On predictive inference for intractable models via approximate Bayesian computation

Approximate Bayesian computation (ABC) is commonly used for parameter estimation and model comparison for intractable simulator-based statistical models whose likelihood function cannot be evaluated. In this paper we instead investigate the feasibility of ABC as a generic approximate method for predictive inference, in particular, for computing the posterior predictive distribution of future observations or missing data of interest. We consider three complementary ABC approaches for this goal, each based on different assumptions regarding which predictive density of the intractable model can be sampled from. The case where only simulation from the joint density of the observed and future data given the model parameters can be used for inference is given particular attention and it is shown that the ideal summary statistic in this setting is minimal predictive sufficient instead of merely minimal sufficient (in the ordinary sense). An ABC prediction approach that takes advantage of a certain latent variable representation is also investigated. We additionally show how common ABC sampling algorithms can be used in the predictive settings considered. Our main results are first illustrated by using simple time-series models that facilitate analytical treatment, and later by using two common intractable dynamic models.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s11222-022-10163-6.

Farmland transfer and esophageal cancer incidence rate: mediation of pollution-related agricultural input intensity

Cancer is a growing global health threat. Examining the determinants of cancer incidence can benefit for cancer treatment and prevention. Farmland transfer relates to the risk factors of esophageal cancer including environmental pollution, services access, and habits. This study characterizes the associations between farmland transfer and esophageal cancer incidence rate (ECI) that integrate mediated effect of pollution-related agricultural input intensity in Xiaoshan District, China. The state-space model is employed to quantify the relationships among farmland transfer, pollution-related agricultural input intensity, and ECI. The results showed that (1) Total effects of the proportion of transferred farmland (TFA) area cause a reduction in the ECI. Besides, the total positive effects of the proportion of transferred farmland cultivated non-grain crop (NGC) and proportion of farmland transferred to non-farmer users (NFU) show a downward trend. (2) The raise of TFA can result in the reduction of chemical fertilizer use intensity. Meanwhile, the raise of NGC and NFU can result in the growth of pollution-related agricultural input intensity. But these increasing effects generally show a downward trend. (3) Increasing chemical fertilizer use intensity and pesticide use intensity results in the rise of esophageal cancer incidence rate as a whole. (4) In general, farmland transfer has positive direct effects on esophageal cancer incidence rate. (5) The average proportions of mediated effects in all state-space models are larger than 10%. These findings can raise land reform policy designers' awareness of the risk of public health since the land transfer markets are emerging rapidly in land reform in many developing countries to improve agricultural production.

Markov genealogy processes

We construct a family of genealogy-valued Markov processes that are induced by a continuous-time Markov population process. We derive exact expressions for the likelihood of a given genealogy conditional on the history of the underlying population process. These lead to a nonlinear filtering equation which can be used to design efficient Monte Carlo inference algorithms. We demonstrate these calculations with several examples. Existing full-information approaches for phylodynamic inference are special cases of the theory.

Fitting stochastic predator-prey models using both population density and kill rate data

Most mechanistic predator-prey modelling has involved either parameterization from process rate data or inverse modelling. Here, we take a median road: we aim at identifying the potential benefits of combining datasets, when both population growth and predation processes are viewed as stochastic. We fit a discrete-time, stochastic predator-prey model of the Leslie type to simulated time series of densities and kill rate data. Our model has both environmental stochasticity in the growth rates and interaction stochasticity, i.e., a stochastic functional response. We examine what the kill rate data brings to the quality of the estimates, and whether estimation is possible (for various time series lengths) solely with time series of population counts or biomass data. Both Bayesian and frequentist estimation are performed, providing multiple ways to check model identifiability. The Fisher Information Matrix suggests that models with and without kill rate data are all identifiable, although correlations remain between parameters that belong to the same functional form. However, our results show that if the attractor is a fixed point in the absence of stochasticity, identifying parameters in practice requires kill rate data as a complement to the time series of population densities, due to the relatively flat likelihood. Only noisy limit cycle attractors can be identified directly from population count data (as in inverse modelling), although even in this case, adding kill rate data - including in small amounts - can make the estimates much more precise. Overall, we show that under process stochasticity in interaction rates, interaction data might be essential to obtain identifiable dynamical models for multiple species. These results may extend to other biotic interactions than predation, for which similar models combining interaction rates and population counts could be developed.

