ABSTRACT
This chapter introduces regression adjustment using a comprehensive framework that includes linear adjustment, as well as more flexible adjustments, such as non-linear models. It presents regression approaches for approximate Bayesian computation (ABC). The chapter explains the main concepts underlying regression adjustment. It also presents a theorem that compares theoretical properties of posterior distributions obtained with and without regression adjustment. The chapter discusses a practical application of regression adjustment in ABC. It shows that regression adjustment shrinks posterior distributions when compared to a standard rejection approach. It explains why regression adjustment shrinks posterior distribution, which is a desirable feature because credibility intervals obtained with rejection methods can be too wide. The chapter considers four different forms of regression adjustment: linear and homoscedastic adjustment, non-linear and homoscedastic adjustment, linear and heteroscedastic adjustment, and non-linear and heteroscedastic adjustment. It suggests that regression approaches for approximate Bayesian computation can be further improved to provide more reliable inference for complex models in biology and ecology.
