arXiv subjects

Christopher Drovandi

Publications and source records attributed to Christopher Drovandi.

At least 55 records · Page 3Linked to original sources

arXiv2023

Being Bayesian in the 2020s: opportunities and challenges in the practice of modern applied Bayesian statistics

Building on a strong foundation of philosophy, theory, methods and computation over the past three decades, Bayesian approaches are now an integral part of the toolkit for most statisticians and data scientists. Whether they are dedicated Bayesians or opportunistic users, applied professionals can now reap many of the benefits afforded by the Bayesian paradigm. In this paper, we touch on six modern opportunities and challenges in applied Bayesian statistics: intelligent data collection, new data sources, federated analysis, inference for implicit models, model transfer and purposeful software products.

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arXiv2022

Strategic model reduction by analysing model sloppiness: a case study in coral calcification

It can be difficult to identify ways to reduce the complexity of large models whilst maintaining predictive power, particularly where there are hidden parameter interdependencies. Here, we demonstrate that the analysis of model sloppiness can be a new invaluable tool for strategically simplifying complex models. Such an analysis identifies parameter combinations which strongly and/or weakly inform model behaviours, yet the approach has not previously been used to inform model reduction. Using a case study on a coral calcification model calibrated to experimental data, we show how the analysis of model sloppiness can strategically inform model simplifications which maintain predictive power. Additionally, when comparing various approaches to analysing sloppiness, we find that Bayesian methods can be advantageous when unambiguous identification of the best-fit model parameters is a challenge for standard optimisation procedures.

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arXiv2022

Efficient inference and identifiability analysis for differential equation models with random parameters

Heterogeneity is a dominant factor in the behaviour of many biological processes. Despite this, it is common for mathematical and statistical analyses to ignore biological heterogeneity as a source of variability in experimental data. Therefore, methods for exploring the identifiability of models that explicitly incorporate heterogeneity through variability in model parameters are relatively underdeveloped. We develop a new likelihood-based framework, based on moment matching, for inference and identifiability analysis of differential equation models that capture biological heterogeneity through parameters that vary according to probability distributions. As our novel method is based on an approximate likelihood function, it is highly flexible; we demonstrate identifiability analysis using both a frequentist approach based on profile likelihood, and a Bayesian approach based on Markov-chain Monte Carlo. Through three case studies, we demonstrate our method by providing a didactic guide to inference and identifiability analysis of hyperparameters that relate to the statistical moments of model parameters from independent observed data. Our approach has a computational cost comparable to analysis of models that neglect heterogeneity, a significant improvement over many existing alternatives. We demonstrate how analysis of random parameter models can aid better understanding of the sources of heterogeneity from biological data.

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arXiv2022

Automatically adapting the number of state particles in SMC$^2$

Sequential Monte Carlo squared (SMC$^2$) methods can be used for parameter inference of intractable likelihood state-space models. These methods replace the likelihood with an unbiased particle filter estimator, similarly to particle Markov chain Monte Carlo (MCMC). As with particle MCMC, the efficiency of SMC$^2$ greatly depends on the variance of the likelihood estimator, and therefore on the number of state particles used within the particle filter. We introduce novel methods to adaptively select the number of state particles within SMC$^2$ using the expected squared jumping distance to trigger the adaptation, and modifying the exchange importance sampling method of \citet{Chopin2012a} to replace the current set of state particles with the new set of state particles. The resulting algorithm is fully automatic, and can significantly improve current methods. Code for our methods is available at https://github.com/imkebotha/adaptive-exact-approximate-smc.

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arXiv2022

Analysis of sloppiness in model simulations: unveiling parameter uncertainty when mathematical models are fitted to data

This work introduces a comprehensive approach to assess the sensitivity of model outputs to changes in parameter values, constrained by the combination of prior beliefs and data. This novel approach identifies stiff parameter combinations strongly affecting the quality of the model-data fit while simultaneously revealing which of these key parameter combinations are informed primarily by the data or are also substantively influenced by the priors. We focus on the very common context in complex systems where the amount and quality of data are low compared to the number of model parameters to be collectively estimated, and showcase the benefits of this technique for applications in biochemistry, ecology, and cardiac electrophysiology. We also show how stiff parameter combinations, once identified, uncover controlling mechanisms underlying the system being modeled and inform which of the model parameters need to be prioritized in future experiments for improved parameter inference from collective model-data fitting.

