  We consider the challenging problem of estimating causal effects from purely
observational data in the bi-directional Mendelian randomization (MR), where
some invalid instruments, as well as unmeasured confounding, usually exist. To
address this problem, most existing methods attempt to find proper valid
instrumental variables (IVs) for the target causal effect by expert knowledge
or by assuming that the causal model is a one-directional MR model. As such, in
this paper, we first theoretically investigate the identification of the
bi-directional MR from observational data. In particular, we provide necessary
and sufficient conditions under which valid IV sets are correctly identified
such that the bi-directional MR model is identifiable, including the causal
directions of a pair of phenotypes (i.e., the treatment and outcome). Moreover,
based on the identification theory, we develop a cluster fusion-like method to
discover valid IV sets and estimate the causal effects of interest. We
theoretically demonstrate the correctness of the proposed algorithm.
Experimental results show the effectiveness of our method for estimating causal
effects in bi-directional MR.
