New monads/MonadRandom
From HaskellWiki
(→Connection to stochastics) 

(One intermediate revision by one user not shown) 
Latest revision as of 15:27, 30 October 2011
A simple monad transformer to allow computations in the transformed monad to generate random values.
[edit] 1 The code
{#LANGUAGE MultiParamTypeClasses, UndecidableInstances #} {#LANGUAGE GeneralizedNewtypeDeriving, FlexibleInstances #} module MonadRandom ( MonadRandom, getRandom, getRandomR, getRandoms, getRandomRs, evalRandT, evalRand, evalRandIO, fromList, Rand, RandT  but not the data constructors ) where import System.Random import Control.Monad.State import Control.Monad.Identity import Control.Monad.Writer import Control.Monad.Reader import Control.Arrow class (Monad m) => MonadRandom m where getRandom :: (Random a) => m a getRandoms :: (Random a) => m [a] getRandomR :: (Random a) => (a,a) > m a getRandomRs :: (Random a) => (a,a) > m [a] newtype RandT g m a = RandT (StateT g m a) deriving (Functor, Monad, MonadTrans, MonadIO) liftState :: (MonadState s m) => (s > (a,s)) > m a liftState t = do v < get let (x, v') = t v put v' return x instance (Monad m, RandomGen g) => MonadRandom (RandT g m) where getRandom = RandT $ liftState random getRandoms = RandT $ liftState $ first randoms . split getRandomR (x,y) = RandT $ liftState $ randomR (x,y) getRandomRs (x,y) = RandT $ liftState $ first (randomRs (x,y)) . split evalRandT :: (Monad m, RandomGen g) => RandT g m a > g > m a evalRandT (RandT x) g = evalStateT x g runRandT :: (Monad m, RandomGen g) => RandT g m a > g > m (a, g) runRandT (RandT x) g = runStateT x g  Boring random monad :) newtype Rand g a = Rand (RandT g Identity a) deriving (Functor, Monad, MonadRandom) evalRand :: (RandomGen g) => Rand g a > g > a evalRand (Rand x) g = runIdentity (evalRandT x g) runRand :: (RandomGen g) => Rand g a > g > (a, g) runRand (Rand x) g = runIdentity (runRandT x g) evalRandIO :: Rand StdGen a > IO a evalRandIO (Rand (RandT x)) = getStdRandom (runIdentity . runStateT x) fromList :: (MonadRandom m) => [(a,Rational)] > m a fromList [] = error "MonadRandom.fromList called with empty list" fromList [(x,_)] = return x fromList xs = do let total = fromRational $ sum (map snd xs) :: Double  total weight cumulative = scanl1 (\(x,q) (y,s) > (y, s+q)) xs  cumulative weights p < liftM toRational $ getRandomR (0.0, total) return $ fst . head . dropWhile (\(x,q) > q < p) $ cumulative
To make use of common transformer stacks involving Rand and RandT, the following definitions may prove useful:
instance (MonadRandom m) => MonadRandom (StateT s m) where getRandom = lift getRandom getRandomR = lift . getRandomR getRandoms = lift getRandoms getRandomRs = lift . getRandomRs instance (MonadRandom m, Monoid w) => MonadRandom (WriterT w m) where getRandom = lift getRandom getRandomR = lift . getRandomR getRandoms = lift getRandoms getRandomRs = lift . getRandomRs instance (MonadRandom m) => MonadRandom (ReaderT r m) where getRandom = lift getRandom getRandomR = lift . getRandomR getRandoms = lift getRandoms getRandomRs = lift . getRandomRs instance (MonadState s m, RandomGen g) => MonadState s (RandT g m) where get = lift get put = lift . put instance (MonadReader r m, RandomGen g) => MonadReader r (RandT g m) where ask = lift ask local f (RandT m) = RandT $ local f m instance (MonadWriter w m, RandomGen g, Monoid w) => MonadWriter w (RandT g m) where tell = lift . tell listen (RandT m) = RandT $ listen m pass (RandT m) = RandT $ pass m
You may also want a MonadRandom instance for IO:
instance MonadRandom IO where getRandom = randomIO getRandomR = randomRIO getRandoms = fmap randoms newStdGen getRandomRs b = fmap (randomRs b) newStdGen
[edit] 2 Connection to stochastics
There is some correspondence between notions in programming and in mathematics:
random generator  ~  random variable / probabilistic experiment 
result of a random generator  ~  outcome of a probabilistic experiment 
Thus the signature
rx :: (MonadRandom m, Random a) => m a
x < rx
In a language without higher order functions and using a random
generator "function" it is not possible to work with random variables, it
is only possible to compute with outcomes, e.g. rand()+rand()
. In a
language where random generators are implemented as objects, computing
with random variables is possible but still cumbersome.
In Haskell we have both options either computing with outcomes
do x < rx y < ry return (x+y)
or computing with random variables
liftM2 (+) rx ry
random variable arithmetic. But there is also some arithmetic on random variables which can not be performed on outcomes. For example, given a function that repeats an action until the result fulfills a certain property (I wonder if there is already something of this kind in the standard libraries)
untilM :: Monad m => (a > Bool) > m a > m a untilM p m = do x < m if p x then return x else untilM p m
we can suppress certain outcomes of an experiment. E.g. if
getRandomR (10,10)
is a uniformly distributed random variable between −10 and 10, then
untilM (0/=) (getRandomR (10,10))
is a random variable with a uniform distribution of {−10, …, −1, 1, …, 10}.