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arXiv2022

Regularized Zero-Variance Control Variates

Zero-variance control variates (ZV-CV) are a post-processing method to reduce the variance of Monte Carlo estimators of expectations using the derivatives of the log target. Once the derivatives are available, the only additional computational effort lies in solving a linear regression problem. Significant variance reductions have been achieved with this method in low dimensional examples, but the number of covariates in the regression rapidly increases with the dimension of the target. In this paper, we present compelling empirical evidence that the use of penalized regression techniques in the selection of high-dimensional control variates provides performance gains over the classical least squares method. Another type of regularization based on using subsets of derivatives, or a priori regularization as we refer to it in this paper, is also proposed to reduce computational and storage requirements. Several examples showing the utility and limitations of regularized ZV-CV for Bayesian inference are given. The methods proposed in this paper are accessible through the R package ZVCV.

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arXiv2022

Monte Carlo twisting for particle filters

We consider the problem of designing efficient particle filters for twisted Feynman--Kac models. Particle filters using twisted models can deliver low error approximations of statistical quantities and such twisting functions can be learnt iteratively. Practical implementations of these algorithms are complicated by the need to (i) sample from the twisted transition dynamics, and (ii) calculate the twisted potential functions. We expand the class of applicable models using rejection sampling for (i) and unbiased approximations for (ii) using a random weight particle filter. We characterise the average acceptance rates within the particle filter in order to control the computational cost, and analyse the asymptotic variance. Empirical results show the mean squared error of the normalising constant estimate in our method is smaller than a memory-equivalent particle filter but not a computation-equivalent filter. Both comparisons are improved when more efficient sampling is possible which we demonstrate on a stochastic volatility model.

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arXiv2022

Improving the Accuracy of Marginal Approximations in Likelihood-Free Inference via Localisation

Likelihood-free methods are an essential tool for performing inference for implicit models which can be simulated from, but for which the corresponding likelihood is intractable. However, common likelihood-free methods do not scale well to a large number of model parameters. A promising approach to high-dimensional likelihood-free inference involves estimating low-dimensional marginal posteriors by conditioning only on summary statistics believed to be informative for the low-dimensional component, and then combining the low-dimensional approximations in some way. In this paper, we demonstrate that such low-dimensional approximations can be surprisingly poor in practice for seemingly intuitive summary statistic choices. We describe an idealized low-dimensional summary statistic that is, in principle, suitable for marginal estimation. However, a direct approximation of the idealized choice is difficult in practice. We thus suggest an alternative approach to marginal estimation which is easier to implement and automate. Given an initial choice of low-dimensional summary statistic that might only be informative about a marginal posterior location, the new method improves performance by first crudely localising the posterior approximation using all the summary statistics to ensure global identifiability, followed by a second step that hones in on an accurate low-dimensional approximation using the low-dimensional summary statistic. We show that the posterior this approach targets can be represented as a logarithmic pool of posterior distributions based on the low-dimensional and full summary statistics, respectively. The good performance of our method is illustrated in several examples.

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arXiv2022

Component-wise iterative ensemble Kalman inversion for static Bayesian models with unknown measurement error covariance

The ensemble Kalman filter (EnKF) is a Monte Carlo approximation of the Kalman filter for high dimensional linear Gaussian state space models. EnKF methods have also been developed for parameter inference of static Bayesian models with a Gaussian likelihood, in a way that is analogous to likelihood tempering sequential Monte Carlo (SMC). These methods are commonly referred to as ensemble Kalman inversion (EKI). Unlike SMC, the inference from EKI is only asymptotically unbiased if the likelihood is linear Gaussian and the priors are Gaussian. However, EKI is significantly faster to run. Currently, a large limitation of EKI methods is that the covariance of the measurement error is assumed to be fully known. We develop a new method, which we call component-wise iterative ensemble Kalman inversion (CW-IEKI), that allows elements of the covariance matrix to be inferred alongside the model parameters at negligible extra cost. This novel method is compared to SMC on three different application examples: a model of nitrogen mineralisation in soil that is based on the Agricultural Production Systems Simulator (APSIM), a model predicting seagrass decline due to stress from water temperature and light, and a model predicting coral calcification rates. On all of these examples, we find that CW-IEKI has relatively similar predictive performance to SMC, albeit with greater uncertainty, and it has a significantly faster run time.

