While Bayesian methods have attracted considerable interest in actuarial science, they are yet to be embraced in large-scaled insurance predictive modeling applications, due to inefficiencies of Bayesian estimation procedures. The paper presents an efficient method that parallelizes Bayesian computation using distributed computing on Apache Spark across a cluster of computers. The distributed algorithm dramatically boosts the speed of Bayesian computation and expands the scope of applicability of Bayesian methods in insurance modeling. The empirical analysis applies a Bayesian hierarchical Tweedie model to a big data of 13 million insurance claim records. The distributed algorithm achieves as much as 65 times performance gain over the non-parallel method in this application. The analysis demonstrates that Bayesian methods can be of great value to large-scaled insurance predictive modeling.

The $l_{0}$-minimisation problem has attracted much attention in recent years with the development of compressive sensing. The spark of a matrix is an important measure that can determine whether a given sparse vector is the solution of an $l_{0}$-minimisation problem. However, its calculation involves a combinatorial search over all possible subsets of columns of the matrix, which is an NP-hard problem. We use Weyl’s theorem to give two new lower bounds for the spark of a matrix. One is based on the mutual coherence and the other on the coherence function. Numerical examples are given to show that the new bounds can be significantly better than existing ones.