One of the most basic objects in a probabilistic model is the indicator \(1_{\cal A}\) of an event \(\cal A\):
\[ 1_{\cal A} = \left\{ \begin{array}{ll} 1 & \mbox{if ${\cal A}$ occurs} \\ 0 & \mbox{if ${\cal A}$ does not occur}. \end{array} \right. \]
Indicators are useful things. They allow us to switch between probability and expectation:
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As the bicentenary year of Benjamin Gompertz's Law draws to a close (Gompertz 1825) it is salutory to recall what calculating involved for Gompertz and his contemporaries. Not much had changed since logarithms had been invented, two centuries before, and arithmometers were still some decades in the future (Richards 2025). Logarithms it had to be.
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In Macdonald & Richards (2025), Stephen and I pointed out some benefits of models built up from instantaneous Bernoulli trials by product-integration (both of which have featured in previous blogs).
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If this and other recent blogs have a historical flavour, the reason is the 200th anniversary of the 1825 paper by Benjamin Gompertz that introduced his eponymous law of mortality. In the course of his own researches, Stephen drew to my attention a short letter published by Wilhelm Lazarus in Journal of the Institute of Actuaries in 1862. It is a remarkable document.
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Our paper on the practical aspects of contemporary mortality modelling for actuaries has was published today by the British Actuarial Journal. The article is free to access at https://www.doi.org/10.1017/S1357321725000121. The preprint is also available.
In 1860 William Makeham published a famous paper. In it he extended Gompertz's mortality law to include a constant term:
\[\mu_x=e^\epsilon+e^{\alpha+\beta x}\qquad(1),\]
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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:
\[\mu_x = e^{\alpha+\beta x}\qquad(1),\]
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As discussed in earlier blogs, trailblazing actuaries Benjamin Gompertz and William Makeham used parametric models for the mortality hazard. However, the data they worked with were typically grouped into wide age ranges, which involves a loss of information if mortality rates are continually increasing.
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Stephen recently questioned whether the hype around AI models for Life Insurance might be a case of The Emperor's New Clothes. In this blog we discuss an important point of difference: whereas in the fable, a youth reveals the expensive "invisible" new clothes have no substance at all, in our scenario, we find precisely the opposite. AI models utilising machine learning are, far from being see-through, simply not transparent enough.
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In Macdonald et al (2018, Section 2.5) we describe the importance of deduplication, i.e. the identification of individuals behind multiple policies. This is a critical step for a statistical model, as lives can be regarded as independent, whereas the mortality experience of two or more policies written on the same life clearly are not.
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