Paper: arXiv 2409.01908

Abstract

This paper proposes three types of Bayesian CART (or BCART) models for aggregate claim amount, namely, frequency-severity models, sequential models and joint models. We propose a general framework for the BCART models applicable to data with multivariate responses, which is particularly useful for the joint BCART models with a bivariate response: the number of claims and aggregate claim amount. To facilitate frequency-severity modeling, we investigate BCART models for the right-skewed and heavy-tailed claim severity data by using various distributions. We discover that the Weibull distribution is superior to gamma and lognormal distributions, due to its ability to capture different tail characteristics in tree models. Additionally, we find that sequential BCART models and joint BCART models, which incorporate dependence between the number of claims and average severity, are beneficial and thus preferable to the frequency-severity BCART models in which independence is assumed. The effectiveness of these models’ performance is illustrated by carefully designed simulations and real insurance data.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 7.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces advanced Bayesian CART models with multivariate responses and specialized distributions (e.g., Weibull, compound Poisson gamma), requiring sophisticated probability theory and computation. It validates these models through both carefully designed simulations and real insurance data analysis, demonstrating a strong empirical foundation for practical implementation.

Research Flowchart

  flowchart TD
  Start["Research Goal:<br/>Propose Bayesian CART models<br/>for aggregate claim modeling"] --> Inputs["Data & Inputs:<br/>Simulated & Real Insurance Data<br/>(Count, Severity, Aggregate)"]

  Inputs --> Methodology["Methodology:<br/>Develop 3 Bayesian CART Frameworks:<br/>1. Frequency-Severity<br/>2. Sequential<br/>3. Joint (Bivariate)"]

  Methodology --> Computation["Computational Process:<br/>MCMC Sampling<br/>(Gibbs & Metropolis-Hastings)<br/>Tree Structure Learning"]

  Computation --> Analysis["Model Analysis:<br/>Compare Distributions:<br/>Weibull vs. Gamma vs. Lognormal<br/>Test Independence Assumptions"]

  Analysis --> Outcomes["Key Findings & Outcomes:<br/>1. Weibull best for heavy-tailed severity<br/>2. Joint/Sequential models outperform<br/>   Frequency-Severity (due to dependence)<br/>3. Effective for multivariate responses"]

Cited References (145)

