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Added by
Ritabrata Dutta
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
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ABCpy is a highly modular scientific library for Approximate Bayesian Computation (ABC) written in Python. The main contribution of this paper is to document a software engineering effort that enables domain scientists to easily apply ABC to their research without being ABC experts; using ABCpy they can easily run large parallel simulations without much knowledge about parallelization. Further, ABCpy enables ABC experts to easily develop new inference schemes and evaluate them in a standardized environment and to extend the library with new algorithms. These benefits come mainly from the modularity of ABCpy. We give an overview of the design of ABCpy and provide a performance evaluation concentrating on parallelization. This points us towards the inherent imbalance in some of the ABC algorithms. We develop a dynamic scheduling MPI implementation to mitigate this issue and evaluate the various ABC algorithms according to their adaptability towards high-performance computing.
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The Bayesian statistical paradigm uses the language of probability to express uncertainty about the phenomena that generate observed data. Probability distributions thus characterize Bayesian analysis, with the rules of probability used to transform prior probability distributions for all unknowns - parameters, latent variables, models - into posterior distributions, subsequent to the observation of data. Conducting Bayesian analysis requires the evaluation of integrals in which these probability distributions appear. Bayesian computation is all about evaluating such integrals in the typical case where no analytical solution exists. This paper takes the reader on a chronological tour of Bayesian computation over the past two and a half centuries. Beginning with the one-dimensional integral first confronted by Bayes in 1763, through to recent problems in which the unknowns number in the millions, we place all computational problems into a common framework, and describe all computational methods using a common notation. The aim is to help new researchers in particular - and more generally those interested in adopting a Bayesian approach to empirical work - make sense of the plethora of computational techniques that are now on offer; understand when and why different methods are useful; and see the links that do exist, between them all.
Approximate Bayesian computation (ABC) is computationally intensive for complex model simulators. To exploit expensive simulations, data-resampling via bootstrapping can be employed to obtain many artificial datasets at little cost. However, when using this approach within ABC, the posterior variance is inflated, thus resulting in biased posterior inference. Here we use stratified Monte Carlo to considerably reduce the bias induced by data resampling. We also show empirically that it is possible to obtain reliable inference using a larger than usual ABC threshold. Finally, we show that with stratified Monte Carlo we obtain a less variable ABC likelihood. Ultimately we show how our approach improves the computational efficiency of the ABC samplers. We construct several ABC samplers employing our methodology, such as rejection and importance ABC samplers, and ABC-MCMC samplers. We consider simulation studies for static (Gaussian, g-and-k distribution, Ising model, astronomical model) and dynamic models (Lotka-Volterra). We compare against state-of-art sequential Monte Carlo ABC samplers, synthetic likelihoods, and likelihood-free Bayesian optimization. For a computationally expensive Lotka-Volterra case study, we found that our strategy leads to a more than 10-fold computational saving, compared to a sampler that does not use our novel approach.
Approximate Bayesian computation methods are useful for generative models with intractable likelihoods. These methods are however sensitive to the dimension of the parameter space, requiring exponentially increasing resources as this dimension grows. To tackle this difficulty, we explore a Gibbs version of the ABC approach that runs component-wise approximate Bayesian computation steps aimed at the corresponding conditional posterior distributions, and based on summary statistics of reduced dimensions. While lacking the standard justifications for the Gibbs sampler, the resulting Markov chain is shown to converge in distribution under some partial independence conditions. The associated stationary distribution can further be shown to be close to the true posterior distribution and some hierarchic
The likelihood-free sequential Approximate Bayesian Computation (ABC) algorithms, are increasingly popular inference tools for complex biological models. Such algorithms proceed by constructing a succession of probability distributions over the parameter space conditional upon the simulated data lying in an $epsilon$--ball around the observed data, for decreasing values of the threshold $epsilon$. While in theory, the distributions (starting from a suitably defined prior) will converge towards the unknown posterior as $epsilon$ tends to zero, the exact sequence of thresholds can impact upon the computational efficiency and success of a particular application. In particular, we show here that the current preferred method of choosing thresholds as a pre-determined quantile of the distances between simulated and observed data from the previous population, can lead to the inferred posterior distribution being very different to the true posterior. Threshold selection thus remains an important challenge. Here we propose an automated and adaptive method that allows us to balance the need to minimise the threshold with computational efficiency. Moreover, our method which centres around predicting the threshold - acceptance rate curve using the unscented transform, enables us to avoid local minima - a problem that has plagued previous threshold schemes.
Many modern statistical applications involve inference for complicated stochastic models for which the likelihood function is difficult or even impossible to calculate, and hence conventional likelihood-based inferential echniques cannot be used. In such settings, Bayesian inference can be performed using Approximate Bayesian Computation (ABC). However, in spite of many recent developments to ABC methodology, in many applications the computational cost of ABC necessitates the choice of summary statistics and tolerances that can potentially severely bias the estimate of the posterior. We propose a new piecewise ABC approach suitable for discretely observed Markov models that involves writing the posterior density of the parameters as a product of factors, each a function of only a subset of the data, and then using ABC within each factor. The approach has the advantage of side-stepping the need to choose a summary statistic and it enables a stringent tolerance to be set, making the posterior less approximate. We investigate two methods for estimating the posterior density based on ABC samples for each of the factors: the first is to use a Gaussian approximation for each factor, and the second is to use a kernel density estimate. Both methods have their merits. The Gaussian approximation is simple, fast, and probably adequate for many applications. On the other hand, using instead a kernel density estimate has the benefit of consistently estimating the true ABC posterior as the number of ABC samples tends to infinity. We illustrate the piecewise ABC approach for three examples; in each case, the approach enables exact matching between simulations and data and offers fast and accurate inference.
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