[Haskell-cafe] Monads with "The" contexts?

[oleg at okmij.org]

Tillmann Rendel has correctly noted that the source of the problem is
the correlation among the random variables. Specifically, our
measurement of Sun's mass and of Mars mass used the same rather than
independently drawn samples of the Earth mass. Sharing (which
supports what Functional-Logic programming calls ``call-time choice'')
is indeed the solution. Sharing has very clear physical intuition: it
corresponds to the collapse of the wave function.

	Incidentally, a better reference than our ICFP09 paper is the
greatly expanded JFP paper
	http://okmij.org/ftp/Computation/monads.html#lazy-sharing-nondet

You would also need a generalization of sharing -- memoization -- to
build stochastic functions. The emphasis is on _function_: when
applied to a value, the function may give an arbitrary sample from a
distribution. However, when applied to the same value again, the
function should return the same sample. The general memo combinator is
implemented in Hansei and is used all the time. The following article
talks about stochastic functions (and correlated variables):

	http://okmij.org/ftp/kakuritu/index.html#all-different

and the following two articles show just two examples of using memo:

	http://okmij.org/ftp/kakuritu/index.html#noisy-or
	http://okmij.org/ftp/kakuritu/index.html#colored-balls

The noisy-or example is quite close to your problem. It deals with the
inference in causal graphs (DAG): finding out the distribution of
conclusions from the distribution of causes (perhaps given
measurements of some other conclusions). Since a cause may contribute
to several conclusions, we have to mind sharing. Since the code works
by back-propagation (so we don't have to sample causes that don't
contribute to the conclusions of interest), we have to use memoization
(actually, memoization of recursively defined functions).