In this chapter we present two spatial dependent models: one based on defining a latent variable for each area, and the other by defining one latent variable for each pair of latent areas. We call the latter the latent edges model. We compare both models with a real data set. Extensions to spatio-temporal constructions are also considered.

In this chapter we define what a conjugate family is in a Bayesian analysis context and develop detailed examples of some cases; in particular, we review the beta and binomial case, the Pareto and inverse Pareto case, the gamma and gamma case and the gamma and Poisson case. We conclude by providing a list of conjugate models.

In this chapter we show how to define temporal dependent sequences using a moving average type of construction. We compare the performance of this construction with a Markov-process type. We finally extend the models to include seasonal and periodic dependencies.

In this chapter we start with some attempts to construct dependence sequences with order larger than one and present a general result to achieve an invariant distribution via a three-level hierarchical model. We finally present some results involving exponential families.

In this chapter we describe a general procedure to construct Markov sequences with invariant distributions. The procedure can be used with conjugate and non-conjugate models and with parametric and nonparametric distributions. We derive several examples in detail and finish with some applications in survival analysis.

In this chapter we introduce the concept of exchangeability and show how to construct exchangeable sequences; we present our first result of how to construct exchangeable sequences and maintain a desirable marginal distribution and provide detailed examples. We finish with an application of exchangeable constructions in a meta analysis. Bugs and R code are provided.

In this chapter we start by reviewing the different types of inference procedures: frequentist, Bayesian, parametric and non-parametric. We introduce notation by providing a list of the probability distributions that will be used later on, together with their first two moments. We review some results on conditional moments and carry out several examples. We review definitions of stochastic processes, stationary processes and Markov processes, and finish by introducing the most common discrete-time stochastic processes that show dependence in time and space.

In this chapter we conclude the book by presenting dependent models for random vectors and for stochastic processes. The types of dependence are exchangeable, Markov, moving average, spatial or a combination of the latter two.

Chapter Preview. In this chapter we approach many of the topics of the previous chapters, but from a Bayesian viewpoint. Initially we cover the foundations of Bayesian inference. We then describe the Bayesian linear and generalized regression models. We concentrate on the regression models with zero-one and count response and illustrate the models with real datasets. We also cover hierarchical prior specifications in the context of mixed models. We finish with a description of a semi-parametric linear regression model with a nonparametric specification of the error term. We also illustrate its advantage with respect to the fully parametric setting using a real dataset.

Introduction

The use of Bayesian concepts and techniques in actuarial science dates back to Whitney (1918) who laid the foundations for what is now called empirical Bayes credibility. He noted that the solution of the problem “depends upon the use of inverse probabilities.” This is the term used by T. Bayes in his original paper (e.g., Bellhouse 2004). However, Ove Lundberg was apparently the first one to realize the importance of Bayesian procedures (Lundberg 1940). In addition, Bailey (1950) put forth a clear and strong argument in favor of using Bayesian methods in actuarial science. To date, the Bayesian methodology has been used in various areas within actuarial science; see, for example, Klugman (1992), Makov (2001), Makov, Smith, and Liu (1996), and Scollnik (2001).