Abstract
This paper uses Shannon’s entropy index to the base 2 to quantify the risk relative to the age at death in terms of bits (i.e. the amount of information revealed by tossing a fair coin). We first provide a simple decomposition of Shannon’s lifetime entropy index that allows us to analyse the determinants of lifetime entropy (in particular its relation with Wiener’s entropy of the event “death at a particular age conditional on survival to that age”) and to study how the risk about the duration of life is resolved as the individual becomes older. Then, using data on 37 countries from the Human Mortality Database, we show that, over the last two centuries, (period) lifetime entropy at birth has exhibited, in all countries, an inverted-U shape pattern with a maximum in the first half of the twentieth century (at 6 bits), and reaches, in the early twenty-first century, 5.6 bits for men and 5.5 bits for women. It is also shown that the entropy age profile shifted from a non-monotonic profile (in the eighteenth and nineteenth centuries) to a strictly decreasing profile (in the twentieth and twenty-first centuries).
Similar content being viewed by others
Notes
Throughout this paper, the maximal age is fixed to 115 years (at that age, the life table is closed, i.e. \(d_{115}=1\)).
See “Appendix” section on the relationship between Shannon’s lifetime entropy index defined to the base 2 and other entropy indicators.
An axiomatization of Wiener’s entropy index can be found in Aczél and Daroczy (1975). Those authors show that Wiener’s entropy \(W(p)=-\,\log _{2}(p)\) is the unique function F(p) satisfying 3 axioms: (1) non-negativity \(F(p)\ge 0\); (2) additivity: \(F(pq)=F(p)+F(q)\); (3) normalization: \(F(1/2)=1.\)
Note that this is only one aspect of mental attitudes towards death. See Godelier (2018) on the representation of death in 14 religious systems of thought.
In his history of mental attitudes towards death, Ariès (1975) distinguishes between 4 steps: (1) death as a familiar event (mort familière); (2) emergence of individualism of death (mort de soi ); (3) emergence of a concern for joint survival (mort de toi); (4) death as a taboo (mort interdite). In the above paragraph, we are concerned with the progressive shift from (1) to (4), and we deliberately leave dimensions (2) and (3) aside.
Note that Shannon’s entropy index is defined for all kinds of distributions, including these for which outcomes are not equally likely. The distribution with equally likely outcomes is just one case among many others. An interesting feature of that particular case is that, for a distribution with \(n\) possible outcomes, it is when those outcomes are equally likely, and thus arise with a probability 1 / n, that entropy is maximal. That property was part of Khinchin’s (1957) early axiomatization of the entropy index (for all bases).
Simple axiomatizations of Shannon’s entropy index defined to the base 2 can be found in Aczél and Daroczy (1975). A previous axiomatization, for more general entropy indexes (any base) can be found in Khinchin (1957). Khinchin (1957) shows that the general entropy index can be derived from three conditions on the (continuous) function \(G\left( p_{1},\ldots ,p_{n}\right) \): (1) \(G\left( \cdot \right) \) takes its maximum at \(\left( \frac{1}{n},\ldots , \frac{1}{n}\right) \); (2) \(G(AB)=G(A)+G(B)\); (3) \(G\left( p_{1},\ldots ,p_{n},0\right) =G\left( p_{1},\ldots ,p_{n}\right) \).
This property was made explicit in Khinchin’s (1957) early axiomatic characterization of the general entropy index. In the case of Shannon’s lifetime entropy index, it is easy to check this by derivating the values of all \(p_{i,k}\) that maximize \(H_{k}\) subject to the constraint that the \(p_{i,k}\) sum up to 1. The solution is given by each \(p_{i,k}=\frac{1}{115-k}.\) When \(k=0\), \(p_{i,k}=\frac{1}{115}\), which yields the maximal lifetime entropy at birth, equal to 6.905 bits.
In line with Khinchin (1957), adding or subtracting a scenario that has no chance to occur does not affect the value of the entropy index.
