Abstract
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 datasets 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 examples arising in the study of infectious disease transmission, demonstrating our methodology for inference on nonlinear dynamic models using genetic sequence data and panel time-series data. We also discuss applicability to nonlinear time-series and spatio-temporal data.
Highlights
This paper develops profile likelihood inference methodology for situations where computationally intensive Monte Carlo methods are employed to evaluate and maximize the likelihood function
If the profile log-likelihood function can be computed with a Monte Carlo error small compared to one unit, carrying out statistical inference from the Monte Carlo profile as if it were the true profile will have relatively small effects on resulting confidence intervals
This paper has focused on likelihood-based confidence intervals
Summary
This paper develops profile likelihood inference methodology for situations where computationally intensive Monte Carlo methods are employed to evaluate and maximize the likelihood function. A useful simplification arises when it is reasonable to treat the distribution of the Monte Carlo bias and error in (1.6) as constant across the statistically relevant region having high-profile likelihood. This leads us to consider a metamodel with bk( y*) 1⁄4 b( y*) and with e1:K( y*) independent and identically distributed (i.i.d.).
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