On the use of Markovian stick-breaking priors

Recently, a ‘Markovian stick-breaking’ process which generalizes the Dirichlet process ( μ , θ ) (\mu , \theta ) with respect to a discrete base space X \mathfrak {X} was introduced. In particular, a sample from from the ‘Markovian stick-breaking’ processs may be represented in stick-breaking form ∑ i ≥ 1 P i δ T i \sum _{i\geq 1} P_i \delta _{T_i} where { T i } \{T_i\} is a stationary, irreducible Markov chain on X \mathfrak {X} with stationary distribution μ \mu , instead of i.i.d. { T i } \{T_i\} each distributed as μ \mu as in the Dirichlet case, and { P i } \{P_i\} is a GEM ( θ ) (\theta ) residual allocation sequence. Although the previous motivation was to relate these Markovian stick-breaking processes to empirical distributional limits of types of simulated annealing chains, these processes may also be thought of as a class of priors in statistical problems. The aim of this work in this context is to identify the posterior distribution and to explore the role of the Markovian structure of { T i } \{T_i\} in some inference test cases.

DERIVING ROBUST BAYESIAN PREMIUMS UNDER BANDS OF PRIOR DISTRIBUTIONS WITH APPLICATIONS

We study the propagation of uncertainty from a class of priors introduced by Arias-Nicolás et al. [(2016) Bayesian Analysis, 11(4), 1107–1136] to the premiums (both the collective and the Bayesian), for a wide family of premium principles (specifically, those that preserve the likelihood ratio order). The class under study reflects the prior uncertainty using distortion functions and fulfills some desirable requirements: elicitation is easy, the prior uncertainty can be measured by different metrics, and the range of quantities of interest is easily obtained from the extremal members of the class. We illustrate the methodology with several examples based on different claim counts models.

Robust Bayesian Credibility Using Semiparametric Models

In performing Bayesian analysis of insurance losses, one usually chooses a parametric conditional loss distribution for each risk and a parametric prior distribution to describe how the conditional distributions vary across the risks. Young (1997) applies techniques from nonparametric density estimation to estimate the prior and uses the estimated model to calculate the predictive mean of future claims given past claims. A shortcoming of this method is that, in estimating the prior, one assumes the average claim amount equals the conditional claim. In this paper, we consider a class of priors obtained by perturbing the one determined nonparametrically, as in Young (1997). We thereby reflect the uncertainty in the prior that arises from the randomness in the claim data. We, then, calculate intervals for the corresponding predictive means. We illustrate our method with data from Dannenburg et al. (1996) and compare the intervals of the predictive means with nonparametric confidence intervals.