Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate Bayesian computation

Many modern statistical applications involve inference for complex stochastic models, where it is easy to simulate from the models, but impossible to calculate likelihoods. Approximate Bayesian computation (ABC) is a method of inference for such models. It replaces calculation of the likelihood by a step which involves simulating artificial data for different parameter values, and comparing summary statistics of the simulated data with summary statistics of the observed data. Here we show how to construct appropriate summary statistics for ABC in a semi-automatic manner. We aim for summary statistics which will enable inference about certain parameters of interest to be as accurate as possible. Theoretical results show that optimal summary statistics are the posterior means of the parameters. Although these cannot be calculated analytically, we use an extra stage of simulation to estimate how the posterior means vary as a function of the data; and we then use these estimates of our summary statistics within ABC. Empirical results show that our approach is a robust method for choosing summary statistics that can result in substantially more accurate ABC analyses than the ad hoc choices of summary statistics that have been proposed in the literature. We also demonstrate advantages over two alternative methods of simulation-based inference.

Time to decide? Simplicity and congruity in comparative judgment

What is the relationship between magnitude judgments relying on directly available characteristics versus probabilistic cues? Question frame was manipulated in a comparative judgment task previously assumed to involve inference across a probabilistic mental model (e.g., “which city is largest” – the “larger” question – versus “which city is smallest” – the “smaller” question). Participants identified either the largest or smallest city (Experiments 1a, 2) or the richest or poorest person (Experiment 1b) in a three-alternative forced choice (3-AFC) task (Experiment 1) or 2-AFC task (Experiment 2). Response times revealed an interaction between question frame and the number of options recognized. When asked the smaller question, response times were shorter when none of the options were recognized. The opposite pattern was found when asked the larger question: response time was shorter when all options were recognized. These task-stimuli congruity results in judgment under uncertainty are consistent with, and predicted by, theories of magnitude comparison which make use of deductive inferences from declarative knowledge.

A Summary of the NATO ASI on Polar Cap Boundary Phenomena

The polar cap boundary is a subject of central importance to current magnetosphere-ionosphere research and its applications in “space weather” activities. The problems are that it has a number of definitions, and that the most physically meaningful definition (namely the open-closed field line boundary) is very difficult to identify in observations. New understanding of the importance of the structure and dynamics of the boundary region made the time right for a meeting reviewing our knowledge in this area. The Advanced Study Institute (ASI) on Svalbard in June 1997 discussed the boundary on both the dayside and the nightside, mapping magnetically to the dayside magnetopause and to tail plasma sheet/lobe interface, respectively. We held a “brainstorming” session, in which different ideas which arose from the presented papers were discussed and developed, and a summary session, in which session convenors gave a personal view of progress that has been made and problems which still need solving. Both were designed as ways of promoting further discussion. This paper attempts to distil some of the themes that emerged from these discussions.