Multivariate Monotone Bayesian Updating

We consider a Bayesian decision problem in which an agent must draw inferences about multiple unobserved variables. This problem appears in different economic settings, for instance in the theory of agency. Assume that both observed and unobserved variables are ordered. We say that the inference process (Bayesian updating) is monotone if higher observations imply higher posterior probabilities. When the parameter space is totally ordered, monotone Bayesian updating is characterized by the monotone likelihood ratio property. We show here that this can be extended to the case in which the parameter space is partially ordered, by defining a generalized monotone likelihood ratio property. However, when the parameter space is multidimensional and endowed with its componentwise order, Bayesian updating is monotone only in trivial cases