In this article, it is proved that for any probability law μ over R with finite first moment and a given deterministic time t>0, there exists a gap diffusion with law μ at the prescribed time t.

The method starts by constructing a discrete time process X on a finite state space, where X_τ has law μ, for a geometric time τ, independent of the diffusion. This argument is developed, using a fixed point theorem, to give conditions for the existence of a process with prescribed law when stopped at an independent time with negative binomial distribution. Reducing the time mesh gives a continuous time diffusion with prescribed law for τ with Gamma distribution. Keeping E[τ]=t fixed, the parameters of the Gamma distribution are altered, giving the prescribed law for the deterministic time. An approximating sequence establishes the result for arbitrary probability measure over R.

Bayesian Networks: An Introductionprovides a self-contained introduction to the theory and applications of Bayesian networks, a topic of interest and importance for statisticians, computer scientists and those involved in modelling complex data sets. The material has been extensively tested in classroom teaching and assumes a basic knowledge of probability, statistics and mathematics. All notions are carefully explained and feature exercises throughout.

Features include:

• An introduction to Dirichlet Distribution, Exponential Families and their applications.
• A detailed description of learning algorithms and Conditional Gaussian Distributions using Junction Tree methods.
• A discussion of Pearl's intervention calculus, with an introduction to the notion of see and do conditioning.
• All concepts are clearly defined and illustrated with examples and exercises. Solutions are provided online.

This book will prove a valuable resource for postgraduate students of statistics, computer engineering, mathematics, data mining, artificial intelligence, and biology.

Researchers and users of comparable modelling or statistical techniques such as neural networks will also find this book of interest.

Supporting high availability by checkpointing and switching to a backup upon failure of a primary has a cost. Trade-off studies help system architects to decide whether higher availability at the cost of higher response time is to strive for. The decision will lead to configuring a fault-tolerant server for best performance. This paper provides a mathematical model employing queuing theory that helps to compute the optimal checkpointing interval for a primary-backup replicated server. The optimization criterion is system availability. The model guides towards the checkpointing interval that is short enough to give low failover time, but long enough to utilize most of the system resources for servicing client requests. The novelty of the work is the detailed modelling of service times, wait times for earlier calls in the queue, and priority of checkpointing calls over client calls within the queues. Studies on the model in Mathematica and validation of a modelling assumption through simulations are included.

[No abstract available]