Wilhelm Lazarus: A century ahead of his time

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. In our recent paper Macdonald & Richards (2025) on contemporary mortality models, we emphasised two decidedly modern features: (A) representing individual life history data as a counting process; and (B) writing survival model likelihoods as product integrals. Both ideas are in the letter, (A) in essence and (B) in rather more than essence.

(A). Lazarus introduces a somewhat vague ‘power of life’ which is some continuously diminishing quantity underlying the number alive at age \(x\) in the life table, \(L(x)\). Sometimes he calls this quantity an ‘intensity’, not to be confused with its modern actuarial meaning. He remarks on a "... peculiarity, which should always be borne in mind."

But the thing suddenly changes when we divide our field of observation in its single parts — the individuals. Of the 814 persons alive a year ago, 8 are dead and 806 alive; in each individual the ‘power of life,’ in the sense in which we have defined this expression, is the same as it has been a year before, and in the 8 people dead it is entirely extinct. While we have observed a decrease of the power of life in the total number considered as a unit, two states opposed to each other appear in the individuals; they must be considered as each having always the same intensity, as there cannot be a question of a gradual passing from the one state to the other in the sense in which we mean life and death.

Lazarus (1862, p.284) (bold emphasis added)

Lazarus is reaching towards the idea of two states, alive and dead, presence in the alive state being marked by a constant value of the ‘power of life’, which drops to zero on transition to the dead state. He says, in essence, that the ‘power of life’ in an individual may be represented by a piecewise-constant jump process. We have only to assign numerical value 1 to the presence of the ‘power of life’ and 0 to its absence, and specify some technical details, and we have the process \(1_{\{N (t)=0\}}\) from Macdonald & Richards (2025); formally, the indicator that the process \(N(t\)) counting the number of deaths has not jumped yet.

(B). This is even more remarkable because it is not vague but is clear mathematics. I quote the key passage below, just adding equation numbers for clarity, and correcting an obvious typo in (2).

If we denote the probability that a person aged \(x\) will live the interval of time \(\Delta\), by \(\displaystyle{\frac{L_{(x+\Delta)}}{L_{(x)}}}\), this is evidently equal to \[\frac{L_{\left( x + \frac{\Delta}{n} \right)}}{L_{\left( x \right)}} \cdot \frac{L_{\left( x + \frac{2 \Delta}{n} \right)}}{L_{\left( x + \frac{\Delta}{n} \right)}} \cdot \frac{L_{\left( x + \frac{3 \Delta}{n} \right)}}{L_{\left( x + \frac{ 2 \Delta}{n} \right)}} \cdots \frac{L_{\left( x + \Delta \right)}}{L_{\left( x + \frac{ (n-1) \Delta}{n} \right)}};\qquad(1)\] therefore, \[\log \frac{L_{(x+\Delta)}}{L_{(x)}} = \log \left[ \frac{L_{\left( x + \frac{\Delta}{n} \right)}}{L_{\left( x \right)}} \right] + \log \left[ \frac{L_{\left( x + \frac{2 \Delta}{n} \right)}}{L_{\left( x + \frac{\Delta}{n} \right)}} \right] + \log \left[ \frac{L_{\left( x + \frac{3 \Delta}{n} \right)}}{L_{\left( x + \frac{ 2 \Delta}{n} \right)}} \right] + \cdots \qquad(2)\] and, if we take \(n\) infinitely large, \[\log \frac{L_{(x+\Delta)}}{L_{(x)}} = \int\limits_x^{x+\Delta} \frac{d.L_{(x)}}{L_{(x)}}.\qquad(3)\] Now \(\displaystyle{\frac{dL_{(x)}}{L_{(x)}}}\) is nothing else but the probability to die in the next moment taken inversely\[\displaystyle{1 - \frac{L_{(x+dx)}}{L_{(x)}} = \frac{L_{(x)} - L_{(x+dx)}}{L_{(x)}} = \frac{-dL_{(x)}}{L_{(x)}}},\] while the logarithm of the probability to live still in the next moment is \(d.L_{(x)}\). \(\ldots\) If we denote the probability to die in the next moment by the derived function \(\phi_{(x)}' \, dx \ldots\)

Lazarus (1862, p.285)

Lazarus has here all the ingredients of the product integral. I make three observations:

1. If we start at the end above, substituting the modern \(\mu_x\) hazard rate notation for Lazarus's \(\phi_{(x)}'\), the statement following (3) says that\[ \frac{-dL_{(x)}}{L_{(x)}} = \mu_x \, dx\]is the probability of death "in the next moment".
2. The same statement lets us write (1) (defining \(h = \Delta/n\) for brevity) as:\[\frac{L_{(x+\Delta)}}{L_{(x)}} \approx \left( 1 - \mu_x \, h \right) \, \left( 1 - \mu_{x+h} \, h \right) \, \left( 1 - \mu_{x+2h} \, h \right) \cdots.\qquad(4)\]
3. Equation (2) uses the first-order Taylor approximation \(\log( 1 + x) \approx x\) for small \(x\).

Then all Lazarus had to do (in 1862, remember) was exponentiate (3) and formally pass to the limit in (4) and he would have had the fundamental product-integration identity from survival analysis:

\[ \prod\limits_0^{\Delta} \big( 1 - \mu_{x+t} \, dt \big) = \exp \left( - \int \limits_0^{\Delta} \mu_{x+t} \, dt  \right). \]

Compare the above with Macdonald & Richards (2025), equation (21) and Appendix B. Except for notation the derivation is almost identical.  The first application of the product-integral is usually credited to Volterra (1887) in connection with solving differential equations. Its first known use in survival analysis may have been in Arley (1943), a PhD thesis from the University of Copenhagen, see Andersen et al. (1993). So, although there is much else in the letter we could discuss, I will end by observing that Lazarus was so far ahead of his time, that it would take a century or so for his ideas to be resurrected.