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arXiv2022

Optimal Bayesian design for model discrimination via classification

Performing optimal Bayesian design for discriminating between competing models is computationally intensive as it involves estimating posterior model probabilities for thousands of simulated datasets. This issue is compounded further when the likelihood functions for the rival models are computationally expensive. A new approach using supervised classification methods is developed to perform Bayesian optimal model discrimination design. This approach requires considerably fewer simulations from the candidate models than previous approaches using approximate Bayesian computation. Further, it is easy to assess the performance of the optimal design through the misclassification error rate. The approach is particularly useful in the presence of models with intractable likelihoods but can also provide computational advantages when the likelihoods are manageable.

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arXiv2022

A Comparison of Likelihood-Free Methods With and Without Summary Statistics

Likelihood-free methods are useful for parameter estimation of complex models with intractable likelihood functions for which it is easy to simulate data. Such models are prevalent in many disciplines including genetics, biology, ecology and cosmology. Likelihood-free methods avoid explicit likelihood evaluation by finding parameter values of the model that generate data close to the observed data. The general consensus has been that it is most efficient to compare datasets on the basis of a low dimensional informative summary statistic, incurring information loss in favour of reduced dimensionality. More recently, researchers have explored various approaches for efficiently comparing empirical distributions in the likelihood-free context in an effort to avoid data summarisation. This article provides a review of these full data distance based approaches, and conducts the first comprehensive comparison of such methods, both qualitatively and empirically. We also conduct a substantive empirical comparison with summary statistic based likelihood-free methods. The discussion and results offer guidance to practitioners considering a likelihood-free approach. Whilst we find the best approach to be problem dependent, we also find that the full data distance based approaches are promising and warrant further development. We discuss some opportunities for future research in this space. Computer code to implement the methods discussed in this paper can be found at https://github.com/cdrovandi/ABC-dist-compare.

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arXiv2022

Modularized Bayesian analyses and cutting feedback in likelihood-free inference

There has been much recent interest in modifying Bayesian inference for misspecified models so that it is useful for specific purposes. One popular modified Bayesian inference method is "cutting feedback" which can be used when the model consists of a number of coupled modules, with only some of the modules being misspecified. Cutting feedback methods represent the full posterior distribution in terms of conditional and sequential components, and then modify some terms in such a representation based on the modular structure for specification or computation of a modified posterior distribution. The main goal of this is to avoid contamination of inferences for parameters of interest by misspecified modules. Computation for cut posterior distributions is challenging, and here we consider cutting feedback for likelihood-free inference based on Gaussian mixture approximations to the joint distribution of parameters and data summary statistics. We exploit the fact that marginal and conditional distributions of a Gaussian mixture are Gaussian mixtures to give explicit approximations to marginal or conditional posterior distributions so that we can easily approximate cut posterior analyses. The mixture approach allows repeated approximation of posterior distributions for different data based on a single mixture fit, which is important for model checks which aid in the decision of whether to "cut". A semi-modular approach to likelihood-free inference where feedback is partially cut is also developed. The benefits of the method are illustrated in two challenging examples, a collective cell spreading model and a continuous time model for asset returns with jumps.

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arXiv2022

Population Calibration using Likelihood-Free Bayesian Inference

In this paper we develop a likelihood-free approach for population calibration, which involves finding distributions of model parameters when fed through the model produces a set of outputs that matches available population data. Unlike most other approaches to population calibration, our method produces uncertainty quantification on the estimated distribution. Furthermore, the method can be applied to any population calibration problem, regardless of whether the model of interest is deterministic or stochastic, or whether the population data is observed with or without measurement error. We demonstrate the method on several examples, including one with real data. We also discuss the computational limitations of the approach. Immediate applications for the methodology developed here exist in many areas of medical research including cancer, COVID-19, drug development and cardiology.

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arXiv2022

The effect of biologically mediated decay rates on modelling soil carbon sequestration in agricultural settings

Microbial biomass carbon (MBC), a crucial soil labile carbon fraction, is the most active component of the soil organic carbon (SOC) that regulates bio-geochemical processes in terrestrial ecosystems. Some studies in the literature ignore the effect of microbial population growth on carbon decomposition rates. In reality, we might expect that the decomposition rate should be related to the population of microbes in the soil and have a positive relationship with the size of the microbial biomass pool. In this study, we explore the effect of microbial population growth on the accuracy of modelling soil carbon sequestration by developing and comparing two soil carbon models that consider a carrying capacity and limit to the growth of the microbial pool. We apply our models to three datasets, two small and one large datasets, and we select the best model in terms of having the best predictive performance through two model selection methods. Through this analysis we reveal that commonly used complex soil carbon models can over-fit in the presence of both small and large time-series datasets, and our simpler model can produce more accurate predictions. We conclude that considering the microbial population growth in a soil carbon model improves the accuracy of a model in the presence of a large dataset.

