A comparative review of dimension reduction methods in approximate Bayesian computation

Approximate Bayesian computation (ABC) methods make use of comparisons between simulated and observed summary statistics to overcome the problem of computationally intractable likelihood functions. As the practical implementation of ABC requires computations based on vectors of summary statistics, rather than full data sets, a central question is how to derive low-dimensional summary statistics from the observed data with minimal loss of information. In this article we provide a comprehensive review and comparison of the performance of the principal methods of dimension reduction proposed in the ABC literature. The methods are split into three nonmutually exclusive classes consisting of best subset selection methods, projection techniques and regularization. In addition, we introduce two new methods of dimension reduction. The first is a best subset selection method based on Akaike and Bayesian information criteria, and the second uses ridge regression as a regularization procedure. We illustrate the performance of these dimension reduction techniques through the analysis of three challenging models and data sets.

Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate Bayesian computation

Many modern statistical applications involve inference for complex stochastic models, where it is easy to simulate from the models, but impossible to calculate likelihoods. Approximate Bayesian computation (ABC) is a method of inference for such models. It replaces calculation of the likelihood by a step which involves simulating artificial data for different parameter values, and comparing summary statistics of the simulated data with summary statistics of the observed data. Here we show how to construct appropriate summary statistics for ABC in a semi-automatic manner. We aim for summary statistics which will enable inference about certain parameters of interest to be as accurate as possible. Theoretical results show that optimal summary statistics are the posterior means of the parameters. Although these cannot be calculated analytically, we use an extra stage of simulation to estimate how the posterior means vary as a function of the data; and we then use these estimates of our summary statistics within ABC. Empirical results show that our approach is a robust method for choosing summary statistics that can result in substantially more accurate ABC analyses than the ad hoc choices of summary statistics that have been proposed in the literature. We also demonstrate advantages over two alternative methods of simulation-based inference.

A generalised Bayesian instrumental variable approach under student t-distributed errors with application

Bayesian analysis is given of an instrumental variable model that allows for heteroscedasticity in both the structural equation and the instrument equation. Specifically, the approach for dealing with heteroscedastic errors in Geweke (1993) is extended to the Bayesian instrumental variable estimator outlined in Rossi et al. (2005). Heteroscedasticity is treated by modelling the variance for each error using a hierarchical prior that is Gamma distributed. The computation is carried out by using a Markov chain Monte Carlo sampling algorithm with an augmented draw for the heteroscedastic case. An example using real data illustrates the approach and shows that ignoring heteroscedasticity in the instrument equation when it exists may lead to biased estimates.

Forecasting with Bayesian multivariate vintage-based VARs

We consider the forecasting of macroeconomic variables that are subject to revisions, using Bayesian vintage-based vector autoregressions. The prior incorporates the belief that, after the first few data releases, subsequent ones are likely to consist of revisions that are largely unpredictable. The Bayesian approach allows the joint modelling of the data revisions of more than one variable, while keeping the concomitant increase in parameter estimation uncertainty manageable. Our model provides markedly more accurate forecasts of post-revision values of inflation than do other models in the literature.

Parametric preference functionals under risk in the gain domain: a Bayesian analysis

The performance of rank dependent preference functionals under risk is comprehensively evaluated using Bayesian model averaging. Model comparisons are made at three levels of heterogeneity plus three ways of linking deterministic and stochastic models: the differences in utilities, the differences in certainty equivalents and contextualutility. Overall, the"bestmodel", which is conditional on the form of heterogeneity is a form of Rank Dependent Utility or Prospect Theory that cap tures the majority of behaviour at both the representative agent and individual level. However, the curvature of the probability weighting function for many individuals is S-shaped, or ostensibly concave or convex rather than the inverse S-shape commonly employed. Also contextual utility is broadly supported across all levels of heterogeneity. Finally, the Priority Heuristic model, previously examined within a deterministic setting, is estimated within a stochastic framework, and allowing for endogenous thresholds does improve model performance although it does not compete well with the other specications considered.

