Uniformity of node level conflict measures in Bayesian hierarchical models based on directed acyclic graphs

Hierarchical models defined by means of directed, acyclic graphs are a power- ful and widely used tool for Bayesian analysis of problems of varying degrees of complexity. A simulation based method for model criticism in such models has been suggested by O'Hagan in the form of a con ict measure based on contrasting separate local information sources about each node in the graph. This measure is however not well calibrated. In order to rectify this, alter- native mutually similar tail probability based measures have been proposed independently, and have been proved to be uniformly distributed under the assumed model in quite general normal models with known covariance matri- ces. In the present paper, exploiting the property of pivotality, we extend this result to a variety of models. An advantage of this is that computationally costly pre-calibration schemes needed for some other suggested methods can be avoided. Another advantage is that non-informative prior distributions can be used when performing model criticism

Node-Level Conflict Measures in Bayesian Hierarchical Models Based on Directed Acyclic Graphs

Over the last decades, Bayesian hierarchical models defined by means of directed, acyclic graphs have become an essential and widely used methodology in the analysis of complex data. Simulation-based model criticism in such models can be based on conflict measures constructed by contrasting separate local information sources about each node in the graph. An initial suggestion of such a measure was not well calibrated. This shortcoming has, however, to a large extent been rectified by subsequently proposed alternative mutually similar tail probability-based measures, which have been proved to be uniformly distributed under the assumed model under various circumstances, and in particular, in quite general normal models with known covariance matrices. An advantage of this is that computationally costly precalibration schemes needed for some other suggested methods can be avoided. Another advantage is that noninformative prior distributions can be used when performing model criticism. In this chapter, we describe the basic framework and review the main uniformity results

Bounds for the availabilities for multistate monotone systems based on decomposition into stochastically independent modules

Multistate monotone systems are used to describe technological or biological systems when the system itself and its components can perform at different operationally meaningful levels. This generalizes the binary monotone systems used in standard reliability theory. In this paper we consider the availabilities and unavailabilities of the system in an interval, i.e. the probabilities that the system performs above or below the different levels throughout the whole interval. In complex systems it is often impossible to calculate these availabilities and unavailabilities exactly, but it is possible to construct lower and upper bounds based on the minimal path and cut vectors to the different levels. In this paper we consider systems which allow a modular decomposition. We analyse in depth the relationship between the minimal path an cut vectors for the system, the modules and the organizing structure. We analyse the extent to which the availability bounds are improved by taking advantage of the modular decomposition. This problem was treated also in Butler (1982) and Funnemark and Natvig (1985), but the treatment was based on an inadequate analysis of the relationship between the different minimal path and cut vectors involved, and as a result was somewhat inaccurate. We also extend to an interval bounds that have previously only been given for availabilities at a fixed point of time