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arXiv2021

Bayesian Detectability of Induced Polarisation in Airborne Electromagnetic Data using Reversible Jump Sequential Monte Carlo

Detection of induced polarisation (IP) effects in airborne electromagnetic (AEM) measurements does not yet have an established methodology. This contribution develops a Bayesian approach to the IP-detectability problem using decoupled transdimensional layered models, and applies an approach novel to geophysics whereby transdimensional proposals are used within the embarrassingly parallelisable and robust static Sequential Monte Carlo (SMC) class of algorithms for the simultaneous inference of parameters and models. Henceforth referring to this algorithm as Reversible Jump Sequential Monte Carlo (RJSMC), the statistical methodological contributions to the algorithm account for adaptivity considerations for multiple models and proposal types, especially surrounding particle impoverishment in unlikely models. Methodological contributions to solid Earth geophysics include the decoupled model approach and proposal of a statistic that use posterior model odds for IP detectability. A case study is included investigating detectability of IP effects in AEM data at a broad scale.

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arXiv2021

Accelerating sequential Monte Carlo with surrogate likelihoods

Delayed-acceptance is a technique for reducing computational effort for Bayesian models with expensive likelihoods. Using a delayed-acceptance kernel for Markov chain Monte Carlo can reduce the number of expensive likelihoods evaluations required to approximate a posterior expectation. Delayed-acceptance uses a surrogate, or approximate, likelihood to avoid evaluation of the expensive likelihood when possible. Within the sequential Monte Carlo framework, we utilise the history of the sampler to adaptively tune the surrogate likelihood to yield better approximations of the expensive likelihood, and use a surrogate first annealing schedule to further increase computational efficiency. Moreover, we propose a framework for optimising computation time whilst avoiding particle degeneracy, which encapsulates existing strategies in the literature. Overall, we develop a novel algorithm for computationally efficient SMC with expensive likelihood functions. The method is applied to static Bayesian models, which we demonstrate on toy and real examples, code for which is available at https://github.com/bonStats/smcdar.

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arXiv2021

Combined parameter and state inference with automatically calibrated ABC

State space models contain time-indexed parameters, termed states, as well as static parameters, simply termed parameters. The problem of inferring both static parameters as well as states simultaneously, based on time-indexed observations, is the subject of much recent literature. This problem is compounded once we consider models with intractable likelihoods. In these situations, some emerging approaches have incorporated existing likelihood-free techniques for static parameters, such as approximate Bayesian computation (ABC) into likelihood-based algorithms for combined inference of parameters and states. These emerging approaches currently require extensive manual calibration of a time-indexed tuning parameter: the acceptance threshold. We design an SMC$^2$ algorithm (Chopin et al., 2013, JRSS B) for likelihood-free approximation with automatically tuned thresholds. We prove consistency of the algorithm and discuss the proposed calibration. We demonstrate this algorithm's performance with three examples. We begin with two examples of state space models. The first example is a toy example, with an emission distribution that is a skew normal distribution. The second example is a stochastic volatility model involving an intractable stable distribution. The last example is the most challenging; it deals with an inhomogeneous Hawkes process.

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arXiv2021

Bayesian inference using synthetic likelihood: asymptotics and adjustments

Implementing Bayesian inference is often computationally challenging in applications involving complex models, and sometimes calculating the likelihood itself is difficult. Synthetic likelihood is one approach for carrying out inference when the likelihood is intractable, but it is straightforward to simulate from the model. The method constructs an approximate likelihood by taking a vector summary statistic as being multivariate normal, with the unknown mean and covariance matrix estimated by simulation for any given parameter value. Our article makes three contributions. The first shows that if the summary statistic satisfies a central limit theorem, then the synthetic likelihood posterior is asymptotically normal and yields credible sets with the correct level of frequentist coverage. This result is similar to that obtained by approximate Bayesian computation. The second contribution compares the computational efficiency of Bayesian synthetic likelihood and approximate Bayesian computation using the acceptance probability for rejection and importance sampling algorithms with a "good" proposal distribution. We show that Bayesian synthetic likelihood is computationally more efficient than approximate Bayesian computation, and behaves similarly to regression-adjusted approximate Bayesian computation. Based on the asymptotic results, the third contribution proposes using adjusted inference methods when a possibly misspecified form is assumed for the covariance matrix of the synthetic likelihood, such as diagonal or a factor model, to speed up the computation. The methodology is illustrated with some simulated and real examples.

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