Dear Oleg, You're right. The points boil down to
That assumption (that the deviations are small) is not stated in types and it is hard to see how can we enforce it. and even if it's small, there's corner cases at df/dx = 0 or df/dx = infinity (as you have mentioned.)
Thanks to your advices, I'll look for other ways to set up
probabilistic computations.
2012/7/19
http://en.pk.paraiso-lang.org/Haskell/Monad-Gaussian What do you think? Will this be a good approach or bad?
I don't think it is a Monad (or even restricted monad, see below). Suppose G a is a `Gaussian' monad and n :: G Double is a random number with the Gaussian (Normal distribution). Then (\x -> x * x) `fmap` n is a random number with the chi-square distribution (of the degree of freedom 1). Chi-square is _not_ a normal distribution. Perhaps a different example is clearer:
(\x -> if x > 0 then 1.0 else 0.0) `fmap` n
has also the type G Double but obviously does not have the normal distribution (since that random variable is discrete).
There are other problems
Let's start with some limitation; we restrict ourselves to Gaussian distributions and assume that the standard deviations are small compared to the scales we deal with.
That assumption is not stated in types and it is hard to see how can we enforce it. Nothing prevents us from writing liftM2 n n in which case the variance will no longer be small compared with the mean.
Just a technical remark: The way G a is written, it is a so-called restricted monad, which is not a monad (the adjective `restricted' is restrictive here). http://okmij.org/ftp/Haskell/types.html#restricted-datatypes
-- Takayuki MURANUSHI The Hakubi Center for Advanced Research, Kyoto University http://www.hakubi.kyoto-u.ac.jp/02_mem/h22/muranushi.html