Parameter inference in small world network disease models with approximate Bayesian Computational methods

Analytically or computationally intractable likelihood functions can arise in complex statistical inferential problems making them inaccessible to standard Bayesian inferential methods. Approximate Bayesian computation (ABC) methods address such inferential problems by replacing direct likelihood evaluations with repeated sampling from the model. ABC methods have been predominantly applied to parameter estimation problems and less to model choice problems due to the added difficulty of handling multiple model spaces. The ABC algorithm proposed here addresses model choice problems by extending Fearnhead and Prangle (2012, Journal of the Royal Statistical Society, Series B 74, 1-28) where the posterior mean of the model parameters estimated through regression formed the summary statistics used in the discrepancy measure. An additional stepwise multinomial logistic regression is performed on the model indicator variable in the regression step and the estimated model probabilities are incorporated into the set of summary statistics for model choice purposes. A reversible jump Markov chain Monte Carlo step is also included in the algorithm to increase model diversity for thorough exploration of the model space. This algorithm was applied to a validating example to demonstrate the robustness of the algorithm across a wide range of true model probabilities. Its subsequent use in three pathogen transmission examples of varying complexity illustrates the utility of the algorithm in inferring preference of particular transmission models for the pathogens.

Wastewater production, like many other engineered and environmental processes, is inherent stochastic in nature and requires the use of complex stochastic models, for example, to predict realistic patterns of down-the-drain chemicals or pharmaceuticals and personal care products. Up until now, a formal method of statistical inference has been lacking for many of those models, where explicit likelihood functions were intractable. In this Article, we investigate Approximate Bayesian Computation (ABC) methods to infer important parameters of stochastic environmental models. ABC methods have been recently suggested to perform model-based inference in a Bayesian setting when model likelihoods are analytically or computationally intractable and have not been applied to environmental systems analysis or water quality modeling before. In a case study, we investigate the performance of three different algorithms to infer the number of wastewater pulses contained in three high-resolution data series of benzotriazole and total nitrogen loads in sewers. We find that all algorithms perform well and that the uncertainty in the inferred number of corresponding wastewater pulses varies between 6% and 28%. In our case, the results are more sensitive to substance characteristics than to catchment properties. Although the application of ABC methods requires careful tuning and attention to detail, they have a great general potential to update stochastic model parameters with monitoring data and improve their predictive capabilities.

Approximate Bayesian methods, or “likelihood-free” methods, have become a key component of modern statistical methodology, providing a framework for inference, prediction, and decision-making. Their versatility lies in their ability to handle parameter estimation, model selection, and uncertainty quantification within a unified probabilistic framework. In this work, Approximate Bayesian Computation (ABC) using an ellipsoidal Nested Sampling (NS) approach is employed to deal with model updating and model comparison issues applied to structural health monitoring. ABC methods rely essentially on the ability to simulate data from a simulator/forward model, bypassing the need to write down an explicit likelihood function, which is always far from trivial. These methods are particularly captivating because of the modelling freedom they provide; in addition, they are flexible in the sense that different discrepancy functions measuring the similarity between the observed modal data and the corresponding model output can be used. However, Bayesian inference methods usually require numerous forward model simulations to generate converged samples. To enhance the computational efficiency of the sampler, the minimum-volume enclosing ellipsoid (MVEE) is incorporated to better enclose the accepted particles and to guide the sampler more efficiently towards the highest probability region. The performance and the robustness of the novel sampler in structural model updating and model selection is demonstrated here via different numerical studies using modal data.

Approximate Bayesian computation (ABC) constitutes a class of computational methods rooted in Bayesian statistics. In all model-based statistical inference, the likelihood function is of central importance, since it expresses the probability of the observed data under a particular statistical model, and thus quantifies the support data lend to particular values of parameters and to choices among different models. For simple models, an analytical formula for the likelihood function can typically be derived. However, for more complex models, an analytical formula might be elusive or the likelihood function might be computationally very costly to evaluate. ABC methods bypass the evaluation of the likelihood function. In this way, ABC methods widen the realm of models for which statistical inference can be considered. ABC methods are mathematically well-founded, but they inevitably make assumptions and approximations whose impact needs to be carefully assessed. Furthermore, the wider application domain of ABC exacerbates the challenges of parameter estimation and model selection. ABC has rapidly gained popularity over the last years and in particular for the analysis of complex problems arising in biological sciences (e.g., in population genetics, ecology, epidemiology, and systems biology).

We consider the context of probabilistic inference of model parameters given error bars or confidence intervals on model output values, when the data is unavailable. We introduce a class of algorithms in a Bayesian framework, relying on maximum entropy arguments and approximate Bayesian computation methods, to generate consistent data with the given summary statistics. Once we obtain consistent data sets, we pool the respective posteriors, to arrive at a single, averaged density on the parameters. This approach allows us to perform accurate forward uncertainty propagation consistent with the reported statistics.