Predicting how many animals will be where: how to build, calibrate and evaluate individual-based models

Individual-based models (IBMs) can simulate the actions of individual animals as they interact with one another and the landscape in which they live. When used in spatially-explicit landscapes IBMs can show how populations change over time in response to management actions. For instance, IBMs are being used to design strategies of conservation and of the exploitation of fisheries, and for assessing the effects on populations of major construction projects and of novel agricultural chemicals. In such real world contexts, it becomes especially important to build IBMs in a principled fashion, and to approach calibration and evaluation systematically. We argue that insights from physiological and behavioural ecology offer a recipe for building realistic models, and that Approximate Bayesian Computation (ABC) is a promising technique for the calibration and evaluation of IBMs. IBMs are constructed primarily from knowledge about individuals. In ecological applications the relevant knowledge is found in physiological and behavioural ecology, and we approach these from an evolutionary perspective by taking into account how physiological and behavioural processes contribute to life histories, and how those life histories evolve. Evolutionary life history theory shows that, other things being equal, organisms should grow to sexual maturity as fast as possible, and then reproduce as fast as possible, while minimising per capita death rate. Physiological and behavioural ecology are largely built on these principles together with the laws of conservation of matter and energy. To complete construction of an IBM information is also needed on the effects of competitors, conspecifics and food scarcity; the maximum rates of ingestion, growth and reproduction, and life-history parameters. Using this knowledge about physiological and behavioural processes provides a principled way to build IBMs, but model parameters vary between species and are often difficult to measure. A common solution is to manually compare model outputs with observations from real landscapes and so to obtain parameters which produce acceptable fits of model to data. However, this procedure can be convoluted and lead to over-calibrated and thus inflexible models. Many formal statistical techniques are unsuitable for use with IBMs, but we argue that ABC offers a potential way forward. It can be used to calibrate and compare complex stochastic models and to assess the uncertainty in their predictions. We describe methods used to implement ABC in an accessible way and illustrate them with examples and discussion of recent studies. Although much progress has been made, theoretical issues remain, and some of these are outlined and discussed.

Bayesian model comparison with un-normalised likelihoods

Models for which the likelihood function can be evaluated only up to a parameter-dependent unknown normalizing constant, such as Markov random field models, are used widely in computer science, statistical physics, spatial statistics, and network analysis. However, Bayesian analysis of these models using standard Monte Carlo methods is not possible due to the intractability of their likelihood functions. Several methods that permit exact, or close to exact, simulation from the posterior distribution have recently been developed. However, estimating the evidence and Bayes’ factors for these models remains challenging in general. This paper describes new random weight importance sampling and sequential Monte Carlo methods for estimating BFs that use simulation to circumvent the evaluation of the intractable likelihood, and compares them to existing methods. In some cases we observe an advantage in the use of biased weight estimates. An initial investigation into the theoretical and empirical properties of this class of methods is presented. Some support for the use of biased estimates is presented, but we advocate caution in the use of such estimates.

Bayesian analysis of uncertainty in the GlobCover 2009 land cover product at climate model grid scale

Land cover data derived from satellites are commonly used to prescribe inputs to models of the land surface. Since such data inevitably contains errors, quantifying how uncertainties in the data affect a model’s output is important. To do so, a spatial distribution of possible land cover values is required to propagate through the model’s simulation. However, at large scales, such as those required for climate models, such spatial modelling can be difficult. Also, computer models often require land cover proportions at sites larger than the original map scale as inputs, and it is the uncertainty in these proportions that this article discusses. This paper describes a Monte Carlo sampling scheme that generates realisations of land cover proportions from the posterior distribution as implied by a Bayesian analysis that combines spatial information in the land cover map and its associated confusion matrix. The technique is computationally simple and has been applied previously to the Land Cover Map 2000 for the region of England and Wales. This article demonstrates the ability of the technique to scale up to large (global) satellite derived land cover maps and reports its application to the GlobCover 2009 data product. The results show that, in general, the GlobCover data possesses only small biases, with the largest belonging to non–vegetated surfaces. In vegetated surfaces, the most prominent area of uncertainty is Southern Africa, which represents a complex heterogeneous landscape. It is also clear from this study that greater resources need to be devoted to the construction of comprehensive confusion matrices.

Future change of climate in South America in the late twenty-first century: intercomparison of scenarios from three regional climate models

