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 α−divergence from the SR-posterior distribution. For 0 ≤ |α| < 1, the result is a Jeffreys-type prior that is proportional to the square root of the determinant of the Godambe information matrix associated to the scoring rule. Some examples are discussed.

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.

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.

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 (1); see, e.g., Robert and Casella (2004). All these techniques use simulation to avoid tailoring analytical work to specific models. However, these methods may present some difficulties, especially when d is large, and may have poor tail behaviour.Parallel with these developments has been the development of analytical higher-order approximations for parametric inference in small sample (see, e.g., Brazzale and Davison, 2008, and references therein). Using higher-order asymptotics it is possible to avoid the difficulties related to numerical methods and obtain accurate approximations of (1) and related quantities (see, e.g., Reid, 1996Reid, , 2003Sweeting, 1996, andBrazzale et al., 2007). These methods are highly accurate in many situations, but are nevertheless underused compared to simulation-based procedures (Brazzale and Davison, 2008).Starting from higher-order tail area approximations for (1) (see Diciccio and Martin, 1991;Reid, 1996and Brazzale et al., 2007, this paper describes the implementation and the use of a sampling scheme that give rise to very accurate computation of marginal posterior densities, and related quantities, such as posterior summaries. The implementation of the proposed higher-order tail area approximation (HOTA) sampling scheme is available at little additional computation cost over simple first-order approximations, and it has the advantage over MCMC methods that samples are drawn independently in much lower computation time. This technique applies to regular models only; precise regularity conditions for their validity are given in Kass et al. (1990).The paper is organized as follows. Section 2 briefly reviews higher-order approximations for the marginal posterior distribution (1), and for the corresponding tail area. Section 3 describes the proposed HOTA sampling scheme and its implementation. Numerical examples and applications are discussed in Section 4. Finally, some concluding remarks are given in Section 5.

Summary We discuss an approach of robust fitting on nonlinear regression models, both in a frequentist and a Bayesian approach, which can be employed to model and predict the contagion dynamics of COVID‐19 in Italy. The focus is on the analysis of epidemic data using robust dose‐response curves, but the functionality is applicable to arbitrary nonlinear regression models.