An approximate likelihood perspective on ABC methods

Abstract

We are living in the big data era, as current technologies and networks allow for the easy and routine collection of data sets in different disciplines. Bayesian Statistics offers a flexible modeling approach which is attractive for describing the complexity of these datasets. These models often exhibit a likelihood function which is intractable due to the large sample size, high number of parameters, or functional complexity. Approximate Bayesian Computational (ABC) methods provides likelihood-free methods for performing statistical inferences with Bayesian models defined by intractable likelihood functions. The vastity of the literature on ABC methods created a need to review and relate all ABC approaches so that scientists can more readily understand and apply them for their own work. This article provides a unifying review, general representation, and classification of all ABC methods from the view of approximate likelihood theory. This clarifies how ABC methods can be characterized, related, combined, improved, and applied for future research. Possible future research in ABC is then suggested.MSC 2010 subject classifications: Primary 60-08, 62F15; secondary 62G05.

Approximate Bayesian Computation Via the Energy Statistic

“…Next, we let w (d, ) be a non-negative, decreasing (in d), and bounded (importance sampling) weight function (cf. Section of [14]), which takes as inputs a data discrepancy measurement d = D (x n , y m ) ≥ 0 and a calibration parameter > 0. Using the weight and discrepancy functions, we can propose the following approximation for (2).…”

“…In this article, we develop upon the discrepancy measurement approach of [17], via the importance sampling ABC (IS-ABC) approach, which makes use of a weight function; see e.g. [14]. In particular, we report on a class of ABC algorithms that utilize the two-sample energy statistic (ES) of [21] (see also [22; 23; 24; 25]).…”

Modelling Preference Data with the Wallenius Distribution

“…The ABC methodology can be considered as a (class of) popular algorithms that achieves posterior simulation by avoiding the computation of the likelihood function: see Beaumont (2010), Marin et al (2012) and Karabatsos and Leisen (2018) for recent surveys. As remarked by Marin et al (2012), the first genuine ABC algorithm was introduced by Pritchard et al (1999) in a population genetics setting.…”

Um estudo comparativo do uso de métodos de calibração bayesiana aproximada em modelos estocásticos

“…A resolução numérica pode ser feita facilmente usando métodos de amostragem do tipo Monte Carlo. Além de serem computacionalmente intensivos, estes métodos requerem o conhecimento explícito da função de verossimilhança (p(y o |θ)), o que nem sempreé possível, seja por não se conhecer de fato a relação entre y o e θ ou por ser intratável na prática por relacionar número grande de amostras e/ou parâmetros ou por complexidade funcional [4]. Nestas situações, a computação bayesiana aproximada (ABC -Approximate Bayesian Computation) tem se tornado uma alternativa atrativa por ser um método "livre de função de verossimilhança".…”

“…As quantidades de simulações executadas e os valores mais prováveis estão indicados em parênteses e abaixo dos gráficos, respectivamente. 2176 × 10 −4 5, 2056 × 10 −4 7, 02 × 10 −4…”