  In this paper we consider a class of {\it conditional McKean-Vlasov SDEs}
(CMVSDE for short). Such an SDE can be considered as an extended version of
McKean-Vlasov SDEs with common noises, as well as the general version of the
so-called {\it conditional mean-field SDEs} (CMFSDE) studied previously by the
authors [1, 14], but with some fundamental differences. In particular, due to
the lack of compactness of the iterated conditional laws, the existing
arguments of Schauder's fixed point theorem do not seem to apply in this
situation, and the heavy nonlinearity on the conditional laws caused by change
of probability measure adds more technical subtleties. Under some structure
assumptions on the coefficients of the observation equation, we prove the
well-posedness of solution in the weak sense along a more direct approach. Our
result is the first that deals with McKean-Vlasov type SDEs involving
state-dependent conditional laws.
