The fundamental 'atom' of mortality modelling

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.

Consider a life currently alive and about to be observed over a small interval of time \(ds\).  If we make \(ds\) small enough, we can span any interval of time with one or more intervals of width \(ds\).  For example, if we make \(ds\) a single day, we can span weeks, months or years by chaining together enough days.  The probability that the life dies in \(ds\) is then approximately \(\mu_xds\), where \(\mu_x\) is the mortality hazard at age \(x\).  We don't need to worry that this is an approximation, since we can make our calculation arbitrarily accurate by reducing the size of \(ds\).  The survival or death of the life over \(ds\) is therefore a Bernoulli trial with likelihood:

\[L=(1-\mu_xds)^{1-d}(\mu_xds)^d,\qquad(1)\]

where \(d\) takes the value 1 if the life dies in \(ds\) and zero otherwise.  Since \((ds)^d\) doesn't affect inference, we can drop it from the likelihood in equation (1):

\[L\propto (1-\mu_xds)^{1-d}(\mu_x)^d.\qquad(2)\]

Assume now that we observe the individual over an interval of length \(t\), which is some multiple of \(ds\). By definition every trial except the last is guaranteed to have \(d=0\).  This means that when we multiply the likelihoods in equation (2) we get the following:

\[L\propto \left(\prod_{s=0}^{t-ds} (1-\mu_{x+s}ds)\right)(\mu_{x+t-ds})^d,\qquad(3)\]

where \(d\) is now the status of the life at the end of interval \(t\) and (loosely) \(s\) steps through \(0, ds, 2ds, 3ds, \ldots, t-ds\). Now we let \(ds\) tend to zero from above, i.e. \(ds\to0^+\), and so \((t-ds)\to t\).  This allows us to replace equation (3) with the following:

\[L\propto \left(\prod_0^t (1-\mu_{x+s}ds)\right)(\mu_{x+t})^d.\qquad(4)\]

The \(\prod\) term in equation (4) is the product integral (that thing again!), so equation (4) can also be written as:

\[L\propto \exp\left(-\int_0^t \mu_{x+s}ds\right)(\mu_{x+t})^d.\qquad(5)\]

Equations (4) and (5) are therefore alternative ways of writing the most fundamental unit of modelling mortality over an interval of \(t\).  We recognise equation (5) as the contribution of a single life to the likelihood of a survival model. Thus, a survival model for individual lives is the most fundamental unit or 'atom' of mortality modelling.  In Macdonald & Richards (2024) we refer to modelling the mortality hazard at the level of the individual as a 'micro' model.  Many other 'macro' models can be built from this.