist assessment, leading to a robust check that validates the model on its generalization. Of course the population distribution is not usually available; thus we use tools like the bootstrap and cross validation to estimate the Pop-PC. Further, we extend Pop-PCs to hierarchical models. We study Pop-PCs on classical regression and a hierarchical model of text. We show that Pop-PCs are robust to overfitting and can be easily deployed on a broad family of models

Comment: Bayesian Checking of the Second Level of Hierarchical Models: Cross-Validated Posterior Predictive Checks Using Discrepancy Measures

Comment: Bayesian Checking of the Second Level of Hierarchical Models [arXiv:0802.0743]Comment: Published in at http://dx.doi.org/10.1214/07-STS235B the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayesian inference for stochastic differential equation mixed effects models of a tumor xenography study

We consider Bayesian inference for stochastic differential equation mixed effects models (SDEMEMs) exemplifying tumor response to treatment and regrowth in mice. We produce an extensive study on how a SDEMEM can be fitted using both exact inference based on pseudo-marginal MCMC and approximate inference via Bayesian synthetic likelihoods (BSL). We investigate a two-compartments SDEMEM, these corresponding to the fractions of tumor cells killed by and survived to a treatment, respectively. Case study data considers a tumor xenography study with two treatment groups and one control, each containing 5-8 mice. Results from the case study and from simulations indicate that the SDEMEM is able to reproduce the observed growth patterns and that BSL is a robust tool for inference in SDEMEMs. Finally, we compare the fit of the SDEMEM to a similar ordinary differential equation model. Due to small sample sizes, strong prior information is needed to identify all model parameters in the SDEMEM and it cannot be determined which of the two models is the better in terms of predicting tumor growth curves. In a simulation study we find that with a sample of 17 mice per group BSL is able to identify all model parameters and distinguish treatment groups.Comment: Minor revision: posterior predictive checks for BSL have ben updated (both theory and results). Code on GitHub has ben revised accordingl