  Approximate Bayesian computation (ABC) has gained popularity in recent years
owing to its easy implementation, nice interpretation and good performance. Its
advantages are more visible when one encounters complex models where maximum
likelihood estimation as well as Bayesian analysis via Markov chain Monte Carlo
demand prohibitively large amount of time. This paper examines properties of
ABC both from a theoretical as well as from a computational point of view.We
consolidate the ABC theory by proving theorems related to its limiting
behaviour. In particular, we consider partial posteriors, which serve as the
first step towards approximating the full posteriors. Also, a new
semi-automatic algorithm of ABC is proposed using sufficient dimension
reduction (SDR) method. SDR has primarily surfaced in the frequentist
literature. But we have demonstrated in this paper that it has connections with
ABC as well.