Section 4 will consider the evolution of lifetime entropy by age, to examine the speed at which the risk about the duration of life is resolved as individuals become older.
It should be stressed that the stylized facts discussed in this section are observed for all countries for which we have long time series (see “Appendix” section for other countries).
One may argue that, in preindustrial times, there was no lifecycle view, because of the high level of mortality, which shortened the time horizon. This is precisely what is captured by the index \(H_{0}\), which computes the expected amount of information revealed by a death at age \(i\ge 0\), while weighting each possible scenario by its probability of occurrence.
Death being less concentrated at some ages, the event of each death reveals, on average, more information than before.
It should be stressed, however, that the increase in the probabilities to reach higher and higher ages also affects the comparison of lifetime entropy, by adding, at ages above 90, extra terms with a relatively low \(s_{i,k}\) leading to a relatively high \(W\left( s_{i,k}\right)\), and also a relatively high \(d_{k}\), leading to low \(W(d_{k})\) at very high ages.
That first observation, which illustrates the discussion in Sect. 2.3, can be made for all countries in the sample for which we have a sufficiently long time series.
This can be explained by the fact that cohort life tables for those cohorts exhibited probabilities to reach ages beyond 95 years that were often close to zero or equal to zero when the life table was closed at a maximum age equal to 95. Note that the fact that life tables were closed at lower ages is not a pure measurement artifact: it reflects that surviving beyond those ages was, at those distant epochs, extremely rare.
That puzzle goes as follows. Despite the large probability (about 30 to 50 percents) to enter a nursing home at the old age (Brown and Finkelstein 2009), and despite the large costs of LTC, few people purchase private LTC insurance. According to Brown et al. (2007), only 9 to 10 percents of the population at risk of LTC purchase a private insurance in the US.
References
Aczél J, Daroczy Z (1975) On Measures of information and their characterizations. Mathematics in science and engineering, vol 115. Elsevier, Amsterdam
Ariès P (1975) Essais sur l’histoire de la mort en occident. Editions du Seuil, Paris
Brouwer W, van Exel N (2005) Expectations regarding length and health related quality of life: some empirical findings. Soc Sci Med 61(5):1083–1094
Brown J, Finkelstein A (2007) Why is the market for long-term care insurance so small? J Public Econ 91(10):1967–1991
Brown J, Finkelstein A (2009) The private market for long term care in the USA review of the evidence. J Risk Insur 76(1):5–29
Brown J, Finkelstein A (2011) Insuring long term care in the United States. J Econ Perspect 25(4):119–142
Brown J, Coe N, Finkelstein A (2007) Medicaid crowd-out of private long-term care insurance demand: evidence from the health and retirement survey. NBER Chapters. In: Tax policy and the economy, volume 21. National Bureau of Economic Research, pp 1–34
Cheung SL-K, Robine JM, Tu EJ-C, Caselli G (2005) Three dimensions of the survival curve: horizontalization, verticalization and longevity extension. Demography 42(2):243–258
Clark G (2007) A farewell to alms. A brief economic history of the world. Princeton University Press, Princeton
de la Croix D, Sommacal A (2009) A theory of medical effectiveness, differential mortality, income inequality and growth for pre-industrial England. Math Popul Stud 16:1–34
Demetrius L (1976) Measures of variability in age-structured populations. J Theor Biol 63:397–404
Dormont B, Samsonz A-L, Fleurbaey M, Luchini S, Schokkaert E, Thebaut C, Van de Voorde C (2018) Individual uncertainty on longevity. Demography 55:1829–1854
