  In this paper we continue the study of conditional Markov chains (CMCs) with
finite state spaces, that we initiated in Bielecki, Jakubowski and
Niew\k{e}g{\l}owski (2014a) in an effort to enrich the theory of CMCs that was
originated in Bielecki and Rutkowski (2004). We provide an alternative
definition of a CMC and an alternative construction of a CMC via a change of
probability measure. It turns out that our construction produces CMCs that are
also doubly stochastic Markov chains (DSMCs), which allows for study of several
properties of CMCs using tools available for DSMCs.
