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"Given that after age 100 the annual probability of dying is about 1/2, the chances of a centenarian living to 122 are incredibly small." Small rant about this

"Given that after age 100 the annual probability of dying is about 1/2, the chances of a centenarian living to 122 are incredibly small." Small rant about this statement. The annual probability of dying is not determined by a fair coin flip. (at over 100 years) It is determined by the physical world. That number, .5 chance of dying is the result of a model to predict the chance that someone over 100 will die. So, to say that "the chances of a centenarian living to 122 are incredibly small.", is to say, the chances of living beyond one hundred are determined by a model's output, and not the physical world. You certainly could say that based on empirical evidence, people hitting over 120 is very unlikely.

munchler 5y ago
The model doesn't even support their conclusion. It's true that the chances of any particular centenarian living that long are "incredibly small", but the picture changes when we think about them as a group: * Chances of any given 100 year-old dying before age 122 (according to this model): 1 - 1/(2 ^ 22) = 1 - 1/4,194,304 = 99.999976%. * Number of centenarians currently alive: ~573,000 * Chances that all living centenarians will die before age 122: (1 - 1/4,194,304) ^ ~573,000 = ~87%. (I've hand-waved a bit here, but it's close enough.) So, according to this model, there's roughly a 13% chance that at least one current centenarian will live to 122. I would actually say this model is too optimistic.
- Someone 5y ago
  https://en.wikipedia.org/wiki/Gompertz–Makeham_law_of_mortality https://en.wikipedia.org/wiki/Gompertz–Makeham_law_of_mortal...: “The Gompertz–Makeham law states that the human death rate is the sum of an age-dependent component (the Gompertz function, named after Benjamin Gompertz), which increases exponentially with age[2] and an age-independent component (the Makeham term, named after William Makeham)” I think it’s possible they badly described this model. Now, that’s an empirical law. There’s evidence that it overestimates probability of dying for centenarians. A recent article (https://www.demographic-research.org/volumes/vol44/52/default.htm https://www.demographic-research.org/volumes/vol44/52/defaul...) uses a different model, compares it with actuarial data, and concludes: “Based on the Bayesian analysis, there is a greater than 99% probability that the current MRAD of 122 will be broken by 2100. We estimate the probabilities that a person lives to at least age 126, 128, or 130 this century, as 89%, 44%, and 13%, respectively.”
- jobigoud 5y ago
  You may have missed the fact that not all these 573K living centenarians are 100 year olds, only half of them are, the rest are already 101 or more, and of those only half of them are 101, the rest are already older.
  - mkl 5y ago
    Agreed. I think the distribution would be like this (using the fact that the infinite sum of reciprocal powers of 2 is 1, and assuming steady-state population): In [1]: print('\n'.join([f'(age={100+i}, number={573000/2**(i+1):.3g})' for i in range(23)])) (age=100, number=2.86e+05) (age=108, number=1.12e+03) (age=116, number=4.37) (age=101, number=1.43e+05) (age=109, number=560) (age=117, number=2.19) (age=102, number=7.16e+04) (age=110, number=280) (age=118, number=1.09) (age=103, number=3.58e+04) (age=111, number=140) (age=119, number=0.546) (age=104, number=1.79e+04) (age=112, number=69.9) (age=120, number=0.273) (age=105, number=8.95e+03) (age=113, number=35) (age=121, number=0.137) (age=106, number=4.48e+03) (age=114, number=17.5) (age=122, number=0.0683) (age=107, number=2.24e+03) (age=115, number=8.74) I.e. the model predicts that on average if there are 573000 centenarians, ~1 is 118, and the probability of reaching 122 is .0683/573000 ≈ 1.19×10⁻⁷. The current real numbers [1] are 11 at 114, 1 at 115, 2 at 116, and 1 at 117 and 118, which are all a bit lower than the model, but not hugely. Surprisingly different from munchler's numbers though, so I may have something wrong. [1] https://en.wikipedia.org/wiki/List_of_the_oldest_living_people https://en.wikipedia.org/wiki/List_of_the_oldest_living_peop...
mFixman 5y ago
And this doesn't consider the prior of the amount of people reaching 100 each year. If a million people become centenarians each year and the yearly probability of death for one was 1/2, we would expect a to see a 122 year old every 4 years.

