Bayesian encompassing specification test under not completely known partial observability

This paper proposes the construction of a Bayesian specification test based on the encompassing principle for the case of partial observability of latent variables. A structural parametric model (null model) is compared against a non-parametric alternative (alternative model) at the level of latent variables. The null extended model is obtained by incorporating the non Euclidean parameter of the alternative model. This extension is defined through a Bayesian Pseudo-True Value, that makes the null model a reduction by sufficiency of the extended model. The same observability process is introduced in both the null and the alternative models; after integrating out the latent variables, a null and alternative statistical models are accordingly obtained. The comparison is made between the posterior measures of the non Euclidean parameter (of the alternative model) in the extended and in the alternative statistical models. The general development is illustrated with an example where only a linear combination of a latent vector is observed; in the example, the partial observability is known upto the vector defining the observed linear combination. Some identifiability issues are treated and the examples hows the operationality and some pitfalls of the proposed test, through a numerical experiment.