Information Matrix

Scientists have long admired a 'neat trick', meaning an ingenious idea for overcoming an obstacle, and definitely not trying to mislead anyone. Just mentioning a 'trick' can mean trouble, though, if it gets the attention of internet trolls. Climate scientist Phil Jones found this out the hard way when he mentioned "Mike's Nature trick" in a private email that was later hacked (Pearce 2010, Chapter 14). The trolls deliberately twisted the meaning of the word 'trick' and Professor Jones's life was changed forever.

Three years ago I covered two examples of repressive regimes hiding the official extent of deaths, but where enterprising researchers deduced the likely truth from other administrative data.  One of those examples was Russia's combat deaths in its war on Ukraine.

A fascinating paper from 2018 contained an unusual description of cancer. From the perspective of a clinical outsider, it was unexpected to see what I'd always lazily thought of as a disease of the cell, portrayed more as a form of societal breakdown:

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:

Aging is an unavoidable double-edged sword. Whilst it remains, as numerous wags have suggested, far better than the alternative, alongside the upsides of wisdom and experience, we often acquire long-term conditions (LTCs). Once these arrive in late-life, they usually hang around to plague us as unwelcome, life-long travelling companions. There are many diseases in this category, including diabetes, hypertension, heart or kidney disease, and neurodegenerative conditions like Parkinsons.

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.

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.

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).

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.

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).