[yummyfajitas]

The real idea is that you have an implicit model, specifically a normal distribution. The variance is one of the parameters of the normal distribution (the other being the mean).

A normal distribution is a good implicit model to choose - the central limit theorem and similar laws suggest that lots of other distributions will asymptotically approach it. But it's not always the right choice - e.g., it's a disaster when you have power law tails, or low frequency high amplitude noise.

[Tycho]

could you give some examples or detail about power law tails?

[yummyfajitas]

So the CLT says the sum of random variables with rapidly decaying tails will approach a normal distribution. There are similar results showing that the sum of slowly decaying random variables approaches a stable distribution:

https://en.wikipedia.org/wiki/Stable_distribution

This makes the stable distribution the right answer under some circumstances.

For different test statistics (e.g. max drawdown), you've got similar fat tailed distributions, e.g. GEV:

https://en.wikipedia.org/wiki/Generalized_extreme_value_dist...

As an example of how you might use slowly decaying distributions, consider this example of Cauchy PCA:

http://arxiv.org/pdf/1412.6506v1.pdf

I'm working on an blog post explaining the use of fat tailed distributions for linear regression in a Bayesian context.