Abstract
AbstractIn this talk, we consider different methods of parameter inference for ABC. Derivations of the asymptotic bias and variance of the standard ABC estimators indicates that ABC may achieve poor performance when the dimension of the summary statistics is large. The linear adjustment introduced by Beaumont et al. (2002) is found to achieve better performance when there is a nearly homoscedastic relationship between the summary statistics and the parameter of interest. To provide a more flexible adjustment method, we propose two innovations. The new method fits 1/a heteroscedatic rather than a homoscedastic regression model and consider 2/non linear instead of linear regression. The new algorithm is compared to the state-of-the-art approximate Bayesian methods and typically provides narrower credibility intervals.
Highlights
Main theoremAsymptotic bias of the estimates of the posterior gj (θ|sobs), j = 0 (rejection), (linear adj.), 2 (quadratic adj.)
C np(sobs )εd where d is the dimension of the summary statistics and n is the number of simulations
When the model θi = m(si ) + i is homoscedastic in the vicinity of sobs, bias≤ bias≤ bias
Summary
Asymptotic bias of the estimates of the posterior gj (θ|sobs), j = 0 (rejection), (linear adj.), 2 (quadratic adj.). C np(sobs )εd where d is the dimension of the summary statistics and n is the number of simulations. The rate at which the minimal MSEs converges to 0 decreases importantly (at least theoretically) as the dimension d of sobs increases. Possible solution Projecting the summary statistics on a lower dimensional subspace. When the model θi = m(si ) + i is homoscedastic in the vicinity of sobs, bias (quadratic adj.)≤ bias (linear adj.)≤ bias (without adj.). Makes the model more homoscedastic : transformations of sum stats and parameters (not pursued here, see Blum JASA 2010). Provides a more flexible regression model : non-linear and heteroscedastic regression
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