Marginal likelihood, also known as model evidence, is a fundamental quantity in Bayesian statistics. It is used for model selection using Bayes factors or for empirical Bayes tuning of prior hyper-parameters. Yet, the calculation of evidence has remained a longstanding open problem in Gaussian graphical models. Currently, the only feasible solutions that exist are for special cases such as the Wishart or G-Wishart, in moderate dimensions. We develop an approach based on a novel telescoping block decomposition of the precision matrix that allows the estimation of evidence by application of Chib's technique under a very broad class of priors under mild requirements. Specifically, the requirements are: (a) the priors on the diagonal terms on the precision matrix can be written as gamma or scale mixtures of gamma random variables and (b) those on the off-diagonal terms can be represented as normal or scale mixtures of normal. This includes structured priors such as the Wishart or G-Wishart, and more recently introduced element-wise priors, such as the Bayesian graphical lasso and the graphical horseshoe. Among these, the true marginal is known in an analytically closed form for Wishart, providing a useful validation of our approach. For the general setting of the other three, and several more priors satisfying conditions (a) and (b) above, the calculation of evidence has remained an open question that this article resolves under a unifying framework.

Formalising a measurement is often preceded by rich discussion of the ideas, and the development of general understanding of the meaning of concepts, often over decades. As a consequence, data required to fit into a representational measurement system is usually readily available and there is a general acceptance of the intention of the measure. This paper reports on a research project which has formalised the measurement of a relatively new body of knowledge, open systems. Open systems rely on standards to guarantee interoperability, portability, scalability and user portability. The Internet is the most successful of the open systems in existence, in terms of interoperability and scalability. The first of four issues was that since the project was a research contract, and because there were very few generally understood notions of measurement foundations (or relations) within open systems, the aims and the requirements of the measurement were formalised into a measurement requirements specification (MRS). A second issue concerns the building of a relational model. Building relations in a representational measurement model is relatively straightforward in the case where the measurement entity has been around. A third issue concerned the use of the measures when the measures were (not surprisingly) complex, and had to be combined by biased combiners, the values of which not all parties would naturally agree. A fourth issue raised in the paper is the extent of the validation of the measures which was required because of the contract.

The need to address system architecture within the computer systems development process is well accepted. Despite this, the process of architecting is still not well understood. In this paper we discuss the need for an architecting process which goes beyond specific architecting methods in order to address issues such as architecture evaluation and tailoring of architectures to specific system requirements. Based on these needs we propose an architecting process which selects from existing methods, bases and performance indices to produce a verifiable architecture. This process provides a framework within which we can develop an architecture which has the maximum (measurable) likelihood of satisfying both functional and nonfunctional system requirements upon implementation.