This presentation is review of the original paper 'Modelling Frontier Mortality using Bayesian Generalized Additive Models' by Hilton et al. Here in our review, we have simulated the HMD data of 10 years of all 19 countries mentioned in the original database and have obtained the result accordingly.

1. Modelling Frontier Mortality using
Bayesian Generalised Additive Models
MTH535A
Submitted by:
Ashish Dhiman(170156)
Gyanendra
Awasthi(201315)
Pankaj Kumar(170455)
In supervision of:
Prof. Arnab Hazra

2. Motivations
● Applications in policy making and development of government and the
private sector.
● Useful in constructing policies regarding pensions, health care, life
insurance and annuity pricing etc that aids the economy to strive and
thrive optimally.
● Informs the government to make informed decision of fronts like
housing , local developments plans, business planning, innovation and
incubation, marketing etc.

3. Life Expectancy
● Best practice is usually defined as the highest value of life expectancy
globally.
● It has shown sustained increase over many decades and national life
expectancies in different states appear to be converging
● Previously suggested limits to life expectancy tended to be breached
not long after they were proposed.
● Our paper continues with a contrary approach referencing the author
indicating that sustainability of the trend was subject to debate lately.

4. Life Expectancy or Mortality? ...
● Period life expectancy is “a very particular and non-linear summary
measure.
● In order to produce population projections, age-specific rates are
needed in any case.
● Log-mortality rates are preferred to capture diversity of patterns in age-
specific change in mortality across countries.
● Steady rates of change in mortality levels produce steady absolute
increases in life expectancy: linear trend of record life expectancy.

5. Life Expectancy or Mortality?
● Change in life expectancy is a weighted sum of age-specific mortality
improvements.
○ Weights change depending on the level of mortality.
○ Linear improvements in mortality constant across age will result in linear life
expectancy increases
● These weights become more emphasised at older ages as mortality
declines over time.
● In practice, the difference between linear life expectancy growth and
constancy in log-mortality improvements appears to be relatively slight.

6. The Mortality Frontier
● Is a schedule of mortality rates that represents the best achievable outcome by a
national population at a given point in time.
● We consider the frontier as a mortality surface that is lower than, but as close as
possible to, the force of mortality for all national populations of a reasonable size.
● Consistent declines in the hypothetical mortality frontier :
○ Regular stream of continuing progress from advances in income, salubrity, nutrition, eduction,
sanitation, and medicine.
○ Mortality at younger ages drops,progress focus shifts at older ages.
○ With technological progress in economics, we might expect a penalty for innovators in terms of
future progress, as they are unable to borrow ideas from more advanced neighbours.

7. Empirical Mortality Frontier Plot
• This is the standard result obtained by the author
• Human Mortality Database (2019) spanning from 1816 to 2016

8. Empirical formula of Central Mortality Rate
𝑚𝑥𝑡 =
𝐷𝑥𝑡
𝑅𝑥𝑡
Dxt denotes the number of deaths of individual aged between x and x + 1
during years t.
Rxt is the exposure to risk during the same group over that periods ,
measured in terms of person-years lived.
Ages may range from 0 to some maximum age X, with the latest year
denoted by T.
• The formula of central mortality rate (m𝑥𝑡 ) is

9. Mortality Frontier And Mortality
Improvement
• Empirical ‘Frontier’ mortality is defined as the best (lowest) mortality rate at
each year and age among all countries for which data are available.
𝑚𝑥𝑡
∗
= 𝑚𝑖𝑛𝑐(𝑚𝑥𝑡𝑐)
where c indicates a particular country.
• Mortality improvement is measured using log mortality ratios (or improvement
factors) defined as
log(𝑚𝑥𝑡)
log(𝑚𝑥,𝑡−1)
.

10. Existing Works using frontier mortality
Many works has attempted to make use of frontier mortality.
● The major endeavour and assumption were towards long term convergence
towards frontier life expectancy.
● Few have modelled frontier life expectancy and the gap between this and
country using log transform.
● Employment of two-gap model to include males and females and therein
ensure forecast coherence.
● One of the interesting analysis was fitted smooth 2d splines to mortality rate
surfaces to identify the location of minimum mortality.

11. Model (Hilton and et al.) Used in the Article
- • Employs the Bayesian Generalised Additive Model(GAM) to capture both the
frontier mortality surface and deviations from it
• GAMs model target quantities as sums of smooth functions of covariates, with
identifying constraints ensuring such smooths are distinguishable
• Objective is to frontier mortality rates using log run-rate of log martality
improvement ratio
• Has modelled mortality schedules of individual country as deviations from this
frontier experience

