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On the approximation of posterior laws in compound loss models by conditional Wasserstein GANs
Aleksandar Arandjelovic, Pavel V. Shevchenko, George Tzougas
TL;DR
Repeated inference for compound loss models becomes computationally demanding outside conjugate cases. The paper addresses this with a conditional Wasserstein GAN driven by sufficient statistics and prior information, finding that one shared generator approximates frequency and severity posteriors across prior mixtures and closely matches an MCMC-based catastrophe-loss forecast. It also quantifies sensitivity of aggregate VaR99.5% to severity-prior mixture weights.
Problem
Non-conjugate compound loss models require repeated numerical posterior inference across policies, scenarios, and prior specifications, while richer actuarial structures reduce analytical tractability.
Method
The paper trains a conditional Wasserstein GAN that generates posterior draws from sufficient statistics, prior mean and coefficient of variation, and prior-family mixture weights.
Results
A single shared generator approximated Poisson-intensity and Pareto-shape posteriors across Gamma, inverse-Gaussian, and lognormal prior mixtures, while the 2026 catastrophe-loss forecast closely matched an extensive MCMC reference.
Takeaways & Limitations
The framework supports rolling aggregate-loss prediction and quantifies how severity-prior mixture weights affect aggregate VaR99.5%.
Takeaways & Limitations
The study is limited to scalar latent parameters, conditionally independent Poisson counts and Pareto severities, fixed severity thresholds, low-dimensional sufficient statistics, and trained conditioning regions and prior families.
Abstract
from arXiv · showhide
Bayesian inference in compound loss models must often be repeated across policies, market scenarios, and prior specifications. Outside conjugate cases, this may require repeated numerical integration or Markov chain Monte Carlo (MCMC). We formulate this problem as amortized posterior approximation and construct a conditional Wasserstein generative adversarial network conditioned on sufficient statistics, prior mean and coefficient of variation, and mixture weights of prior families. Notably, a single shared generator is able to approximate the posterior laws of both the Poisson intensity and the Pareto shape parameter under mixtures of Gamma, inverse-Gaussian, and lognormal priors. We assess the approximation by simulation-based calibration and by comparisons with analytical posteriors, deterministic quadrature, and extensive MCMC simulations. In an application to data on extreme natural catastrophe losses, we produce rolling one-year posterior predictive distributions, and examine the effects of heavy-tailed severity and prior-family uncertainty on aggregate tail risk.
1 Introduction
The paper frames repeated Bayesian inference in compound loss models as amortized approximation of conditional posterior laws. It develops a conditional Wasserstein GAN to support flexible posterior and predictive inference across conditioning values, prior families, and catastrophe-risk applications.
- Motivation: Non-conjugate actuarial models support richer structures but require numerical posterior inference, diagnostics, sensitivity analyses, and predictive checks.Conjugate models offer analytical tractability but constrain compatible likelihood–prior choices; richer features such as heavy-tailed severities generally remove closed-form posteriors.
- Amortized posterior approximation: Amortized inference replaces repeated pointwise posterior computation with one conditional approximation evaluated at new conditioning values.The conditioning information includes sample size, sufficient statistics, and actuarial inputs such as prior hyperparameters, covariates, or stress scenarios.
- Conditional Wasserstein GAN: A conditional Wasserstein GAN represents the posterior through a generator and learns it by minimizing an approximation to the averaged 1-Wasserstein distance.The generator maps independent noise and conditioning values to a distributional approximation rather than only selected posterior functionals.
- Model design: The framework uses sufficient statistics to reduce conditioning dimension and generates posterior draws for claim-frequency and severity parameters.Posterior predictive loss distributions are obtained by mixing the claims law with respect to the generated posterior distribution.
- Evaluation: A single conditional generator covers Poisson frequency and Pareto severity under mixtures of Gamma, inverse-Gaussian, and lognormal priors, with conjugate and non-conjugate benchmarks.Accuracy is assessed using analytical posteriors where available, and deterministic quadrature and MCMC otherwise.
- Application: The application infers prior-family mixture weights from extreme natural-catastrophe data, produces rolling one-year aggregate-loss predictions, and examines heavy-tailed severity and prior uncertainty.The study focuses on how these modeling choices affect catastrophe tail risk.
2 Problem formulation
The paper formulates posterior inference for latent claim-frequency and severity parameters under conditional actuarial models, emphasizing sufficient-statistic conditioning and repeated updating across priors and observations. It distinguishes analytically tractable conjugate cases from non-conjugate settings requiring numerical posterior inference.
- Model setup: The framework models a latent parameter ϑ governing conditionally independent observations, with actuarial information ξ determining the prior and conditional observation laws.The setup covers frequency and severity models, including hierarchical and Bühlmann–Straub formulations.
- Posterior prediction: Approximating the full posterior law is relevant because posterior predictive loss mixtures are not generally determined by posterior means or variances alone.The full law directly supports predictive quantiles and tail probabilities.
- Conjugate benchmark: Conjugate Gamma priors yield Gamma posteriors for both the Poisson intensity and Pareto shape parameter, with parameters updated by observed counts or log-excess sums.The corresponding posterior laws are Gamma(αξ + y+, βξ + n) and Gamma(αξ + n, βξ + t_n).
- Non-conjugate inference: Lognormal priors are non-conjugate because nonlinear terms prevent the transformed posterior densities from belonging to the Gaussian family.Posterior quantities must therefore generally be determined by numerical integration or simulation for each conditioning value.
