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New monads/MonadRandom

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(Connection to stochastics)
 
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A simple monad transformer to allow computations in the transformed monad to generate random values.
 
A simple monad transformer to allow computations in the transformed monad to generate random values.
 
==The code==
 
==The code==
<haskell>
+
<haskell>{-#LANGUAGE MultiParamTypeClasses, UndecidableInstances #-}
{-# LANGUAGE MultiParamTypeClasses, UndecidableInstances, GeneralizedNewtypeDeriving, FlexibleInstances #-}
+
{-#LANGUAGE GeneralizedNewtypeDeriving, FlexibleInstances #-}
 
 
 
module MonadRandom (
 
module MonadRandom (
Line 26: Line 26:
 
import Control.Monad.Reader
 
import Control.Monad.Reader
 
import Control.Arrow
 
import Control.Arrow
+
 
class (Monad m) => MonadRandom m where
 
class (Monad m) => MonadRandom m where
getRandom :: (Random a) => m a
+
getRandom :: (Random a) => m a
getRandoms :: (Random a) => m [a]
+
getRandoms :: (Random a) => m [a]
getRandomR :: (Random a) => (a,a) -> m a
+
getRandomR :: (Random a) => (a,a) -> m a
 
getRandomRs :: (Random a) => (a,a) -> m [a]
 
getRandomRs :: (Random a) => (a,a) -> m [a]
 
 
newtype (RandomGen g) => RandT g m a = RandT (StateT g m a)
+
newtype RandT g m a = RandT (StateT g m a)
 
deriving (Functor, Monad, MonadTrans, MonadIO)
 
deriving (Functor, Monad, MonadTrans, MonadIO)
 
 
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instance (Monad m, RandomGen g) => MonadRandom (RandT g m) where
 
instance (Monad m, RandomGen g) => MonadRandom (RandT g m) where
getRandom = RandT . liftState $ random
+
getRandom = RandT $ liftState random
getRandoms = RandT . liftState $ first randoms . split
+
getRandoms = RandT $ liftState $ first randoms . split
getRandomR (x,y) = RandT . liftState $ randomR (x,y)
+
getRandomR (x,y) = RandT $ liftState $ randomR (x,y)
getRandomRs (x,y) = RandT . liftState $
+
getRandomRs (x,y) = RandT $ liftState $
 
first (randomRs (x,y)) . split
 
first (randomRs (x,y)) . split
 
 
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runRand :: (RandomGen g) => Rand g a -> g -> (a, g)
 
runRand :: (RandomGen g) => Rand g a -> g -> (a, g)
runRand (Rand x) g = runIdentity (runRandT x g)
+
runRand (Rand x) g = runIdentity (runRandT x g)
 
 
 
evalRandIO :: Rand StdGen a -> IO a
 
evalRandIO :: Rand StdGen a -> IO a
 
evalRandIO (Rand (RandT x)) = getStdRandom (runIdentity . runStateT x)
 
evalRandIO (Rand (RandT x)) = getStdRandom (runIdentity . runStateT x)
+
 
fromList :: (MonadRandom m) => [(a,Rational)] -> m a
 
fromList :: (MonadRandom m) => [(a,Rational)] -> m a
 
fromList [] = error "MonadRandom.fromList called with empty list"
 
fromList [] = error "MonadRandom.fromList called with empty list"
 
fromList [(x,_)] = return x
 
fromList [(x,_)] = return x
fromList xs = do let s = fromRational $ sum (map snd xs) -- total weight
+
fromList xs = do
cs = scanl1 (\(x,q) (y,s) -> (y, s+q)) xs -- cumulative weight
+
let total = fromRational $ sum (map snd xs) :: Double -- total weight
p <- liftM toRational $ getRandomR (0.0,s :: Double)
+
cumulative = scanl1 (\(x,q) (y,s) -> (y, s+q)) xs -- cumulative weights
return . fst . head $ dropWhile (\(x,q) -> q < p) cs
+
p <- liftM toRational $ getRandomR (0.0, total)
  +
return $ fst . head . dropWhile (\(x,q) -> q < p) $ cumulative
 
</haskell>
 
</haskell>
   
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</haskell>
 
</haskell>
 
   
 
== Connection to stochastics ==
 
== Connection to stochastics ==
Line 149: Line 148:
 
In Haskell we have both options either computing with outcomes
 
In Haskell we have both options either computing with outcomes
 
<haskell>
 
<haskell>
do x <- rx
+
do x <- rx
y <- ry
+
y <- ry
return (x+y)
+
return (x+y)
 
</haskell>
 
</haskell>
 
or computing with random variables
 
or computing with random variables

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
can be considered as "
rx
is a random variable". In the do-notation the line
x <- rx
means that "
x
is an outcome of
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
This means that
liftM
like functions convert ordinary arithmetic into

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}.

[edit] 3 See also