Specification of distributions by combinations of marginal and conditional distributions

A k-dimensional density function is determined by certain combinations of marginal and conditional densities. The present paper identifies all possible such specifications. Gelman and Speed (1993) have treated the finite discrete case of this problem. The present paper extends their work to a more general setting. Appropriate compatibility conditions are described as are sufficient conditions for uniqueness of the resulting density which must, in general, be checked for integrability. An example of the application of the conditions is included.