Separating the effects of climate, bycatch, predation and harvesting on tītī (Ardenna grisea) population dynamics in New Zealand: A model-based assessment

A suite of factors may have contributed to declines in the tītī (sooty shearwater; Ardenna grisea) population in the New Zealand region since at least the 1960s. Recent estimation of the magnitude of most sources of non-natural mortality has presented the opportunity to quantitatively assess the relative importance of these factors. We fit a range of population dynamics models to a time-series of relative abundance data from 1976 until 2005, with the various sources of mortality being modelled at the appropriate part of the life-cycle. We present estimates of effects obtained from the best-fitting model and using model averaging. The best-fitting models explained much of the variation in the abundance index when survival and fecundity were linked to the Southern Oscillation Index, with strong decreases in adult survival, juvenile survival and fecundity being related to El Niño-Southern Oscillation (ENSO) events. Predation by introduced animals, harvesting by humans, and bycatch in fisheries also appear to have contributed to the population decline. It is envisioned that the best-fitting models will form the basis for quantitative assessments of competing management strategies. Our analysis suggests that sustainability of the New Zealand tītī population will be most influenced by climate, in particular by how climate change will affect the frequency and intensity of ENSO events in the future. Removal of the effects of both depredation by introduced predators and harvesting by humans is likely to have fewer benefits for the population than alleviating climate effects.

Uncovering ecological state dynamics with hidden Markov models

Ecological systems can often be characterised by changes among a finite set of underlying states pertaining to individuals, populations, communities or entire ecosystems through time. Owing to the inherent difficulty of empirical field studies, ecological state dynamics operating at any level of this hierarchy can often be unobservable or 'hidden'. Ecologists must therefore often contend with incomplete or indirect observations that are somehow related to these underlying processes. By formally disentangling state and observation processes based on simple yet powerful mathematical properties that can be used to describe many ecological phenomena, hidden Markov models (HMMs) can facilitate inferences about complex system state dynamics that might otherwise be intractable. However, HMMs have only recently begun to gain traction within the broader ecological community. We provide a gentle introduction to HMMs, establish some common terminology, review the immense scope of HMMs for applied ecological research and provide a tutorial on implementation and interpretation. By illustrating how practitioners can use a simple conceptual template to customise HMMs for their specific systems of interest, revealing methodological links between existing applications, and highlighting some practical considerations and limitations of these approaches, our goal is to help establish HMMs as a fundamental inferential tool for ecologists.

Inference on high-dimensional implicit dynamic models using a guided intermediate resampling filter

We propose a method for inference on moderately high-dimensional, nonlinear, non-Gaussian, partially observed Markov process models for which the transition density is not analytically tractable. Markov processes with intractable transition densities arise in models defined implicitly by simulation algorithms. Widely used particle filter methods are applicable to nonlinear, non-Gaussian models but suffer from the curse of dimensionality. Improved scalability is provided by ensemble Kalman filter methods, but these are inappropriate for highly nonlinear and non-Gaussian models. We propose a particle filter method having improved practical and theoretical scalability with respect to the model dimension. This method is applicable to implicitly defined models having analytically intractable transition densities. Our method is developed based on the assumption that the latent process is defined in continuous time and that a simulator of this latent process is available. In this method, particles are propagated at intermediate time intervals between observations and are resampled based on a forecast likelihood of future observations. We combine this particle filter with parameter estimation methodology to enable likelihood-based inference for highly nonlinear spatiotemporal systems. We demonstrate our methodology on a stochastic Lorenz 96 model and a model for the population dynamics of infectious diseases in a network of linked regions.

Choices and trade-offs in inference with infectious disease models

Inference using mathematical models of infectious disease dynamics can be an invaluable tool for the interpretation and analysis of epidemiological data. However, researchers wishing to use this tool are faced with a choice of models and model types, simulation methods, inference methods and software packages. Given the multitude of options, it can be challenging to decide on the best approach. Here, we delineate the choices and trade-offs involved in deciding on an approach for inference, and discuss aspects that might inform this decision. We provide examples of inference with a dataset of influenza cases using the R packages pomp and rbi.

Inference of the generalized-growth model via maximum likelihood estimation: A reflection on the impact of overdispersion

Recently, the generalized-growth model was introduced as a flexible approach to characterize growth dynamics of disease outbreaks during the early ascending phase. In this work, by using classical maximum likelihood estimation to obtain parameter estimates, we evaluate the impact of varying levels of overdispersion on the inference of the growth scaling parameter through comparing Poisson and Negative binomial models. In particular, under exponential and sub-exponential growth scenarios, we evaluate, via simulations, the error rate of making an incorrect characterization of early outbreak growth patterns. Simulation results show that the ability to correctly identify early outbreak growth patterns can be affected by overdispersion even when accounted for using the Negative binomial model. We exemplify our findings using data on five different outbreaks. Overall, our results show that estimates should be interpreted with caution when data are overdispersed.