- Sufficient statistics: For fixed n and ξ, both posterior and posterior predictive laws depend on the observations only through the sufficient statistic s_n, including in non-conjugate cases.This reduction motivates learning a posterior map from compact conditioning information rather than the full observation history.
- Sufficient statistics: For Poisson frequency, the sufficient statistic is the total count y+, while for Pareto severity it is the log-excess sum t_n.Thus conditioning can be reduced from the full history to c = (n, y+, ξ) or c = (n, t_n, ξ).
3 Conditional Wasserstein generative adversarial networks
The section formulates conditional posterior approximation with Wasserstein GANs, using conditional generators and neural critics to approximate law-valued targets across conditioning values. It also introduces hybrid k-packs to address sparse repeated conditioning values in empirical training while preserving the original objective's minimizers.
- Wasserstein formulation: The 1-Wasserstein distance provides a dual optimization over 1-Lipschitz test functions, replacing optimization over couplings with a tractable critic-based formulation.This dual form underlies the WGAN objective.
- WGAN formulation: WGANs represent candidate laws as pushforwards of fixed noise through neural generators, while neural critics restrict the 1-Lipschitz function class.The generated law is induced by the generator's pushforward of the noise distribution.
- Conditional formulation: Conditional WGANs use a generator Gθ(c, Z) to define an approximating law Qθ,c for each conditioning value c, with an integrated conditional Wasserstein objective over the conditioning distribution.The conditioning value includes sufficient statistics and actuarial information such as prior hyperparameters, covariates, or stress scenarios.
- Empirical challenge: Finite batches may contain few or no repeated conditioning values, limiting a restricted critic's ability to learn conditional variation directly.This issue is especially pronounced when the conditioning distribution is non-atomic.
- Hybrid k-packs: Hybrid k-packs jointly evaluate one draw from the target law and k − 1 draws from the generated law at the same conditioning value.The construction generalizes the k = 2 case and allows arbitrary k ≥1.
- Hybrid k-packs: The hybrid k-pack and original conditional Wasserstein objectives differ only by a factor 1/k, so they share minimizing sequences and, when they exist, minimizers.The factor is attributed to the choice of metric on the product space.
4 Simulation study
The simulation study evaluates a shared cWGAN for frequency and severity posteriors under mixtures of three prior families, using calibration, posterior-functional comparisons, and Wasserstein errors. It also applies the framework to rolling catastrophe-loss prediction and examines tail-risk behavior under heavy-tailed severity and prior uncertainty.
- Simulation design: The study evaluates one shared generator for Poisson frequency and Pareto severity posteriors under Gamma, inverse-Gaussian, and lognormal prior mixtures.The conditioning includes prior moments and mixture weights, while the likelihood form permits a common generator across both tasks.
- Catastrophe application: The catastrophe application uses 370 extreme events above the xmin = 2.50 billion USD threshold, with rolling prior recalibration through 2025.The dataset covers drought, earthquake, flood, storm, and wildfire records from 2000–2025.
- Calibration: 50,000 prior-predictive replications per task with L = 1023 posterior draws support simulation-based calibration diagnostics.The diagnostics compare empirical rank behavior with the discrete uniform reference.
- Calibration: 0.24% and 0.50% are the supremum ECDF deviations for frequency and severity parameter ranks, respectively, with both curves inside the 95% simultaneous null bands.Relative posterior-functional errors have interquartile ranges within [−1%, 1%] and central 95% ranges within [−4%, 4%].
- Approximation accuracy: 0.2%–0.5% are the median normalized W1 errors, while the 97.5% quantiles remain below 2.5% across the Gamma–inverse-Gaussian–lognormal mixture simplex.These errors are assessed across mixture-simplex vertices, edges, and interior points.
- Catastrophe application: At the 2025 cutoff, the 2026 forecast has median 128.2 billion USD and 95% and 97.5% quantiles of 582.1 and 985.6 billion USD, closely matching MCMC values of 127.8, 580.8, and 986.4 billion USD.Across sixteen rolling forecasts, every realized aggregate loss lies within its central 95% predictive interval.
5 Conclusion
The paper develops a shared conditional Wasserstein GAN for amortized posterior approximation in compound loss models and evaluates it computationally and on catastrophe-loss data. The application produces rolling forecasts and quantifies prior-mixture sensitivity, while the method remains bounded by its modeling and training scope.
- 5 Conclusion: A single shared cWGAN approximates posterior laws for Poisson intensity and Pareto shape across conditioning values.Conditioning uses sample size, a sufficient statistic, prior mean and coefficient of variation, and prior-family mixture weights.
- 5 Conclusion: The 2026 predictive distribution closely matches an extensive MCMC reference, and aggregate VaR99.5% sensitivity to severity-prior mixture weights is quantified.The catastrophe-loss application uses rolling one-year aggregate-loss forecasts and examines prior-family uncertainty in tail risk.
- 5 Conclusion: The cWGAN study is limited to scalar latent parameters, conditionally independent Poisson counts and Pareto severities, a fixed threshold, and low-dimensional sufficient statistics.It excludes general cross-peril and frequency–severity dependence, covariates, censoring, truncation, reporting delays, and richer multilevel portfolio structures.
- 5 Conclusion: Approximation accuracy depends on the conditioning region, training prior families, neural-network architecture, and adversarial-optimization hyperparameters.The paper identifies multivariate samplers and learned permutation-invariant summaries as extensions beyond these boundaries.