Information Matrix

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:

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

I rediscovered my Faculty of Actuaries diploma recently, having misplaced it in a house move some years ago. It testifies to my knowledge of ‘the doctrine of probabilities’, which is nice. But not long after I received it, Prof Hans Bühlmann classified actuaries like me as ‘Actuaries of the First Kind’ and said:

Contrary to [the Actuary] of the First Kind in life assurance, whose methods were essentially deterministic, [the Actuary of the Second Kind] had to master the skills of probabilistic thinking.

Of all the actuary's standard formulae derived from the life table, none is more important in survival modelling than:

\[{}_tp_x = \exp\left(-\int_0^t\mu_{s+s}ds\right).\qquad(1)\]

In survival-model work there is a fundamental relationship between the \(t\)-year survival probability from age \(x\), \({}_tp_x\), and the force of mortality, \(\mu_x\):

\[{}_tp_x = \exp\left(-\int_0^t\mu_{x+s}ds\right).\qquad(1)\]