  1. Collective risk models with dependence uncertainty (2017)
  2. Bayesian total loss estimation using shared random effects (2015)
  3. Predicting the frequency and amount of health care expenditures (2011)
  4. Tweedie gradient boosting for extremely unbalanced zero-inflated data (2022)
  5. A T weedie compound Poisson model in reproducing kernel Hilbert space (2023)
  6. Regression for copula-linked compound distributions with applications in modeling aggregate insurance claims (2020)
  7. Bayesian model selection (1998)
  8. Bayesian treed generalized linear models (2003)
and 137 more
  1. Theory of Posterior Concentration for Generalized Bayesian Additive Regression Trees (2023) — link
  2. Variable selection with ABC Bayesian forests (2021)
  3. Variable selection for BART : an application to gene regulation (2014)
  4. Actuarial intelligence in auto insurance: Claim frequency modeling with driving behavior features and improved boosted trees (2022)
  5. Identifying representative trees from ensembles (2012)
  6. Continuous-time birth-death MCMC for Bayesian regression tree models (2020)
  7. Posterior concentration for B ayesian regression trees and forests (2020)
  8. Bayesian regression tree ensembles that adapt to smoothness and sparsity (2018)
  9. Insurance Analytics with Tree-Based Models (2019)
  10. Accounting for shared covariates in semi-parametric B ayesian additive regression trees (2021) — link
  11. Imbalanced learning for insurance using modified loss functions in tree-based models (2022)
  12. Effective Statistical Learning Methods for Actuaries (2019)
  13. Machine learning in P & C insurance: A review for pricing and reserving (2020)
  14. A Course in Credibility Theory and its Applications (2005)
  15. Stability of decision trees and logistic regression (2019)
  16. Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-malus Systems (2007)
  17. Boosting insights in insurance tariff plans with tree-based machine learning methods (2021)
  18. Auto insurance premium calculation using generalized linear models (2015)
  19. Fitting T weedie’s c ompound P oisson model to insurance claims data (1994)
  20. Multivariate frequency-severity regression models in insurance (2016)
  21. A dependent frequency-severity approach to modeling longitudinal insurance claims (2019)
  22. A mixed copula model for insurance claims and claim sizes (2012)
  23. Joint regression analysis of correlated data using G aussian copulas (2009)
  24. Dependence modeling of frequency-severity of insurance claims using waiting time (2023)
  25. The balance property in neural network modelling (2022)
  26. Bias regularization in neural network models for general insurance pricing (2020)
  27. Bayesian analysis of zero-inflated distributions (2003)
  28. Branching Processes (2004)
  29. Bayesian regression trees for high-dimensional prediction and variable selection (2018)
  30. Semiparametric mixed-scale models using shared B ayesian forests (2020)
  31. Multinomial probit B ayesian additive regression trees (2016)
  32. Efficient M etropolis– H astings proposal mechanisms for B ayesian regression tree models (2016)
  33. bart M achine: Machine learning with B ayesian additive regression trees (2013) — link
  34. The art of data augmentation (2001)
  35. The calculation of posterior distributions by data augmentation (1987)
  36. Reversible jump MCMC computation and B ayesian model determination (1995)
  37. Seeking efficient data augmentation schemes via conditional and marginal augmentation (1999)
  38. Understanding predictive information criteria for B ayesian models (2014)
  39. The deviance information criterion: 12 years on (2014)
  40. Asymptotic equivalence of B ayes cross validation and widely applicable information criterion in singular learning theory. (2010)
  41. Lognormal and gamma mixed negative binomial regression (2012)
  42. Non-life Insurance: Mathematics & Statistics (2022)
  43. Data Analytics for Non-Life Insurance Pricing (January 9, 2023). (2022)
  44. Statistical Foundations of Actuarial Learning and its Applications (2022)
  45. Delta boosting implementation of negative binomial regression in actuarial pricing (2020)
  46. Classification and Regression Trees (1984)
  47. Bayesian CART model search (1998)
  48. A B ayesian CART algorithm (1998)
  49. evtree: Evolutionary learning of globally optimal classification and regression trees in R (2011)
  50. Non-life Insurance Pricing with Generalized Linear Models (2010)
  51. BART : B ayesian additive regression trees (2010)
  52. Bayesian credibility for GLM s (2018)
  53. Managing multiple models (2001)
  54. Zero-inflated P oisson ( ZIP ) distribution: parameter estimation and applications to model data from natural calamities (2014)
  55. Generalized linear models (1972)
  56. Generalized additive models: some applications (1987)
  57. Statistical motor rating: making effective use of your data (1992)
  58. Modelling the claims process in the presence of covariates (1994)
  59. Testing for asymmetric information in insurance markets (2000)
  60. A data driven binning strategy for the construction of insurance tariff classes (2018)
  61. An exploratory technique for investigating large quantities of categorical data (1980)
  62. Tree-structured classification via generalized discriminant analysis: Comment (1988)
  63. Random forests (2001)
  64. Bayesian treed models (2002)
  65. Bayesian CART : Prior specification and posterior simulation (2007)
  66. Bayesian treed Gaussian process models with an application to computer modeling (2008)
  67. tgp: an R package for Bayesian nonstationary, semiparametric nonlinear regression and design by treed G aussian process models (2007)
  68. Categorical inputs, sensitivity analysis, optimization and importance tempering with tgp version 2, an R package for treed Gaussian process models (2009)
