Robust approximate Bayesian inference

We discuss an approach for deriving robust posterior distributions from $M$-estimating functions using Approximate Bayesian Computation (ABC) methods. In particular, we use $M$-estimating functions to construct suitable summary statistics in ABC algorithms. The theoretical properties of the robust posterior distributions are discussed. Special attention is given to the application of the method to linear mixed models. Simulation results and an application to a clinical study demonstrate the usefulness of the method. An R implementation is also provided in the robustBLME package.Comment: This is a revised and personal manuscript version of the article that has been accepted for publication by Journal of Statistical Planning and Inferenc

A note on marginal posterior simulation via higher-order tail area approximations

We explore the use of higher-order tail area approximations for Bayesian simulation. These approximations give rise to an alternative simulation scheme to MCMC for Bayesian computation of marginal posterior distributions for a scalar parameter of interest, in the presence of nuisance parameters. Its advantage over MCMC methods is that samples are drawn independently with lower computational time and the implementation requires only standard maximum likelihood routines. The method is illustrated by a genetic linkage model, a normal regression with censored data and a logistic regression model

Approximate Bayesian Computation with composite score functions

Both Approximate Bayesian Computation (ABC) and composite likelihood methods are useful for Bayesian and frequentist inference, respectively, when the likelihood function is intractable. We propose to use composite likelihood score functions as summary statistics in ABC in order to obtain accurate approximations to the posterior distribution. This is motivated by the use of the score function of the full likelihood, and extended to general unbiased estimating functions in complex models. Moreover, we show that if the composite score is suitably standardised, the resulting ABC procedure is invariant to reparameterisations and automatically adjusts the curvature of the composite likelihood, and of the corresponding posterior distribution. The method is illustrated through examples with simulated data, and an application to modelling of spatial extreme rainfall data is discussed.Comment: Statistics and Computing (final version

Recent Advances in Approximate Bayesian Computation Methods

The Bayesian approach to statistical inference in fundamentally probabilistic. Exploiting the internal consistency of the probability framework, the posterior distribution extracts the relevant information in the data, and provides a complete and coherent summary of post data uncertainty. However, summarising the posterior distribution often requires the calculation of awkward multidimensional integrals. A further complication with the Bayesian approach arises when the likelihood functions is unavailable. In this respect, promising advances have been made by theory of Approximate Bayesian Computations (ABC). This thesis focuses on computational methods for the approximation of posterior distributions, and it discusses six original contributions. The first contribution concerns the approximation of marginal posterior distributions for scalar parameters. By combining higher-order tail area approximation with the inverse transform sampling, we define the HOTA algorithm which draws independent random sample from the approximate marginal posterior. The second discusses the HOTA algorithm with pseudo-posterior distributions, \eg, posterior distributions obtained by the combination of a pseudo-likelihood with a prior within Bayes' rule. The third contribution extends the use of tail-area approximations to contexts with multidimensional parameters, and proposes a method which gives approximate Bayesian credible regions with good sampling coverage properties. The forth presents an improved Laplace approximation which can be used for computing marginal likelihoods. The fifth contribution discusses a model-based procedure for choosing good summary statistics for ABC, by using composite score functions. Lastly, the sixth contribution discusses the choice of a default proposal distribution for ABC that is based on the notion of quasi-likelihood

Approximate Bayesian Computation by Modelling Summary Statistics in a Quasi-likelihood Framework

Approximate Bayesian Computation (ABC) is a useful class of methods for Bayesian inference when the likelihood function is computationally intractable. In practice, the basic ABC algorithm may be inefficient in the presence of discrepancy between prior and posterior. Therefore, more elaborate methods, such as ABC with the Markov chain Monte Carlo algorithm (ABC-MCMC), should be used. However, the elaboration of a proposal density for MCMC is a sensitive issue and very difficult in the ABC setting, where the likelihood is intractable. We discuss an automatic proposal distribution useful for ABC-MCMC algorithms. This proposal is inspired by the theory of quasi-likelihood (QL) functions and is obtained by modelling the distribution of the summary statistics as a function of the parameters. Essentially, given a real-valued vector of summary statistics, we reparametrize the model by means of a regression function of the statistics on parameters, obtained by sampling from the original model in a pilot-run simulation study. The QL theory is well established for a scalar parameter, and it is shown that when the conditional variance of the summary statistic is assumed constant, the QL has a closed-form normal density. This idea of constructing proposal distributions is extended to non constant variance and to real-valued parameter vectors. The method is illustrated by several examples and by an application to a real problem in population genetics.Comment: Published at http://dx.doi.org/10.1214/14-BA921 in the Bayesian Analysis (http://projecteuclid.org/euclid.ba) by the International Society of Bayesian Analysis (http://bayesian.org/

Improved Laplace approximation for marginal likelihoods

Statistical applications often involve the calculation of intractable multidimensional integrals. The Laplace formula is widely used to approximate such integrals. However, in high-dimensional or small sample size problems, the shape of the integrand function may be far from that of the Gaussian density, and thus the standard Laplace approximation can be inaccurate. We propose an improved Laplace approximation that reduces the asymptotic error of the standard Laplace formula by one order of magnitude, thus leading to third-order accuracy. We also show, by means of practical examples of various complexity, that the proposed method is extremely accurate, even in high dimensions, improving over the standard Laplace formula. Such examples also demonstrate that the accuracy of the proposed method is comparable with that of other existing methods, which are computationally more demanding. An R implementation of the improved Laplace approximation is also provided through the R package iLaplace available on CRAN.Comment: 24 page

A note on approximate Bayesian credible sets based on modified loglikelihood ratios

Asymptotic arguments are widely used in Bayesian inference, and in recent years there has been considerable developments of the so-called higher-order asymptotics. This theory provides very accurate approximations to posterior distributions, and to related quantities, in a variety of parametric statistical problems, even for small sample sizes. The aim of this contribution is to discuss recent advances in approximate Bayesian computations based on the asymptotic theory of modified loglikelihood ratios, both from theoretical and practical point of views. Results on third-order approximations for univariate posterior distributions, also in the presence of nuisance parameters, are reviewed and a new formula for a vector parameter of interest is presented. All these approximations may routinely be applied in practice for Bayesian inference, since they require little more than standard likelihood quantities for their implementation, and hence they may be available at little additional computational cost over simple first-order approximations. Moreover, these approximations give rise to a simple simulation scheme, alternative to MCMC, for Bayesian computation of marginal posterior distributions for a scalar parameter of interest. In addition, they can be used for testing precise null hypothesis and to define accurate Bayesian credible sets. Some illustrative examples are discussed, with particular attention to the use of matching priors

SERRS Multiplexing with Multivalent Nanostructures for the Identification and Enumeration of Epithelial and Mesenchymal Cells

Liquid biopsy represents a new frontier of cancer diagnosis and prognosis, which allows the isolation of tumor cells released in the blood stream. The extremely low abundance of these cells needs appropriate methodologies for their identification and enumeration. Herein we present a new protocol based on surface enhanced resonance Raman scattering (SERRS) gold multivalent nanostructures to identify and enumerate tumor cells with epithelial and mesenchimal markers. The validation of the protocol is obtained with spiked samples of peripheral blood mononuclear cells (PBMC). Gold nanostructures are functionalized with SERRS labels and with antibodies to link the tumor cells. Three types of such nanosystems were simultaneously used and the protocol allows obtaining the identification of all individual tumor cells with the help of a Random Forest ensemble learning method