  This book chapter introduces regression approaches and regression adjustment
for Approximate Bayesian Computation (ABC). Regression adjustment adjusts
parameter values after rejection sampling in order to account for the imperfect
match between simulations and observations. Imperfect match between simulations
and observations can be more pronounced when there are many summary statistics,
a phenomenon coined as the curse of dimensionality. Because of this imperfect
match, credibility intervals obtained with regression approaches can be
inflated compared to true credibility intervals. The chapter presents the main
concepts underlying regression adjustment. A theorem that compares theoretical
properties of posterior distributions obtained with and without regression
adjustment is presented. Last, a practical application of regression adjustment
in population genetics shows that regression adjustment shrinks posterior
distributions compared to rejection approaches, which is a solution to avoid
inflated credibility intervals.
