Approximate Bayesian Computations to fit and compare insurance loss models

Approximate Bayesian Computation (ABC) is a statistical learning technique to calibrate and select models by comparing observed data to simulated data. This technique bypasses the use of the likelihood and requires only the ability to generate synthetic data from the models of interest. We apply ABC to fit and compare insurance loss models using aggregated data. We present along the way how to use ABC for the more common claim counts and claim sizes data. A state-of-the-art ABC implementation in Python is proposed. It uses sequential Monte Carlo to sample from the posterior distribution and the Wasserstein distance to compare the observed and synthetic data. MSC 2010 : 60G55, 60G40, 12E10.

[1]  Stuart A. Klugman,et al.  Loss Models: From Data to Decisions , 1998 .

[2]  C. Robert,et al.  ABC likelihood-free methods for model choice in Gibbs random fields , 2008, 0807.2767.

[3]  Yanan Fan,et al.  Handbook of Approximate Bayesian Computation , 2018 .

[4]  S. Sisson,et al.  A comparative review of dimension reduction methods in approximate Bayesian computation , 2012, 1202.3819.

[5]  Edward W. Frees,et al.  Predicting the Frequency and Amount of Health Care Expenditures , 2011 .

[6]  Richard G. Everitt,et al.  A rare event approach to high-dimensional approximate Bayesian computation , 2016, Statistics and Computing.

[7]  Lena Jaeger,et al.  Loss Models: From Data to Decisions , 2006 .

[8]  Paul Fearnhead,et al.  Constructing summary statistics for approximate Bayesian computation: semi‐automatic approximate Bayesian computation , 2012 .

[9]  A. E. Renshaw,et al.  Modelling the Claims Process in the Presence of Covariates , 1994 .

[10]  Michael P. H. Stumpf,et al.  Simulation-based model selection for dynamical systems in systems and population biology , 2009, Bioinform..

[11]  Paul Fearnhead,et al.  Semi-automatic selection of summary statistics for ABC model choice , 2013, Statistical applications in genetics and molecular biology.

[12]  Mark A. Beaumont,et al.  Approximate Bayesian Computation Without Summary Statistics: The Case of Admixture , 2009, Genetics.

[13]  Peng Shi,et al.  Dependent frequency–severity modeling of insurance claims , 2015 .

[14]  Rudolf Grübel,et al.  Decompounding: an estimation problem for Poisson random sums , 2003 .

[15]  M. Bøgsted,et al.  Decompounding random sums: a nonparametric approach , 2010 .

[16]  Gareth W. Peters,et al.  Bayesian Inference, Monte Carlo Sampling and Operational Risk. , 2006 .

[17]  Arnaud Doucet,et al.  An adaptive sequential Monte Carlo method for approximate Bayesian computation , 2011, Statistics and Computing.

[18]  Adam M. Johansen,et al.  A simple approach to maximum intractable likelihood estimation , 2013 .

[19]  Peter Spreij,et al.  A kernel type nonparametric density estimator for decompounding , 2005, math/0505355.

[20]  John Salvatier,et al.  Probabilistic programming in Python using PyMC3 , 2016, PeerJ Comput. Sci..

[21]  Mathieu Gerber,et al.  Approximate Bayesian computation with the Wasserstein distance , 2019, Journal of the Royal Statistical Society: Series B (Statistical Methodology).

[22]  John K Kruschke,et al.  Bayesian data analysis. , 2010, Wiley interdisciplinary reviews. Cognitive science.

[23]  Peter E. Rossi,et al.  A bayesian approach to testing the arbitrage pricing theory , 1991 .

[24]  Christian Genest,et al.  Generalized linear models for dependent frequency and severity of insurance claims , 2015 .

[25]  M. Blum.  Approximate Bayesian Computation: A Nonparametric Perspective , 2009, 0904.0635.

[26]  P. Spreij,et al.  A non-parametric Bayesian approach to decompounding from high frequency data , 2015, Statistical Inference for Stochastic Processes.

[27]  A. Coca,et al.  Efficient nonparametric inference for discretely observed compound Poisson processes , 2015, 1512.08472.

[28]  Gareth W. Peters,et al.  Chain ladder method: Bayesian bootstrap versus classical bootstrap , 2010 .