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Antonio Eleuteri (Royal Liverpool University Hospital, UK), Azzam Taktak (Royal Liverpool University Hospital, UK), Bertil Damato (Royal Liverpool University Hospital, UK), Angela Douglas (Liverpool Women’s Hospital, UK) and Sarah Coupland (Royal Liverpool University Hospital, UK)

DOI: 10.4018/978-1-59904-849-9.ch060

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TopLet *T* denote an absolutely continuous positive random variable, with distribution function *P*, representing the time of occurrence of an event. The survival function, *S*(*t*), is defined as:*S*(*t*)=Pr(*T>t*),that is, the probability of surviving beyond time *t*. We shall generally assume that the survival function also depends on a set of covariates, represented by the vector *x* (which can itself be assumed to be a random variable). An important function related to the survival function is the *hazard rate* (Cox & Oakes, 1984), defined as:*h _{r}* (

In many survival analysis applications we do not directly observe realisations of the random variable *T*; therefore we must deal with a missing data problem. The most common form of missingness is *right censoring*, i.e., we observe realisations of the random variable:*Z=*min(*T,C*),where *C* is a random variable whose distribution is usually unknown. We shall use a censoring indicator *d* to denote whether we have observed an event (*d=*1) or not (*d=*0). It can be shown that inference does not depend on the distribution of *C* (Cox & Oakes, 1984).

Bayesian Inference: Inference rules which are based on application of Bayes’ theorem and the basic laws of probability calculus.

Random Variable: Measurable function from a sample space to the measurable space of possible values of the variable.

Survival Analysis: Statistical analysis of data represented in terms of realisation of point events. In medical applications usually the point event is the death of an individual, or recurrence of a disease.

Hyperparameter: Parameter in a hierarchical problem formulation. In Bayesian inference, the parameters of a prior.

Neural Networks: A graphical representation of a nonlinear function. Usually represented as a directed acyclic graph. Neural networks can be trained to find nonlinear relationships in data, and are used in applications such as robotics, speech recognition, signal processing or medical diagnosis.

Prior Distribution: Probabilistic representation of prior knowledge.

Censoring: Mechanism which precludes observation of an event. A form of missing data.

Posterior Distribution: Probabilistic representation of knowledge, resulting from combination of prior knowledge and observation of data.

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