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whuber
whuber
3:13 PM
@NikeDattani Interesting. I would be surprised not to find some version of this procedure omnipresent among surveyors by the end of the 18th century; and I would not be at all surprised to find special cases in use among Islamic geodesists and astronomers as early as the 10th century CE. I wonder, then, what the OP would understand a full statement of this "law" to be?
 
Nike Dattani
Nike Dattani
3:51 PM
Since I am the OP, perhaps I can answer that. My understanding of the "law of propagation of error" is the formula given in the screenshots in the question.

I understand that it only applies when the variables are independent (correlations are 0). It seems it relies on some calculus-type approximations too, and when I see the term "variance" I'm assuming it's the variance of a Gaussian, but I haven't been able to dig deeply into this, largely because this "law" is not treated in the way that things like the "fundamental theorem of calculus" is (meaning that it's been a challenge to figure …
 
whuber
whuber
4:10 PM
@NikeDattani (Sorry, I didn't notice you were the OP, or I would have addressed you directly!)
 
Nike Dattani
Nike Dattani
@whuber That's no problem at all! The username and picture is certainly very different :)
 
whuber
whuber
@NikeDattani The formula is unrelated to Normal distributions. It easily could have been formulated eons ago for (a) special functions, such as quadratics, or (b) arbitrary functions, but not using notation of the differential calculus. That's why I wonder about how broadly this formula is to be interpreted.
As an example of a special case, there must be something like this law present for linear combinations (of arbitrarily many variables) in Bernoulli's Ars Conjectandi, published posthumously in 1715.
 
Nike Dattani
Nike Dattani
@whuber let me try to understand your last message correctly. Does the first sentence have anything to do with the rest of the message?
 
whuber
whuber
@NikeDattani Sorry--they are separate thoughts.
 
Nike Dattani
Nike Dattani
I see
Because the "normal distribution" part was to do with the standard deviation factors in the formula, and the rest of the thought seemed to be about the function whose derivatives appear in the formula.
 
whuber
whuber
4:16 PM
If the formula is meant in a narrower sense--it must involve concepts of partial differentiation of arbitrary multivariate nonlinear functions--then I suspect Gauss was using it by 1818 in his survey of the Hanover area.
 
Nike Dattani
Nike Dattani
I think if the function is f=A + B, then the uncertainty is sqrt(s_A^2 + s_B^2) where s_A and s_B are the uncertainties in A and B.
 
whuber
whuber
@NikeDattani The power of the formula is that it makes only mild distributional assumptions and is a statement only about low order moments.
 
Nike Dattani
Nike Dattani
@whuber What are the assumptions about the distributions, and what is meant by low order moments?
 
whuber
whuber
It has restrictions: the function f must be differentiable in a neighorhood of its mean and its first-order Taylor series around that mean must be an excellent approximation with high probability. In other words, if f changes a lot in a non-linear way as its arguments are varied throughout their likely values, then this formula may be wildly wrong.
The "low order moments" are the mean and variance.
 
Nike Dattani
Nike Dattani
Ok mean = 1st order, variance = 2nd order. That's what you mean?
"is a statement only about low order moments" -> isn't it only a statement about the 2nd order moments?
 
whuber
whuber
4:21 PM
@NikeDattani Yes. But in higher dimensions these have interesting interpretations. Returning to geodesy, for instance, the first two moments of the earth's mass provide the familiar ellipsoid of revolution model used to approximate its true, hugely complicated shape.
@NikeDattani The variance is a central moment and, as such, is usually thought of as depending on the first moment as well.
 
Nike Dattani
Nike Dattani
Ok that makes sense since the mean is part of the formula to calculate the variance
 
whuber
whuber
@NikeDattani That's right. But there is a formula that doesn't require the mean ;-).
At any rate, conceptually the moments (of distributions as well as solid bodies) are thought to provide additional, subtler corrections to initial crude approximations.
I need to return to work -- thank you for your interesting question.
 
Nike Dattani
Nike Dattani
I just want to make sure I understand properly your concern about how "narrow" or "general" the law is interpreted.

If f=A + B, then the uncertainty is sqrt(s_A^2 + s_B^2) where s_A and s_B are the uncertainties in A and B. You are saying that for f being quadratic, the "law of propagation of uncertainty" was probably known eons ago, and that for f being linear the law would be known since Bernoulli's *Ars Conjectandi* but for f being a general differentiable function it would be known maybe not until Gauss's 1818 survey of Hanover?
 

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