Monte Carlo profile confidence intervals

Monte Carlo methods to evaluate and maximize the likelihood function enable the construction of confidence intervals and hypothesis tests, facilitating scientific investigation using models for which the likelihood function is intractable. When Monte Carlo error can be made small, by sufficiently exhaustive computation, then the standard theory and practice of likelihood-based inference applies. As data become larger, and models more complex, situations arise where no reasonable amount of computation can render Monte Carlo error negligible. We develop profile likelihood methodology to provide frequentist inferences that take into account Monte Carlo uncertainty. We investigate the role of this methodology in facilitating inference for computationally challenging dynamic latent variable models. We present three examples arising in the study of infectious disease transmission. These three examples demonstrate our methodology for inference on nonlinear dynamic models using genetic sequence data, panel time series data, and spatiotemporal data. We also discuss applicability to nonlinear time series analysis.