Candace Wu

Inference on the minimum clinically important difference, or MCID, is an important practical problem in medicine. The basic idea is that a treatment being statistically significant may not lead to an improvement in the patients' well-being. The MCID is defined as a threshold such that, if a diagnostic measure exceeds this threshold, then the patients are more likely to notice an improvement. Typical formulations use an underspecified model, which makes a genuine Bayesian solution out of reach. Here, for a challenging personalized MCID problem, where the practically-significant threshold depends on patients' profiles, we develop a novel generalized posterior distribution, based on a working binary quantile regression model, that can be used for estimation and inference. The advantage of this formulation is two-fold: we can theoretically control the bias of the misspecified model and it has a latent variable representation which we can leverage for efficient Gibbs sampling. To ensure that the generalized Bayes inferences achieve a level of frequentist reliability, we propose a variation on the so-called generalized posterior calibration algorithm to suitably tune the spread of our proposed posterior.

In prediction problems, it is common to model the data-generating process and then use a model-based procedure, such as a Bayesian predictive distribution, to quantify uncertainty about the next observation. However, if the posited model is misspecified, then its predictions may not be calibrated---that is, the predictive distribution's quantiles may not be nominal frequentist prediction upper limits, even asymptotically. Rather than abandoning the comfort of a model-based formulation for a more complicated non-model-based approach, here we propose a strategy in which the data itself helps determine if the assumed model-based solution should be adjusted to account for model misspecification. This is achieved through a generalized Bayes formulation where a learning rate parameter is tuned, via the proposed generalized predictive calibration (GPrC) algorithm, to make the predictive distribution calibrated, even under model misspecification. Extensive numerical experiments are presented, under a variety of settings, demonstrating the proposed GPrC algorithm's validity, efficiency, and robustness.