Would stochastic calculus be a useful approach in actuarial prediction of life expectancies of mice?
(And this is why I am pleased to see this high on HN.)
Because you have as many questions (loci) as you have segments that you can reasonably expect to divide time into (changing the time of death by 1/50th of a mouse lifespan would be impossible to detect unless I am wrong?), and because the time intervals are not that numerous, and also because you wouldn't really have a model for the interaction of the state variables and would be using model-free statistical methods, I think you would get all of the value there is to get out of noncontinuous methods.
https://en.wikipedia.org/wiki/Lasso_(statistics)
Since these are ages, I wouldn't assume an underlying Gaussian distribution. Making that change isn't as hard as you think.
https://en.wikipedia.org/wiki/Generalized_linear_model
As Always: Consult your friendly neighborhood statistician
It's been a number of years since I've looked at these things, but https://www.theactuary.com/2024/04/04/bayesian-revolution and https://arxiv.org/abs/2310.14888 are recent articles that may be relevant.
I think stochastic calculus looks at a system whose output value is a smooth/real value. Basically, it is for modeling systems like random walks where there is a little bit of random up-and-down jumping in each interval. However, if you are basically looking time versus dead-or-alive, your output is binary and time-of-death is really all the info you get and you wouldn't need/want a random walk model, just a more ordinary statistical model. Maybe if there was some other variable besides dead-or-alive you were measuring or aware of a stochastic model could help then (which is a bit like saying "if we had bacon, we could have bacon-and-eggs, if we had eggs").
Also, if what you're saying is you have 50*X bytes of information that all influence life expectancy, it sounds like a challenging problem. But also it's kind of Taylor-made for neural networks; many discreet inputs versus a single smooth output. You might try a neural network and linear model and see how much better the neural network is - then you could determine if more complex-than-linear interactions were occurring.
In more practical terms, if I were to approach this problem, I'd discretize it in time and apply classical ml to predict "chance to die during month X assuming you survived that long" and fit it to data - that'd be much easier to spot errors and potential issues with your data.
I'd go for the stochastic calculus or actual survival analysis only if you wanted to prove/draw a connection between some pre-existing mathematical properly such as memory-less-ness and a physical/biological properly of a system such as behavior of certain proteins (that'd be insanely cool, but rather hard, esp if data is limited). In my (very vague) understanding, that's what finance papers that use stochastic analysis do - they make a mathematical assumption about some universal mathematical properly of a system (if markets were always near optimal with probability of deviation decaying as XYZ, the world economy would react this way to these things), and then prove that it actually fits the data.
Happy to chat more, sounds like a fun project :)
For example, your DNA loci of interest could have a state (methylated or unmethylated). And you could come up with a stochastic process where death occurs when a function of methylation changes at those loci (e.g. a linear model) crosses a threshold (first passage in stochastic process jargon).
Omer Karin & Uri Alon have published a similar concept to explain how the decreased capacity of immune cells to remove senescent cells leads to a Gompertz-like law of longevity, something that originates from actuarial studies! Their model is simpler as they deal with a univariate problem [1].
That said, here are some folks trying to use SDEs to model cells, they even have a "dW" on their logo. This is a long way from predicting age of death, but it might eventually give insights into the exact mechanism. Also I think they're starting with bacteria and yeast, so mice might be a way off.
eg https://www.soa.org/globalassets/assets/Files/static-pages/r...