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 a couple ways to sample
- In a monad, using a
sampleFrom DevRandom (uniform 1 100) :: IO Int
- As a pure function transforming a functional RNG:
sampleState (uniform 1 100) :: StdGen -> (Int, StdGen)
RVar - samples the random variable from the provided
source of entropy.
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, and then it can be sampled as usual:
do rw <- rwalkIO x <- sampleFrom DevURandom rw y <- sampleFrom DevURandom 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, but there is a gotcha lurking here:
inherit the extreme generality of
lift, so there will almost always
need to be an explicit type signature lurking somewhere in any client
code making use of
RVarT with MTL types. In this example, the
inferred type of
start would be too general to be practical, so the
rwalk explicitly fixes it to
in this case
sample could be replaced with
\x -> runRVarTWith MTL.lift x StdRandom.
rwalk :: Int -> Double -> StdGen -> ([Double], StdGen) rwalk count start gen = evalState (runStateT (sample (replicateM count rwalkState)) gen) start
RVarT, sampling the random variable it defines.
Lift context allows random variables to be defined using a minimal
underlying functor (
Identity is sufficient for "conventional" random
variables) and then sampled in any monad into which the underlying functor
can be embedded (which, for
Identity, is all monads).
The lifting 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.
runRVarT but allowing a user-specified lift operation. 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