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Expressive Mortality Models through Gaussian Process Kernels

Mike Ludkovski, Jimmy Risk

TL;DR

A flexible Gaussian process (GP) framework for learning the covariance structure of Age- and Year-specific mortality surfaces is developed and a genetic programming algorithm is designed to search for the most expressive kernel for a given population.

Abstract

We develop a flexible Gaussian Process (GP) framework for learning the covariance structure of Age- and Year-specific mortality surfaces. Utilizing the additive and multiplicative structure of GP kernels, we design a genetic programming algorithm to search for the most expressive kernel for a given population. Our compositional search builds off the Age-Period-Cohort (APC) paradigm to construct a covariance prior best matching the spatio-temporal dynamics of a mortality dataset. We apply the resulting genetic algorithm (GA) on synthetic case studies to validate the ability of the GA to recover APC structure, and on real-life national-level datasets from the Human Mortality Database. Our machine-learning based analysis provides novel insight into the presence/absence of Cohort effects in different populations, and into the relative smoothness of mortality surfaces along the Age and Year dimensions. Our modelling work is done with the PyTorch libraries in Python and provides an in-depth investigation of employing GA to aid in compositional kernel search for GP surrogates.

Expressive Mortality Models through Gaussian Process Kernels

TL;DR

A flexible Gaussian process (GP) framework for learning the covariance structure of Age- and Year-specific mortality surfaces is developed and a genetic programming algorithm is designed to search for the most expressive kernel for a given population.

Abstract

We develop a flexible Gaussian Process (GP) framework for learning the covariance structure of Age- and Year-specific mortality surfaces. Utilizing the additive and multiplicative structure of GP kernels, we design a genetic programming algorithm to search for the most expressive kernel for a given population. Our compositional search builds off the Age-Period-Cohort (APC) paradigm to construct a covariance prior best matching the spatio-temporal dynamics of a mortality dataset. We apply the resulting genetic algorithm (GA) on synthetic case studies to validate the ability of the GA to recover APC structure, and on real-life national-level datasets from the Human Mortality Database. Our machine-learning based analysis provides novel insight into the presence/absence of Cohort effects in different populations, and into the relative smoothness of mortality surfaces along the Age and Year dimensions. Our modelling work is done with the PyTorch libraries in Python and provides an in-depth investigation of employing GA to aid in compositional kernel search for GP surrogates.
Paper Structure (24 sections, 14 equations, 8 figures, 15 tables, 1 algorithm)

This paper contains 24 sections, 14 equations, 8 figures, 15 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Gaussian Process Models for Mortality
  3. Gaussian Process Regression
  4. GP Kernels
  5. Kernel Families
  6. Connections to Mortality Modelling
  7. Genetic Programming for GPs
  8. BIC and Bayes Factors for GPs
  9. GA Kernel Representation
  10. GA Operations and Hyperparameters
  11. GP Hyperparameter Optimization
  12. Synthetic Mortality Kernel Recovery
  13. Synthetic Results
  14. Results on Human Mortality Database data
  15. Robustness Check
  16. ...and 9 more sections

Figures (8)

  • Figure 1: Representative compositional kernels and GA operations. Bolded red ellipses indicate the node of $\kappa$ (or $\xi$) that was chosen for mutation or crossover.
  • Figure 2: Frequency of appearance of different kernels in JPN Female models.
  • Figure 3: Properties of the top 100 kernels found by GA.
  • Figure 4: Predictions from the top 10 kernels in ${\cal K}_f$ for JPN Females Age 65. Left: predictive mean and 90% posterior interval from the top-10 kernels. For comparison we also display (black plusses) the 5 observed log-mortality rates during 2014--2019. Right: 4 sample paths from 3 representative kernels.
  • Figure 5: Summary statistics of best kernels proposed by GA as a function of generation $g$.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Remark 1
  • Remark 2
  • Remark 3