Monte Carlo profile confidence intervals for dynamic systems - PubMed (original) (raw)

Monte Carlo profile confidence intervals for dynamic systems

E L Ionides et al. J R Soc Interface. 2017 Jul.

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.

Keywords: likelihood-based inference; panel data; phylodynamic inference; sequential Monte Carlo; spatio-temporal data; time series.

© 2017 The Author(s).

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Conflict of interest statement

We declare we have no competing interests.

Figures

Figure 1.

Figure 1.

The effect of bias on confidence intervals for a quadratic profile log-likelihood function. (a) The blue dotted quadratic represents a log-likelihood profile. The maximum-likelihood estimator of the profiled parameter is _ϕ_3, with corresponding log likelihood formula image. The 95% CI [_ϕ_1, _ϕ_5] is constructed, via the horizontal and vertical blue dotted lines, as the set of parameter values with profile log likelihood higher than ℓ* − 1.92. The red quadratic is the sum of the blue dotted quadratic and linear bias (black dashed line). Horizontal and vertical red lines construct the resulting approximate confidence interval [_ϕ_2, _ϕ_6] and point estimate _ϕ_4. (b) The same construction, but with higher curvature of the profile log likelihood leading to diminishing effect of the bias on the confidence interval. (Online version in colour.)

Figure 2.

Figure 2.

Profile likelihood for an infectious disease transmission parameter inferred from genetic data on pathogens. The smoothed profile likelihood and corresponding MCAP 95% CI are shown as solid red lines. The quadratic approximation in a neighbourhood of the maximum is shown as a blue dotted line. (Online version in colour.)

Figure 3.

Figure 3.

Profile likelihood for a nonlinear partially observed Markov process model for a panel of time series of historical state-level polio incidence in the USA. The smoothed profile likelihood and corresponding MCAP 95% CI are shown as solid red lines. The quadratic approximation in a neighbourhood of the maximum is shown as a dotted blue line. (Online version in colour.)

Figure 4.

Figure 4.

Profile construction for the toy model. The exact profile and its asymptotic 95% CI are constructed with black dashed lines. Points show Monte Carlo profile evaluations. The MCAP is constructed in solid red lines, using the default λ = 0.75 smoothing parameter. The quadratic approximation used to calculate the MCAP profile cut-off is shown as a dotted blue line. (Online version in colour.)

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