New monads/MonadRandom

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A simple monad transformer to allow computations in the transformed monad to generate random values. {-#LANGUAGE MultiParamTypeClasses, UndecidableInstances #-} {-#LANGUAGE GeneralizedNewtypeDeriving, FlexibleInstances #-}

module MonadRandom (

   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

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}.

See also