Prepaid parameter estimation without likelihoods

In various fields, statistical models of interest are analytically intractable and inference is usually performed using a simulation-based method. However elegant these methods are, they are often painstakingly slow and convergence is difficult to assess. As a result, statistical inference is greatly hampered by computational constraints. However, for a given statistical model, different users, even with different data, are likely to perform similar computations. Computations done by one user are potentially useful for other users with different data sets. We propose a pooling of resources across researchers to capitalize on this. More specifically, we preemptively chart out the entire space of possible model outcomes in a prepaid database. Using advanced interpolation techniques, any individual estimation problem can now be solved on the spot. The prepaid method can easily accommodate different priors as well as constraints on the parameters. We created prepaid databases for three challenging models and demonstrate how they can be distributed through an online parameter estimation service. Our method outperforms state-of-the-art estimation techniques in both speed (with a 23,000 to 100,000-fold speed up) and accuracy, and is able to handle previously quasi inestimable models.

Interesting nonlinear models are often analytically intractable. As a result, statistical inference has to rely on massive, time-intensive, simulations. The main idea of our method is to avoid the redundancy of similar computations that typically occur when different researchers independently fit the same model to their particular dataset. Instead, we propose to pool computational resources across the researchers interested in any given model. The prepaid method starts with an extensive simulation of datasets across the parameter space. The simulated data are compressed into summary statistics, and the relation to the parameters is learned using machine learning techniques. This results in a parameter estimation machine that produces accurate estimates very quickly (a 23,000 to 100,000-fold speed up compared to traditional methods).

Panel Data Analysis via Mechanistic Models

Panel data, also known as longitudinal data, consist of a collection of time series. Each time series, which could itself be multivariate, comprises a sequence of measurements taken on a distinct unit. Mechanistic modeling involves writing down scientifically motivated equations describing the collection of dynamic systems giving rise to the observations on each unit. A defining characteristic of panel systems is that the dynamic interaction between units should be negligible. Panel models therefore consist of a collection of independent stochastic processes, generally linked through shared parameters while also having unit-specific parameters. To give the scientist flexibility in model specification, we are motivated to develop a framework for inference on panel data permitting the consideration of arbitrary nonlinear, partially observed panel models. We build on iterated filtering techniques that provide likelihood-based inference on nonlinear partially observed Markov process models for time series data. Our methodology depends on the latent Markov process only through simulation; this plug-and-play property ensures applicability to a large class of models. We demonstrate our methodology on a toy example and two epidemiological case studies. We address inferential and computational issues arising due to the combination of model complexity and dataset size. Supplementary materials for this article are available online.

Two-scale dispersal estimation for biological invasions via synthetic likelihood

Biological invasions reshape environments and affect the ecological and economic welfare of states and communities. Such invasions advance on multiple spatial scales, complicating their control. When modeling stochastic dispersal processes, intractable likelihoods and autocorrelated data complicate parameter estimation. As with other approaches, the recent synthetic likelihood framework for stochastic models uses summary statistics to reduce this complexity; however, it additionally provides usable likelihoods, facilitating the use of existing likelihood-based machinery. Here, we extend this framework to parameterize multi-scale spatio-temporal dispersal models and compare existing and newly developed spatial summary statistics to characterize dispersal patterns. We provide general methods to evaluate potential summary statistics and present a fitting procedure that accurately estimates dispersal parameters on simulated data. Finally, we apply our methods to quantify the short and long range dispersal of Chagas disease vectors in urban Arequipa, Peru, and assess the feasibility of a purely reactive strategy to contain the invasion.

Modeling and inference for infectious disease dynamics: a likelihood-based approach

Likelihood-based statistical inference has been considered in most scientific fields involving stochastic modeling. This includes infectious disease dynamics, where scientific understanding can help capture biological processes in so-called mechanistic models and their likelihood functions. However, when the likelihood of such mechanistic models lacks a closed-form expression, computational burdens are substantial. In this context, algorithmic advances have facilitated likelihood maximization, promoting the study of novel data-motivated mechanistic models over the last decade. Reviewing these models is the focus of this paper. In particular, we highlight statistical aspects of these models like overdispersion, which is key in the interface between nonlinear infectious disease modeling and data analysis. We also point out potential directions for further model exploration.

Monte Carlo profile confidence intervals for dynamic systems

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.