  A powerful technique for solving combinatorial optimization problems is to
reduce the search space without compromising the solution quality by exploring
intrinsic mathematical properties of the problems. For the maximum weight
independent set (MWIS) problem, using an upper bound lemma which says the
weight of any independent set not contained in the MWIS is bounded from above
by the weight of the intersection of its closed neighbor set and the MWIS, we
give two extension theorems -- independent set extension theorem and vertex
cover extension theorem. With them at our disposal, two types of causal
inference techniques (CITs) are proposed on the assumption that a vertex is
strongly reducible (included or not included in all MWISs) or reducible
(contained or not contained in a MWIS). One is a strongly reducible
state-preserving technique, which extends a strongly reducible vertex into a
vertex set where all vertices have the same strong reducibility. The other, as
a reducible state-preserving technique, extends a reducible vertex into a
vertex set with the same reducibility as that vertex and creates some weighted
packing constraints to narrow the search space. Numerical experiments show that
our CITs can help reduction algorithms find much smaller remaining graphs,
improve the ability of exact algorithms to find the optimal solutions and help
heuristic algorithms produce approximate solutions of better quality. In
particular, detailed tests on $12$ representative graphs generated from
datasets in Network Data Repository demonstrate that, compared to the
state-of-the-art algorithms, the size of remaining graphs is further reduced by
more than 32.6%, and the number of solvable instances is increased from 1 to 5.