Regional climate change projections for the last half of the twenty-first century have been produced for South America, as part of the CREAS (Cenarios REgionalizados de Clima Futuro da America do Sul) regional project. Three regional climate models RCMs (Eta CCS, RegCM3 and HadRM3P) were nested within the HadAM3P global model. The simulations cover a 30-year period representing present climate (1961-1990) and projections for the IPCC A2 high emission scenario for 2071-2100. The focus was on the changes in the mean circulation and surface variables, in particular, surface air temperature and precipitation. There is a consistent pattern of changes in circulation, rainfall and temperatures as depicted by the three models. The HadRM3P shows intensification and a more southward position of the subtropical Pacific high, while a pattern of intensification/weakening during summer/winter is projected by the Eta CCS/RegCM3. There is a tendency for a weakening of the subtropical westerly jet from the Eta CCS and HadRM3P, consistent with other studies. There are indications that regions such of Northeast Brazil and central-eastern and southern Amazonia may experience rainfall deficiency in the future, while the Northwest coast of Peru-Ecuador and northern Argentina may experience rainfall excesses in a warmer future, and these changes may vary with the seasons. The three models show warming in the A2 scenario stronger in the tropical region, especially in the 5A degrees N-15A degrees S band, both in summer and especially in winter, reaching up to 6-8A degrees C warmer than in the present. In southern South America, the warming in summer varies between 2 and 4A degrees C and in winter between 3 and 5A degrees C in the same region from the 3 models. These changes are consistent with changes in low level circulation from the models, and they are comparable with changes in rainfall and temperature extremes reported elsewhere. In summary, some aspects of projected future climate change are quite robust across this set of model runs for some regions, as the Northwest coast of Peru-Ecuador, northern Argentina, Eastern Amazonia and Northeast Brazil, whereas for other regions they are less robust as in Pantanal region of West Central and southeastern Brazil.

Using non-homogeneous Poisson models with multiple change-points to estimate the number of ozone exceedances in Mexico City

In this paper, we consider some non-homogeneous Poisson models to estimate the probability that an air quality standard is exceeded a given number of times in a time interval of interest. We assume that the number of exceedances occurs according to a non-homogeneous Poisson process (NHPP). This Poisson process has rate function lambda(t), t >= 0, which depends on some parameters that must be estimated. We take into account two cases of rate functions: the Weibull and the Goel-Okumoto. We consider models with and without change-points. When the presence of change-points is assumed, we may have the presence of either one, two or three change-points, depending of the data set. The parameters of the rate functions are estimated using a Gibbs sampling algorithm. Results are applied to ozone data provided by the Mexico City monitoring network. In a first instance, we assume that there are no change-points present. Depending on the adjustment of the model, we assume the presence of either one, two or three change-points. Copyright (C) 2009 John Wiley & Sons, Ltd.

Longitudinal Poisson modeling: an application for CD4 counting in HIV-infected patients

In this paper, we present different ofrailtyo models to analyze longitudinal data in the presence of covariates. These models incorporate the extra-Poisson variability and the possible correlation among the repeated counting data for each individual. Assuming a CD4 counting data set in HIV-infected patients, we develop a hierarchical Bayesian analysis considering the different proposed models and using Markov Chain Monte Carlo methods. We also discuss some Bayesian discrimination aspects for the choice of the best model.

Heteroscedastic Nonlinear Regression Models

In this article, we present a generalization of the Bayesian methodology introduced by Cepeda and Gamerman (2001) for modeling variance heterogeneity in normal regression models where we have orthogonality between mean and variance parameters to the general case considering both linear and highly nonlinear regression models. Under the Bayesian paradigm, we use MCMC methods to simulate samples for the joint posterior distribution. We illustrate this algorithm considering a simulated data set and also considering a real data set related to school attendance rate for children in Colombia. Finally, we present some extensions of the proposed MCMC algorithm.

A Bayesian Analysis in the Presence of Covariates for Multivariate Survival Data: An example of Application

In this paper, we introduce a Bayesian analysis for survival multivariate data in the presence of a covariate vector and censored observations. Different ""frailties"" or latent variables are considered to capture the correlation among the survival times for the same individual. We assume Weibull or generalized Gamma distributions considering right censored lifetime data. We develop the Bayesian analysis using Markov Chain Monte Carlo (MCMC) methods.

Use of Bayesian Methods for Multivariate Bioequivalence Measures

In this paper, we introduce a Bayesian analysis for bioequivalence data assuming multivariate pharmacokinetic measures. With the introduction of correlation parameters between the pharmacokinetic measures or between the random effects in the bioequivalence models, we observe a good improvement in the bioequivalence results. These results are of great practical interest since they can yield higher accuracy and reliability for the bioequivalence tests, usually assumed by regulatory offices. An example is introduced to illustrate the proposed methodology by comparing the usual univariate bioequivalence methods with multivariate bioequivalence. We also consider some usual existing discrimination Bayesian methods to choose the best model to be used in bioequivalence studies.

Using stochastic volatility models to analyse weekly ozone averages in Mexico City

In this paper we make use of some stochastic volatility models to analyse the behaviour of a weekly ozone average measurements series. The models considered here have been used previously in problems related to financial time series. Two models are considered and their parameters are estimated using a Bayesian approach based on Markov chain Monte Carlo (MCMC) methods. Both models are applied to the data provided by the monitoring network of the Metropolitan Area of Mexico City. The selection of the best model for that specific data set is performed using the Deviance Information Criterion and the Conditional Predictive Ordinate method.