There are various analysis methods for CT perfusion (CTP). Although the advantages of Bayesian estimation algorithms have been newly suggested, comparisons with other analysis methods on clinical data are still limited. In this study, we compared the Bayesian estimation method with the singular value decomposition (SVD) method in the evaluation of patients with acute cerebral infarction and examined its usefulness. CTP data from 13 patients with acute stroke were analyzed using the SVD and Bayesian estimation methods implemented in Vitrea. Evaluation of visual clarity of the ischemic area and quantitative values of the healthy side-affected side ratio using the mean values of the left and right region of interest (ROI) on the images were compared using the SVD and Bayesian estimation methods. In visual evaluation, there were significant differences in CBV in four cases, and in CBF, MTT, and TTP in many cases. The healthy side-affected side ratio of the SVD and Bayesian estimation methods were as follows: CBF 1.19, 1.84; CBV 1.09, 1.02; MTT 1.12, 1.79; and TTP 1.48, 1.19. For CBF and MTT, the Bayesian estimation method had a larger ratio of the healthy side to the affected side, and for TTP, the SVD method had a larger ratio of the test side to the affected side. We suggest that the Bayesian estimation method is more useful than the SVD method for assessing CBF and MTT in CTP analysis of patients with acute stroke.

The characteristics of the propagation of forest fires under the influence of firebrands and the interaction zone (lx, ly) due to radiation are examined using the model of Small World Network. We analyze the distribution of the connections in a Small World Network and the cluster coefficient that represent the mathematical properties of the network. The used model is a stochastic model for predicting the behavior of wildfires. It is a variant of the social Small World Network, initially proposed by Watts and Strogatz, which allows the creation of more clusters and connections over long distances. This model was successfully applied to the spread of diseases and is characterized by a strong performance in clusters and a Poisson distribution of connections. The Model of Small World Network has also been adapted to study the spread of forest fires where it can include connections beyond nearest neighbors due to radiation from the flames or fire surges induced by firebrands which other propagation models cannot. It has been validated by experimental results of real fires. The main goal of this paper consists to investigate the most robust measures of network topology for a heterogeneous and∕or homogeneous system near the percolation threshold.

In time-series analysis, state-space models (SSMs) have been widely used to estimate the conditional probability distributions of hidden variables and parameter values, as well as to understand structures that can generate the data. For example, the Kalman filter is used to analytically calculate the conditional probability distribution on linear SSMs in terms of minimizing the variance, and several extensions, such as the unscented Kalman filter and particle filter, have been applied to calculate the approximate distribution on nonlinear SSMs. Recently, the approximate Bayesian computation (ABC) has been applied to such time-series filtering to handle intractable likelihoods; however, it remains problematic with respect to consistently achieving a reduction of the estimation bias, evaluating the validity of the models, and dealing with replicated observations. To address these problems, in this paper, we propose a novel method combined with the kernel ABC to perform filtering, parameter estimation, and the model evaluation in SSMs. Simulation studies show that the proposed method produces inference that is comparable to other ABC methods, with the advantage of not requiring a careful calibration of the ABC threshold. In addition, we evaluate the performance of the model-selection capability using true and competitive models on synthetic data from nonlinear SSMs. Finally, we apply the proposed method to real data in rat circadian oscillations, and demonstrated the usefulness in practical situations.

With larger data at their disposal, scientists are emboldened to tackle complex questions that require sophisticated statistical models. It is not unusual for the latter to have likelihood functions that elude analytical formulations. Even under such adversity, when one can simulate from the sampling distribution, Bayesian analysis can be conducted using approximate methods such as Approximate Bayesian Computation (ABC) or Bayesian Synthetic Likelihood (BSL). A significant drawback of these methods is that the number of required simulations can be prohibitively large, thus severely limiting their scope. In this paper we design perturbed MCMC samplers that can be used within the ABC and BSL paradigms to significantly accelerate computation while maintaining control on computational efficiency. The proposed strategy relies on recycling samples from the chain’s past. The algorithmic design is supported by a theoretical analysis while practical performance is examined via a series of simulation examples and data analyses.

We estimate the parameters of a stochastic process model for a macroparasite population within a host using approximate Bayesian computation (ABC). The immunity of the host is an unobserved model variable and only mature macroparasites at sacrifice of the host are counted. With very limited data, process rates are inferred reasonably precisely. Modeling involves a three variable Markov process for which the observed data likelihood is computationally intractable. ABC methods are particularly useful when the likelihood is analytically or computationally intractable. The ABC algorithm we present is based on sequential Monte Carlo, is adaptive in nature, and overcomes some drawbacks of previous approaches to ABC. The algorithm is validated on a test example involving simulated data from an autologistic model before being used to infer parameters of the Markov process model for experimental data. The fitted model explains the observed extra-binomial variation in terms of a zero-one immunity variable, which has a short-lived presence in the host.

