Bayesian optimal designs for linear regression models

Bayesian optimal designs for estimation and prediction in linear regression models are considered. For a large class of optimality criteria $\Phi$, a general Kiefer\u27s type equivalence theorem is proved. This equivalence theorem is used to derive a Bayesian version of Elfving\u27s Theorem for the c-optimality criterion and the class of prior precision matrices R for which the Bayesian c-optimal designs are supported by the points of the classical c-optimal design is characterized. It is also proved that the Bayesian c-optimal design, for n large enough, is always supported at the same support points of the classical c-optimal design \xi\sp\* if \xi\sp\* is supported at exactly k distinct points and for a large class of prior precision matrices R if \xi\sp\* is supported at 1 $\leq$ m $\u3c$ k points. Duality theory is then used to give another proof of Elfving\u27s Theorem for Bayesian c-optimality and conditions under which a one point design is Bayesian c-optimum are given. Emphasis is laid on the geometry inherent in the Bayesian c-optimal design problem and the parallelism between classical and Bayesian c-optimal design theory is illustrated. A number of examples are given to illustrate how Elfving\u27s Theorem can be used to construct Bayesian c-optimal designs. The geometry - duality approach is extended for the $\Psi$-optimality criterion and a matrix analog of the geometric result of Elfving is derived and its applications are discussed

Statistical Model Checking QoS properties of Systems with SBIP ⋆

Abstract. BIP is a component-based framework supporting rigorous design of embedded systems. This paper presents SBIP, an extension of BIP that relies on a new stochastic semantics that enables verification of large-size systems by using Statistical Model Checking. The approach is illustrated on several industrial case studies.