Performing exact posterior inference in complex generative models is often difficult or impossible due to an expensive to evaluate or intractable likelihood function. Approximate Bayesian computation (ABC) is an inference framework that constructs an approximation to the true likelihood based on the similarity between the observed and simulated data as measured by a predefined set of summary statistics. Although the choice of appropriate problem-specific summary statistics crucially influences the quality of the likelihood approximation and hence also the quality of the posterior sample in ABC, there are only few principled general-purpose approaches to the selection or construction of such summary statistics. In this paper, we develop a novel framework for this task using kernel-based distribution regression. We model the functional relationship between data distributions and the optimal choice (with respect to a loss function) of summary statistics using kernel-based distribution regression. We show that our approach can be implemented in a computationally and statistically efficient way using the random Fourier features framework for large-scale kernel learning. In addition to that, our framework shows superior performance when compared to related methods on toy and real-world problems.
A third, and last, category is comprised of methods that construct summary statistics using regression from either the full dataset or a set of candidate statistics, e.g. . provides a general overview of such approaches, while we discuss the aforementioned method in more detail. The semi-automatic ABC (SA-ABC) method @cite_24 focuses on deriving summary statistics that will allow inference about certain parameters of interest to be as accurate as possible. focus on the construction of summary statistics that allow inference to be accurate with respect to a specific loss function. They show that the true posterior mean of the model parameters is the optimal choice of summary statistics under the quadratic loss function. As this quantity cannot be analytically calculated, they estimate it by fitting a regression model from simulated data. In particular, given simulated data @math , a linear model @math is fitted; here, @math is taken to be either the identity function or a power function. The resulting estimates @math are used as the summary statistics in ABC.
@cite_24
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The choice of summary statistics is a crucial step in approximate Bayesian computation (ABC). Since statistics are often not sufficient, this choice involves a trade-off between loss of information and reduction of dimensionality. The latter may increase the efficiency of ABC. Here, we propose an approach for choosing summary statistics based on boosting, a technique from the machine-learning literature. We consider different types of boosting and compare them to partial least-squares regression as an alternative. To mitigate the lack of sufficiency, we also propose an approach for choosing summary statistics locally, in the putative neighborhood of the true parameter value. We study a demographic model motivated by the reintroduction of Alpine ibex (Capra ibex) into the Swiss Alps. The parameters of interest are the mean and standard deviation across microsatellites of the scaled ancestral mutation rate (θanc = 4Neu) and the proportion of males obtaining access to matings per breeding season (ω). By simulation, we assess the properties of the posterior distribution obtained with the various methods. According to our criteria, ABC with summary statistics chosen locally via boosting with the L2-loss performs best. Applying that method to the ibex data, we estimate θ^anc≈1.288 and find that most of the variation across loci of the ancestral mutation rate u is between 7.7 × 10−4 and 3.5 × 10−3 per locus per generation. The proportion of males with access to matings is estimated as ω^≈0.21, which is in good agreement with recent independent estimates.