Erlis Ruli

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.

Marginal posterior simulation via higher-order tail area approximations

In this paper we explore the use of higherorder tail area approximations for Bayesian simulation. These approximations give rise to alternative simulation schemes to MCMC for Bayesian computation of marginal posterior distributions for a scalar parameter of interest, in the presence of nuisance parameters. Their advantage over MCMC methods is that samples are drawn independently and much lower computational time is needed. The methods are illustrated by a genetic linkage model, a normal regression with censored data and a logistic regression model.

Grp94 in complexes with IgG is a soluble diagnostic marker of gastrointestinal tumors and displays immune-stimulating activity on peripheral blood immune cells

Glucose-regulated protein94 (Grp94), the most represented endoplasmic reticulum (ER)-resident heat shock protein (HSP), is a tumor antigen shared by different types of solid and hematological tumors. The tumor-specific feature of Grp94 is its translocation from the ER to the cell surface where it displays pro-oncogenic functions. This un-physiological location has important implications for both the tumor pathology and anti-tumor therapy. We wanted to address the question of whether Grp94 could be measured as liquid marker in cancer patients in order to make predictions of diagnostic and therapeutic relevance for the tumor. To this aim, we performed an in-depth investigation on patients with primary tumors of the gastrointestinal (GI) tract, using different methodological approaches to detect Grp94 in tumor tissues, plasma and peripheral blood mononuclear cells (PBMCs). Results indicate that Grp94 is not only the antigen highly expressed in any tumor tissue and in cells of tumor infiltrates, mostly B lymphocytes, but it is also found in the circulation. However, the only form in which Grp94 was detected in the plasma of any patients and in B lymphocytes induced to proliferate, was that of stable complexes with Immunoglobulin (Ig)G. Using a specific immune-enzyme assay to measure plasma Grp94-IgG complexes, we showed that Grp94-IgG complexes were significantly increased in cancer patients compared to healthy control subjects, serving as diagnostic tumor biomarker. Results also demonstrate that the stimulation of patient PBMCs with Grp94-IgG complexes led to an increased secretion of inflammatory cytokines that might drive a potentially beneficial anti-tumor effect.

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.

Higher-order Bayesian Approximations for Pseudo-posterior Distributions

The theory of higher-order asymptotics provides accurate approximations to posterior distributions for a scalar parameter of interest, and to the corresponding tail area, for practical use in Bayesian analysis. The aim of this article is to extend these approximations to pseudo-posterior distributions, e.g., posterior distributions based on a pseudo-likelihood function and a suitable prior, which are proved to be particularly useful when the full likelihood is analytically or computationally infeasible. In particular, from a theoretical point of view, we derive the Laplace approximation for a pseudo-posterior distribution, and for the corresponding tail area, for a scalar parameter of interest, also in the presence of nuisance parameters. From a computational point of view, starting from these higher-order approximations, we discuss the higher-order tail area (HOTA) algorithm useful to approximate marginal posterior distributions, and related quantities. Compared to standard Markov chain Monte Carlo methods, the main advantage of the HOTA algorithm is that it gives independent samples at a negligible computational cost. The relevant computations are illustrated by two examples.

Objective Bayesian inference with proper scoring rules

Standard Bayesian analyses can be difficult to perform when the full likelihood, and consequently the full posterior distribution, is too complex or even impossible to specify or if robustness with respect to data or to model misspecifications is required. In these situations, we suggest to resort to a posterior distribution for the parameter of interest based on proper scoring rules. Scoring rules are loss functions designed to measure the quality of a probability distribution for a random variable, given its observed value. Important examples are the Tsallis score and the Hyvärinen score, which allow us to deal with model misspecifications or with complex models. Also the full and the composite likelihoods are both special instances of scoring rules. The aim of this paper is twofold. Firstly, we discuss the use of scoring rules in the Bayes formula in order to compute a posterior distribution, named SR-posterior distribution, and we derive its asymptotic normality. Secondly, we propose a procedure for building default priors for the unknown parameter of interest that can be used to update the information provided by the scoring rule in the SR-posterior distribution. In particular, a reference prior is obtained by maximizing the average

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

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Optimal B-robust posterior distributions for operational risk

α-divergence from the SR-posterior distribution. For