Easterlin R (1999) How beneficient is the market? a look at the modern history of mortality. Eur Rev Econ Hist 3:257–294
Edwards R (2013) The cost of uncertain life span. J Popul Econ 26(4):1485–1522
Edwards R, Tuljapurkar S (2005) Inequality in life spans and a new perspective on mortality convergence across industrialized countries. Popul Dev Rev 31(4):645–674
Fogel R (1994) Economic growth, population theory and physiology: the bearing of long-term processes on the making of economic policy. Am Econ Rev 84(3):369–395
Galor O (2012) The demographic transition: causes and consequences. Cliometrica 6(1):1–28
Gigerenzer G, Hertwig R, Van Den Broek E, Fasolo B, Katsikopoulos K (2005) A 30 percent chance of rain tomorrow: how does the public understand probabilistic weather forecasts? Risk Anal 25(3):623–629
Godelier M (2018) La mort et ses au-delà. Editions du CNRS, Paris
Hamermesh D (1985) Expectations, life expectancy, and economic behavior. Q J Econ 100(2):389–408
Hill G (1993) The entropy of the survival curve: an alternative measure. Can Stud Popul 20(1):43–57
Human Mortality Database (2016) University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). www.mortality.org
Jankélévitch W (1977) La mort. Flammarion, Paris
Kannisto V (2000) Measuring the compression of mortality. Demogr Res 3(6):1
Keyfitz N (1977) Applied mathematical demography. Wiley, New York
Khinchin AI (1957) Mathematical foundations of information theory, 2017th edn. Dover, London
Lan Karen Cheung S, Robine JM (2007) Increase in common longevity and the compression of mortality: the case of Japan. Popul Stud 61(1):85–97
Lanciani R (1892) Gambling and cheating in ancient rome. N Am Rev 155(428):97–105
Lee R (2003) The demographic transition: three centuries of fundamental change. J Econ Perspect 17(4):167–190
Mirowsky J (1999) Subjective life expectancy in the US: correspondence to actuarial estimates by age, sex and race. Soc Sci Med 49(7):967–979
Nagnur D (1986) Rectangularization of the survival curve and entropy: the Canadian experience, 1921–1981. Can Stud Popul 13(1):83–102
Noymer A, Coleman C (2014) A universal pattern of the evolution of life table entropy and life expectancy. In: Extended abstracts. Population Association of America, Boston, pp 1–7
Pierce J (1980) An introduction to information theory. Symbols, signals and noise. Dover, London
Shannon C (1948) A mathematical theory of communication. Bell Syst Tech J 27(3):379–423
Smits J, Monden C (2009) Length of life inequality around the globe. Soc Sci Med 68(6):1114–1123
Tuljapurkar S, Edwards R (2011) Variance in death and its implications for modeling and forecasting mortality. Demogr Res 24:497
Tversky A, Kahneman D (1975) Judgement under uncertainty: heuristics and biases. In: Wendt D, Vlek CA (eds) Utility, probability and human decision making. Springer, Berlin, pp 141–162
Van Raalte A, Caswell H (2013) Perturbation analysis of indices of lifespan variability. Demography 50(5):1615–1640
Vaupel J, Zhang Z, van Raalte A (2011) Life expectancy and disparity: an international comparison of life table data. BMJ Open 1(1):e000128
Wiener N (1965) Cybernetics or control and communication in the animal and the machine. MIT Press, Cambridge
Wilmoth J, Horiuchi S (1999) Rectangularization revisited: variability of age at death within human populations. Demography 36(4):475–495
Wrigley EA (1969) History and population. Weidenfeld & Nicolson, London
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors would like to thank Bertrand Villeneuve, Joel Sobel, Carlo-Giovanni Camarda, Aline Désesquelles, Magali Barbieri and participants of the Mortality Unit Seminar at INED, as well as two anonymous referees, for their helpful comments and suggestions on this paper.
Appendix
Appendix
1.1 Related entropy indicators
Among indicators of verticalization of survival curves, entropy indexes have also been widely used to measure the variance of the age at death.
