# Difference between revisions of "New monads/MonadRandom"

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

```   MonadRandom,
getRandom,
getRandomR,
getRandoms,
getRandomRs,
evalRandT,
evalRand,
evalRandIO,
fromList,
Rand, RandT -- but not the data constructors
) where

```

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}.