  69. A review of tree-based B ayesian methods (2017)
  70. Log-linear B ayesian additive regression trees for multinomial Logistic and count regression models (2021)
  71. Bayesian additive regression trees: A review and look forward (2020)
  72. An index which distinguishes between some important exponential families (1984)
  73. Fitting T weedie’s compound P oisson model to insurance claims data: dispersion modelling (2002)
  74. evtree: Evolutionary learning of globally optimal classification and regression trees in R (2014)
  75. Deviance information criteria for missing data models (2006)
  76. Estimation of finite mixture distributions through B ayesian sampling (1994)
  77. Addressing imbalanced insurance data through zero-inflated P oisson regression with boosting (2021)
  78. An introduction to statistical computing: a simulation-based approach (2013)
  79. Bagging predictors (1996)
  80. Bayesian measures of model complexity and fit (2002)
  81. Comparison of hierarchical Bayesian models for overdispersed count data using DIC and Bayes’ factors (2009)
  82. Insurance premium prediction via gradient tree-boosted T weedie compound P oisson models (2018)
  83. Generalised linear models for aggregate claims: t o T weedie or not? (2015)
  84. Autocalibration and T weedie-dominance for insurance pricing with machine learning (2021)
  85. Generalized linear models for dependent frequency and severity of insurance claims (2016)
  86. Hybrid Tree-based Models for Insurance Claims (2020) — link
  87. Performing likelihood ratio tests with multiply-imputed data sets (1992)
  88. Decision tree approaches for zero-inflated count data (2006)
  89. insurance D ata: a c ollection of Insurance Datasets Useful in Risk Classification in Non-life insurance (2014) — link
  90. rpart: Recursive Partitioning and Rregression Trees (2019) — link
  91. A comparison between P oisson and zero-inflated P oisson regression models with an application to number of black spots in C orriedale sheep (2008)
  92. Local bias adjustment, duration-weighted probabilities, and automatic construction of tariff cells (2022)
  93. Deviance information criterion for B ayesian model selection: Justification and variation (2017)
  94. Bayesian Additive Regression Trees, Computational Approaches (2014)
  95. Nonparametric machine learning and efficient computation with B ayesian additive regression trees: the BART R package (2021)
  96. XBART : A ccelerated B ayesian additive regression trees (2019)
  97. Hierarchical priors for Bayesian CART shrinkage (2000)
  98. A primer on copulas for count data (2007)
  99. Bayesian approaches to copula modelling (2011) — link
  100. Estimation of copula models with discrete margins via B ayesian data augmentation (2012)
  101. Dependent frequency-severity modeling of insurance claims (2015)
  102. Investigating dependence between frequency and severity via simple generalized linear models (2019)
  103. Bayesian CART models for insurance claims frequency — → 2303.01923
  104. Making T weedie’s compound P oisson model more accessible (2021)
  105. Fitting T weedie’s c ompound P oisson model to pure premium with the EM algorithm (2024)
  106. Modeling the frequency and severity of auto insurance claims using statistical distributions (2018)
  107. On a bonus-malus system where the claim frequency distribution is geometric and the claim severity distribution is P areto (2005)
  108. Predictive modeling in actuarial science (2014)
  109. Statistical foundations of actuarial learning and its applications (2023)
  110. Stochastic claims reserving methods in insurance (2008)
  111. The weibull distribution: a handbook (2008)
  112. A compendium of conjugate priors (1997)
  113. Selecting among weibull, lognormal and gamma distributions using complete and censored smaples (1982)
  114. optimx: e xpanded Replacement and Extension of the ``optim’’ Function (2023) — link
  115. Cyber claim analysis using Generalized P areto regression trees with applications to insurance (2021)
  116. Spatial modelling of claim frequency and claim size in non-life insurance (2007)
  117. Objective criteria for the evaluation of clustering methods (1971)
  118. Comparing partitions (1985)
  119. The impact of random models on clustering similarity (2017) — link
  120. A novel algorithm and web-based tool for comparing two alternative phylogenetic trees (2006)
  121. fossil: Palaeoecological and Palaeogeographical Analysis Tools (2020) — link
  122. dist R forest: r andom Forests with Distribution-Based Loss Functions (2020) — link
  123. fitdistrplus: h elp to Fit of a Parametric Distribution to Non-Censored or Censored Data (2023) — link
  124. Copula in a multivariate mixed discrete-continuous model (2016)
  125. DTreeSim: A new approach to compute decision tree similarity using re-mining (2017)
  126. Prediction with missing data via B ayesian additive regression trees (2015)
  127. Bayesian Additive Regression Trees for Multivariate Responses (2023) — link
  128. Computational Actuarial Science with R (2014)
  129. Insurance Pricing using Bayesian Tree Models (2024)
  130. Modeling dependent yearly claim totals including zero claims in private health insurance (2012)
  131. Total loss estimation using copula-based regression models (2013)
  132. Tail negative dependence and its applications for aggregate loss modeling (2015)
  133. Collective risk models with dependence (2019)
  134. On copula-based collective risk models: from elliptical copulas to vine copulas (2021)
  135. A copula transformation in multivariate mixed discrete-continuous models (2021)
  136. Collective risk models with FGM dependence (2024)
  137. On hybrid tree-based methods for short-term insurance claims (2023)

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