12. Likelihood and Use of Bayesian
Hierarchical Framework
• Age specific death counts (𝐷𝑥𝑡) are given negative- binomial distribution.
𝐷𝑥𝑡 = 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑏𝑖𝑛𝑜𝑚𝑖𝑎𝑙 𝑚𝑥𝑡𝑅𝑥𝑡, exp 𝜑
• The log mortality frontier log(𝑚𝑥𝑡) is modelled as:
log 𝑚𝑥𝑡 = 𝑓 𝑥, 𝑡 + 𝑔+
𝑥, 𝑡, 𝑐 + 𝐾𝑡𝑐
𝑓 𝑥, 𝑡 = 𝑠𝜇 𝑥 + 𝑠𝛽 𝑥 𝑡
𝑔+ 𝑥, 𝑡, 𝑐 = 𝑠𝛾
𝑐 𝑥 exp(ℎ(𝑥, 𝑡, 𝑐)
ℎ 𝑥, 𝑡, 𝑐 = 𝑠𝛿
𝑐
𝑥 𝑡 + 𝑠𝜆
𝑐
𝑥 𝑡2

13. Priors and Use of Bayesian Hierarchical
Framework
● In Lee Carter Model ℎ 𝑥, 𝑡, 𝑐 = 𝑠𝛿
𝑐
(𝑥)𝑘𝑡𝑐
○ This model assumes age specific mortality rates either converge to or diverge from the frontier; the direction of
change cannot reverse
○ No longer need to include κtc
● In Currie et. Al. Model ℎ 𝑥, 𝑡, 𝑐 = 𝑠𝜂
𝑐
(𝑥, 𝑡)
○ Provides even greater degree of flexibility
● In this model, all smooth terms are modelled using penalized B-Splines.
𝑠𝜇 𝑥 = 𝐵𝑓(𝑥)𝝁
𝑠𝛽 𝑥 = 𝐵𝑓(𝑥)𝜷
𝑠𝛾
𝑐
𝑥 = 𝐵𝑔(𝑥)𝜸𝒄𝑠𝛿
𝑐
𝑥 = 𝐵𝑔(𝑥)𝜹𝒄
𝑠𝜆
𝑐
𝑥 = 𝐵𝑔(𝑥)𝝀𝒄
𝑠𝜂
𝑐
𝑥, 𝑡 = (𝐵𝑔 𝑥 ⨂𝐵𝑙 𝑡 )𝜼𝒄

14. Specifications of each term used in model
● 𝑓 𝑥, 𝑡 = frontier mortality term
● 𝑔+ 𝑥, 𝑡, 𝑐 =country specific term ensuring that all countries must lie above
the frontier
● 𝐾𝑡𝑐=country-specific period effects term capture year to year variation
● 𝑠𝜇 𝑥 = denotes overall pattern of frontier log mortality
● 𝑠𝛽 𝑥 = denotes age specific pattern of mortality improvement factors
● 𝑠𝛾
𝑐 𝑥 = age specific deviations from frontier
● exp ℎ 𝑥, 𝑡, 𝑐 = denotes changes in deviation over time
● 𝑠𝛿
𝑐
𝑥 = controls the rate of decline or increase of deviations from frontier

15. Model Specification…
● For sake of iterating the elements of frontier model, smooth age-specific
patterns of mortality is included.
● The improvements with respect to smooth age-specific mortality had
been equally put as elements of the model.
The country-specific element is constrained to be positive:
● The coefficients on the age pattern of country-specific deviations are
forced to be positive that ensures the frontier lies below all country
specific surfaces.
● Different choices possible for function describing time evolution of
deviations

16. Model Specification
Without further priors and constraints, the model is unidentified, as the frontier
could lie anywhere below the country specific rates:
● Coefficients on country-specific deviations are penalised so that smaller
values are favoured i.e. Double exponential priors on the the deviations sc
ɣ
● Period effects are constrained to sum to zero and have zero linear and
quadratic components
● In addition, standard smoothness penalties are employed for all spline
coefficients
● Linear and quadratic age-specific functions are trialled for h(x,t)

17. Data And Exploration
● Human Mortality Database(2019) from across the 19 developed countries has
been used
● As it provides the opportunity to jointly model the frontier and individual
country rates
● For modelling 10 years of data from 1996 to 2006 of above countries has
been taken
● Female data only : Men are unlikely to contribute to the frontier given their
higher mortality

18. Exploration and Evaluation
● The linear variant of proposed model and comparator model has been used
where each country is fitted independently
● We assume greater stability in the frontier than in country-specific mortality
● For evaluation RMSE(Room mean square estimation) is chosen as the metric
of comparison.

19. Results: Mortality Frontier Plots
● Log-mortality appears to have declined in the choosen years (1996-2006)
● The rate of decline varies for different ages.
● Empirical frontier log-mortality is not smooth, with considerable variability for
young ages(0-30).
● Restricting ourselves to more recent years, we can observe the pattern of
decline in empirical frontier mortality over time for particular ages.

20. Results
● Country : Denmark
● Duration : 10 years
(1996-2006)
An agreeable estimate of
Mortality Frontier and
Posterior Rate.

21. France and England(&Wales)
France England and Wales

22. Japan, Scotland, USA, West Germany#
# Ordered clockwise