Stephen Richards

When we first wrote our survival-modelling software in late 2005, we had to decide how to represent dates for the purpose of calculating exposure times.  We decided to adopt a real-valued approach, e.g. 14th March 1968 would be represented as 1968.177596 (the fractional part is \(\frac{31+29+14}{366}\), since 1968 is a leap year).

In 1860 William Makeham published a famous paper. In it he extended Gompertz's mortality law to include a constant term:

\[\mu_x=e^\epsilon+e^{\alpha+\beta x}\qquad(1),\]

Today is the 200th anniversary of Benjamin Gompertz's reading of his famous paper before the Royal Society of London.  Generations of actuaries and demographers are familiar with his law of mortality:

\[\mu_x = e^{\alpha+\beta x}\qquad(1),\]

In my previous blog I described a real case where so-called artificial intelligence (AI) would have struggled to spot data problems that a (suspicious) human could find.  But what if the input data are clean and reliable?

Regular readers of this blog (both of them) will have noticed how often we advocate that actuaries use the Kaplan-Meier estimator in their mortality analysis.  While parametric survival models are best for multi-factor models, the Kaplan-Meier estimate is exceptionally useful for visualisation, communication and data-quality checking.

There is emerging hype about the application of artificial intelligence (AI) to mortality analysis, specifically the use of machine learning via neural networks. In this blog I provide a counter-example that illustrates why the human element is an absolutely indispensable part of actuarial work, and why I think it always will be.

The World Health Organization (WHO) makes available a one-page checklist for use by surgical teams. The WHO claims that this checklist has made "significant reduction in both morbidity and mortality" and is "now used by a majority of surgical providers around the world".  For example, the checklist is used by surgical teams in NHS England.

In Richards & Macdonald (2024) we advocate that actuaries use the Kaplan-Meier estimate of the survival curve.  This is not just because it is an excellent visual communication tool, but also because it is a particularly useful data-quality check.

The short answer for mortality work is that your Poisson model is never truly Poisson. The longer answer is that the true distribution has a similar likelihood, so you will get the same answer from treating it like Poisson.  Your model is pseudo-Poisson, but not actually Poisson.

In a recent blog, I looked at the most fundamental unit of observation in a mortality study, namely an individual life. But is there such a thing as a fundamental unit of modelling mortality?  In Macdonald & Richards (2024) we argue that there is, namely an infinitesimal Bernoulli trial based on the mortality hazard.