The entropy indicator that is the closest to the one studied in this paper is the lifetime entropy index developed by Hill (1993), and which was applied to measure lifetime entropy in Canada in the twentieth century. Hill’s (1993) entropy of the age at death is defined as:
The unique difference between Shannon’s lifetime entropy index \(H_{k}\) and the \(HI_{k}\) index lies in the fact that \(HI_{k}\) is based on the natural logarithm, and thus consists of an entropy index defined to the base e, whereas Shannon’s entropy index relies on the logarithm defined to the base 2. One can thus rewrite Shannon’s lifetime entropy index in terms of Hill’s entropy index as follows:
This formula shows that Shannon’s entropy index defined to the base 2 can be rewritten in terms of Hill’s entropy index, while rescaling the levels in such a way as to quantify lifetime entropy in terms of bits. This rescaling suggests that, if one focuses only on the growth patterns of lifetime entropy over time, the obtained growth rates are invariant to the measure unit chosen, so that measuring lifetime entropy by means of \(H_{k}\) or \(HI_{k}\) does not make a difference.
However, focusing on the growth patterns of lifetime entropy is not the only possible use of that measure; one may want to be able to interpret the levels of the measure of life riskiness at a particular point in time or at a particular age. From that perspective, adopting the base 2 rather than the base e allows to quantify risk about the duration of life in a metric that makes life riskiness commensurable with the risk involved in tossing a fair coin. On the contrary, the measurement unit of the \(HI_{k}\) index is less easy to connect with common situations of risk. Moreover, relying on the base 2 rather than on the base e allows us also to decompose the lifetime entropy index in terms of Wiener’s entropy of the single event “death at age \(k\) conditionally on survival to age \(k\)”. Wiener’s entropy being defined to the base 2, it is more natural, when constructing a lifetime measure of the risk about the duration of life, to rely also on the base 2, without any re-normalization.
Having compared our indicator with Hill’s entropy indicator, it is also useful to compare Shannon’s lifetime entropy index with Keyfitz’s entropy, which is the most widely used entropy index in demography (see Keyfitz 1977; Demetrius 1976; Nagnur 1986; Noymer and Coleman 2014). Keyfitz’s entropy is:
The denominator of Keyfitz’s entropy is merely the life expectancy at age \(k\) , whereas the numerator aggregates, along the entire lifecycle, the logarithmic transform of the unconditional survival probabilities to the different ages of life.
It should be stressed that Keyfitz’s life table entropy or population entropy is not, mathematically speaking, an entropy index. It measures the reactivity or elasticity of life expectancy to a proportional change of the strength of age-specific mortality. As such, it differs substantially from the lifetime entropy index that we propose in this paper. Actually, Shannon’s lifetime entropy index can be rewritten in terms of Keyfitz’s entropy as:
where \(L_{k}\) is the life expectancy at age \(k\). Given the important difference between Shannon’s lifetime entropy and Keyfitz’s population entropy, it is not surprising that these two indicators exhibit quite different patterns over time.
To illustrate those differences, Fig. 8 below compares, for the period life tables for France (males), the pattern of Shannon’s lifetime entropy index to the base 2 (left scale) with Keyfitz’s entropy index (right scale). The comparison of those two curves is made difficult by the fact that these indicators have completely different scales. Whereas the Keyfitz entropy index relies, over the period, between 0.15 and 0.6, the Shannon entropy index varies between 5.4 and 6.2. Having stressed this difficulty, one can nonetheless make some observations regarding those two patterns. First, if we focus on the second part of the twentieth century, the two entropy indexes exhibit both a declining trend. However, if we focus on the period between 1890 and the Second World War, we can see that the two indicators show completely different patterns. Whereas Keyfitz’s entropy exhibits a declining trend, Shannon’s entropy exhibits an increasing trend. Thus the two indicators exhibit quite different patterns, which is not surprising given that these indicators measure two different things.
1.2 Additional material
Rights and permissions
About this article
Cite this article
Meyer, P., Ponthiere, G. Human lifetime entropy in a historical perspective (1750–2014). Cliometrica 14, 129–167 (2020). https://doi.org/10.1007/s11698-019-00185-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11698-019-00185-y