  We study whether and how can we model a joint distribution $p(x,z)$ using two
conditional models $p(x|z)$ and $q(z|x)$ that form a cycle. This is motivated
by the observation that deep generative models, in addition to a likelihood
model $p(x|z)$, often also use an inference model $q(z|x)$ for extracting
representation, but they rely on a usually uninformative prior distribution
$p(z)$ to define a joint distribution, which may render problems like posterior
collapse and manifold mismatch. To explore the possibility to model a joint
distribution using only $p(x|z)$ and $q(z|x)$, we study their compatibility and
determinacy, corresponding to the existence and uniqueness of a joint
distribution whose conditional distributions coincide with them. We develop a
general theory for operable equivalence criteria for compatibility, and
sufficient conditions for determinacy. Based on the theory, we propose a novel
generative modeling framework CyGen that only uses the two cyclic conditional
models. We develop methods to achieve compatibility and determinacy, and to use
the conditional models to fit and generate data. With the prior constraint
removed, CyGen better fits data and captures more representative features,
supported by both synthetic and real-world experiments.