munchler 5y ago

The model doesn't even support their conclusion. It's true that the chances of any particular centenarian living that long are "incredibly small", but the picture changes when we think about them as a group: * Chances of any given 100 year-old dying before age 122 (according to this model): 1 - 1/(2 ^ 22) = 1 - 1/4,194,304 = 99.999976%. * Number of centenarians currently alive: ~573,000 * Chances that all living centenarians will die before age 122: (1 - 1/4,194,304) ^ ~573,000 = ~87%. (I've hand-waved a bit here, but it's close enough.) So, according to this model, there's roughly a 13% chance that at least one current centenarian will live to 122. I would actually say this model is too optimistic.

Someone 5y ago
https://en.wikipedia.org/wiki/Gompertz–Makeham_law_of_mortality https://en.wikipedia.org/wiki/Gompertz–Makeham_law_of_mortal...: “The Gompertz–Makeham law states that the human death rate is the sum of an age-dependent component (the Gompertz function, named after Benjamin Gompertz), which increases exponentially with age[2] and an age-independent component (the Makeham term, named after William Makeham)” I think it’s possible they badly described this model. Now, that’s an empirical law. There’s evidence that it overestimates probability of dying for centenarians. A recent article (https://www.demographic-research.org/volumes/vol44/52/default.htm https://www.demographic-research.org/volumes/vol44/52/defaul...) uses a different model, compares it with actuarial data, and concludes: “Based on the Bayesian analysis, there is a greater than 99% probability that the current MRAD of 122 will be broken by 2100. We estimate the probabilities that a person lives to at least age 126, 128, or 130 this century, as 89%, 44%, and 13%, respectively.”
jobigoud 5y ago
You may have missed the fact that not all these 573K living centenarians are 100 year olds, only half of them are, the rest are already 101 or more, and of those only half of them are 101, the rest are already older.
- mkl 5y ago
  Agreed. I think the distribution would be like this (using the fact that the infinite sum of reciprocal powers of 2 is 1, and assuming steady-state population): In [1]: print('\n'.join([f'(age={100+i}, number={573000/2**(i+1):.3g})' for i in range(23)])) (age=100, number=2.86e+05) (age=108, number=1.12e+03) (age=116, number=4.37) (age=101, number=1.43e+05) (age=109, number=560) (age=117, number=2.19) (age=102, number=7.16e+04) (age=110, number=280) (age=118, number=1.09) (age=103, number=3.58e+04) (age=111, number=140) (age=119, number=0.546) (age=104, number=1.79e+04) (age=112, number=69.9) (age=120, number=0.273) (age=105, number=8.95e+03) (age=113, number=35) (age=121, number=0.137) (age=106, number=4.48e+03) (age=114, number=17.5) (age=122, number=0.0683) (age=107, number=2.24e+03) (age=115, number=8.74) I.e. the model predicts that on average if there are 573000 centenarians, ~1 is 118, and the probability of reaching 122 is .0683/573000 ≈ 1.19×10⁻⁷. The current real numbers [1] are 11 at 114, 1 at 115, 2 at 116, and 1 at 117 and 118, which are all a bit lower than the model, but not hugely. Surprisingly different from munchler's numbers though, so I may have something wrong. [1] https://en.wikipedia.org/wiki/List_of_the_oldest_living_people https://en.wikipedia.org/wiki/List_of_the_oldest_living_peop...

mFixman 5y ago

And this doesn't consider the prior of the amount of people reaching 100 each year. If a million people become centenarians each year and the yearly probability of death for one was 1/2, we would expect a to see a 122 year old every 4 years.