/multiABC

Use individual ABC analyses for multiple data set, and then combine these inferences.

Approximate Bayesian Computation (ABC) for multiple data sets

Question

How to calibrate a model to multiple data sets?

This is related to meta-analysis where we may have more than one study informing model parameters. A possible way of approaching this problem could be split in to three

  • Combine the data sets before doing a single calibration
  • Perform separate, multiple calibrations for each data set and then combine the set of model parameters afterwards
  • Use each data set one at a time in a sequential fashion

This last approach is what we will pursue in this work. One sequential approach is to use the ABC posterior after analysing the $i$ th data set as a prior for analysing the $(i+1)$ th data set.

This is an idea taken from

Fearnhead, P. and Prangle, D. (2012) ‘Constructing summary statistics for approximate Bayesian computation: Semi-automatic approximate Bayesian computation’, Journal of the Royal Statistical Society. Series B: Statistical Methodology, 74(3), pp. 419–474. doi: 10.1111/j.1467-9868.2011.01010.x.

Description

Calculate a set of approximate likelihoods, one for each data set.

$$ p(\theta \mid s_1) = \int \pi(y \mid \theta) K[(S(y) - s_1)/h] \mbox{d}y $$

$$ p(\theta \mid s_2) = \int \pi(y \mid \theta) K[(S(y) - s_2)/h] \mbox{d}y $$

$$ \vdots $$

Then sequentially update each posterior

$$ \pi_{ABC}(\theta \mid s_1) \propto \pi(\theta) p(\theta \mid s_1) $$

$$ \pi_{ABC}(\theta \mid s_1, s_2) \propto \pi_{ABC}(\theta \mid s_1) p(\theta \mid s_2) $$

$$ \vdots $$