An opaque type modeling a "random variable" - a value
which depends on the outcome of some random event.
can be conveniently defined by an imperative-looking style:
normalPair = do u <- stdUniform t <- stdUniform let r = sqrt (-2 * log u) theta = (2 * pi) * t x = r * cos theta y = r * sin theta return (x,y)
OR by a more applicative style:
logNormal = exp <$> stdNormal
Once defined (in any style), there are several ways to sample
- In a monad, using a
runRVar (uniform 1 100) DevRandom :: IO Int
- In a monad, using a
sampleRVar (uniform 1 100) :: State PureMT Int
- As a pure function transforming a functional RNG:
sampleState (uniform 1 100) :: StdGen -> (Int, StdGen)
sampleState = runState . sampleRVar)
RVar - samples the random variable from the provided
source of entropy.
data RVarT m a
A random variable with access to operations in an underlying monad. Useful examples include any form of state for implementing random processes with hysteresis, or writer monads for implementing tracing of complicated algorithms.
rwalkIO :: IO (RVarT IO Double) rwalkIO d = do lastVal <- newIORef 0 let x = do prev <- lift (readIORef lastVal) change <- rvarT StdNormal let new = prev + change lift (writeIORef lastVal new) return new return x
To run the random walk it must first be initialized, after which it can be sampled as usual:
do rw <- rwalkIO x <- sampleRVarT rw y <- sampleRVarT rw ...
The same random-walk process as above can be implemented using MTL types
as follows (using
import Control.Monad.Trans as MTL):
rwalkState :: RVarT (State Double) Double rwalkState = do prev <- MTL.lift get change <- rvarT StdNormal let new = prev + change MTL.lift (put new) return new
Invocation is straightforward (although a bit noisy) if you're used to MTL:
rwalk :: Int -> Double -> StdGen -> ([Double], StdGen) rwalk count start gen = flip evalState start . flip runStateT gen . sampleRVarTWith MTL.lift $ replicateM count rwalkState
RVarT, sampling the random variable it defines.
The first argument lifts the base monad into the sampling monad. This operation must obey the "monad transformer" laws:
lift . return = return lift (x >>= f) = (lift x) >>= (lift . f)
One example of a useful non-standard lifting would be one that takes
State s to another monad with a different state representation (such as
IO with the state mapped to an
embedState :: (Monad m) => m s -> (s -> m ()) -> State s a -> m a embedState get put = \m -> do s <- get (res,s) <- return (runState m s) put s return res
The ability to lift is very important - without it, every
RVar would have
to either be given access to the full capability of the monad in which it
will eventually be sampled (which, incidentally, would also have to be
monomorphic so you couldn't sample one
RVar in more than one monad)
or functions manipulating
RVars would have to use higher-ranked
types to enforce the same kind of isolation and polymorphism.