Inverse problems arise everywhere we have indirect measurement. Regularization and Bayesian inference methods are two main approaches to handle inverse problems. Bayesian inference approach is more general and has much more tools for developing efficient methods for difficult problems. In this chapter, first, an overview of the Bayesian parameter estimation is presented, then we see the extension for inverse problems. The main difficulty is the great dimension of unknown quantity and the appropriate choice of the prior law. The second main difficulty is the computational aspects. Different approximate Bayesian computations and in particular the variational Bayesian approximation (VBA) methods are explained in details.

As cardiac cell models become increasingly complex, a correspondingly complex ‘genealogy’ of inherited parameter values has also emerged. The result has been the loss of a direct link between model parameters and experimental data, limiting both reproducibility and the ability to re-fit to new data. We examine the ability of approximate Bayesian computation (ABC) to infer parameter distributions in the seminal action potential model of Hodgkin and Huxley, for which an immediate and documented connection to experimental results exists. The ability of ABC to produce tight posteriors around the reported values for the gating rates of sodium and potassium ion channels validates the precision of this early work, while the highly variable posteriors around certain voltage dependency parameters suggests that voltage clamp experiments alone are insufficient to constrain the full model. Despite this, Hodgkin and Huxley's estimates are shown to be competitive with those produced by ABC, and the variable behaviour of posterior parametrized models under complex voltage protocols suggests that with additional data the model could be fully constrained. This work will provide the starting point for a full identifiability analysis of commonly used cardiac models, as well as a template for informative, data-driven parametrization of newly proposed models.

BackgroundModelling disease outbreaks often involves integrating the wealth of data that are gathered during modern outbreaks into complex mathematical or computational models of transmission. Incorporating these data into simple compartmental epidemiological models is often challenging, requiring the use of more complex but also more efficient computational models. In this paper we introduce a new framework that allows for a more systematic and user-friendly way of building and running epidemiological models that efficiently handles disease data and reduces much of the boilerplate code that usually associated to these models. We introduce the framework by developing an SIR model on a simple network as an example.ResultsWe develop Broadwick, a modular, object-oriented epidemiological framework that efficiently handles large epidemiological datasets and provides packages for stochastic simulations, parameter inference using Approximate Bayesian Computation (ABC) and Markov Chain Monte Carlo (MCMC) methods. Each algorithm used is fully customisable with sensible defaults that are easily overridden by custom algorithms as required.ConclusionBroadwick is an epidemiological modelling framework developed to increase the productivity of researchers by providing a common framework with which to develop and share complex models. It will appeal to research team leaders as it allows for models to be created prior to a disease outbreak and has the ability to handle large datasets commonly found in epidemiological modelling.

OBJECTIVES/GOALS: Evaluate the migration and immune suppressive functions of CCR2+/CX3CR1+ myeloid-derived suppressor cells (MDSCs). Integrate experimental data and biologically relevant mathematical models of infiltrating MDSCs in the context of glioblastoma (GBM). METHODS/STUDY POPULATION: CCR2+/CX3CR1+ cells were enriched from bone marrow obtained from CCR2(+/RFP)/CX3CR1(+/GFP) glioma-bearing mice to evaluate their immune-suppressive phenotype and ability to migrate to CCL2 and CCL7. Fluorescent imaging and quantification were performed on a range of tumor sizes to acquire vasculature, tumor, T cell, and MDSC densities. A system of ordinary differential equations was constructed to represent the temporal dynamics of glioma cells, T cells, and MDSCs within the tumor microenvironment. The Approximate Bayesian Computation method was used to determine probability distributions of important parameters, such as the suppression rate of T cells by MDSCs. RESULTS/ANTICIPATED RESULTS: CCR2+/CX3CR1+ M-MDSCs isolated from the bone marrow of tumor-bearing mice suppress CD8+ T cell proliferation and IFNγ production. CCR2+/CX3CR1+ cells migrate to recombinant and KR158B glioma sourced CCL2 and CCL7. Parameter values determined by the Approximate Bayesian Computation method agreed with parameter values from experimental data. This result further validated the structure and results of the mathematical model when performing computer simulations; thus, we can predict CCR2+/CX3CR1+ M-MDSC infiltration over time. DISCUSSION/SIGNIFICANCE: The immune-suppressive microenvironment in GBM contributes to poor outcomes despite standard of care. This study integrates biological and mathematical models to better understand infiltrating immune-suppressive cells, namely CCR2+/CX3CR1+ M-MDSCs. Future directions include modeling immunotherapies.

SummaryThe marginalized particle filtering (MPF) is a powerful technique reducing the number of particles necessary to effectively estimate hidden states of state‐space models. This paper alleviates the assumption of a fully known and computationally tractable observation model. Exploiting the recent developments in the theory of approximate Bayesian computation (ABC) filtration, an ABC counterpart of MPF is proposed, applicable when the observation model is too complex to be evaluated analytically or even numerically, but it is still possible to sample from it by plugging in the state. The novelty is 2‐fold. First, ABC methods have not been used in marginalized filtering yet. Second, a new multivariate robust method for evaluation of particle weights is proposed. The goal of this paper is to demonstrate the idea on the background of the MPF with a particular accent on exposition.