Bayes assisted model assessment

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The problem of assessing model assumptions has two features of current importance. First, statisticians continue to develop myriad new ad hoc procedures. Second we are making more and more use of unidentifiable models. The resulting difficulties are handled by making uncheckable assumptions; these assumptions are often couched in the language of rates of convergence as the number of parameters, p, grows with some measure of sample size, say n. High dimensional models and inference procedures often showcase both. I am engaged in a long term program to try to reduce the extent to which procedures have only such ad hoc motivations and to understand in practical terms the impact of the non-identifiability. I am particularly focussed on model assessment and validation. I consider statistical models in which there are assumptions about some function. In goodness-of-fit, my main area of work, these assumptions are about distribution functions or densities but the ideas are also relevant to regression models (the regression function) or multivariate analysis (assumptions about a copula) or time series (assumptions about a spectral distribution). The main thrust of my research is then to assess the assumptions. My current goal is to use Bayesian ideas to find and evaluate frequentist tests. Objectives of the proposed research program: we will be developing and assessing hypothesis tests of modelling assumptions. Our framework starts with a finite dimensional parametric specification for some function. The null hypothesis is then either a parametric model or a semi-parametric specification in which it is the parametric specification of the function which is to be checked. The alternative is then specified via an infinite dimensional nonparametric model extending the null model. We put priors on the alternative seeking to capture the nature of reasonable alternatives, then describe and study the resulting optimal tests. The priors are treated as a tool in developing the test, as a tool in comparing tests and, by computing posteriors after rejection of the hypothesized model, as a guide to how to modify the model. We will be trying to use these tools to study well known tests with a view to learning what, if any, priors would lead to such tests. We want tests for distributional assumptions: assumptions about the distribution of the data, or of the errors in a regression model or for latent variables (random effects such as frailties). We will work to develop tests for regression functions, for link functions, for spectral densities, and many more such. More broadly still I hope to use priors to develop and evaluate frequentist procedures for multistep inferential processes. We are also working on applying prior distributions in much smaller problems such as on-off experiments, replicating our work on particle discovery in high energy physics in a much simpler framework in order to combine good frequency theory properties with seriously informative priors.