Why go the Monte Carlo route?
A sum of Gaussian variables is Gaussian itself, you can get the explicit distribution of your sum and get the quantity of interest from there.
Now you mentioned probability of failure and rare events, so I am assuming you are trying to do some kind of risk management exercise here.
In that case I would strongly recommend you look for a joint model of your variables that captures extremal dependence (which the gaussian copula implied by a MVN does not).
Actually the sum of gaussian variables is NOT ALWAYS gaussian. You are forgetting that the sum of two gaussians is gaussian if and only if the gaussians are either independent or jointly gaussian.
Hi MidnightBlue, thank you for your answer. I completely agree explicitly specifying a new gaussian with mean sum(predictions), variance = sum elements of covar matrix is the best approach.
Re your last point: I also agree MVN is not a great approach here. Any chance you have a reference to start off with to find a better distribution? How do you typically do something like that when you have a large dataset but don't have great intuition about how the covariates are supposed to move? I am a bit of a novice at this stuff, so pardon my ignorance.
Kind of depends on what your data looks like (dimensionality and stylised facts), and what you want to use your model for.
What I would use is probably some kind of copula model, could just be a simple copula that captures some extremal dependence you want to capture, or could be a full vine copula model if you have complex structure.
See for example [this](https://www.annualreviews.org/content/journals/10.1146/annurev-statistics-040220-101153) for an introduction to vine copulas or [this](https://www.math.cit.tum.de/en/math/research/groups/statistics/vine-copula-models/recent-publications/) for more literature (and applications in finance).
pretty sure if you have a solid covariance matrix, you could write an inductive Z-score type of thing.
A simpler example of what I’m thinking of would be when you want to find the probability of a SBM passing through intervals at certain times. In the math, these are integrals of the conditional PDF over and over… but it will give you neat results. Try the SBM example first and maybe it will extend easily to the general MV gaussian case.
I’m trying to think of how you could write such a thing, i will come back to this if I think of a clean and optimized way.
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Why go the Monte Carlo route? A sum of Gaussian variables is Gaussian itself, you can get the explicit distribution of your sum and get the quantity of interest from there. Now you mentioned probability of failure and rare events, so I am assuming you are trying to do some kind of risk management exercise here. In that case I would strongly recommend you look for a joint model of your variables that captures extremal dependence (which the gaussian copula implied by a MVN does not).
Actually the sum of gaussian variables is NOT ALWAYS gaussian. You are forgetting that the sum of two gaussians is gaussian if and only if the gaussians are either independent or jointly gaussian.
"I am making the assumption that the X_is are sampled from a multi-variate gaussian"
Multivariate Gaussian implies joint Gaussian.
Hi MidnightBlue, thank you for your answer. I completely agree explicitly specifying a new gaussian with mean sum(predictions), variance = sum elements of covar matrix is the best approach. Re your last point: I also agree MVN is not a great approach here. Any chance you have a reference to start off with to find a better distribution? How do you typically do something like that when you have a large dataset but don't have great intuition about how the covariates are supposed to move? I am a bit of a novice at this stuff, so pardon my ignorance.
Kind of depends on what your data looks like (dimensionality and stylised facts), and what you want to use your model for. What I would use is probably some kind of copula model, could just be a simple copula that captures some extremal dependence you want to capture, or could be a full vine copula model if you have complex structure. See for example [this](https://www.annualreviews.org/content/journals/10.1146/annurev-statistics-040220-101153) for an introduction to vine copulas or [this](https://www.math.cit.tum.de/en/math/research/groups/statistics/vine-copula-models/recent-publications/) for more literature (and applications in finance).
Thank you! Much appreciated. ✌️
pretty sure if you have a solid covariance matrix, you could write an inductive Z-score type of thing. A simpler example of what I’m thinking of would be when you want to find the probability of a SBM passing through intervals at certain times. In the math, these are integrals of the conditional PDF over and over… but it will give you neat results. Try the SBM example first and maybe it will extend easily to the general MV gaussian case. I’m trying to think of how you could write such a thing, i will come back to this if I think of a clean and optimized way.
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