-- Hoogle documentation, generated by Haddock -- See Hoogle, http://www.haskell.org/hoogle/ -- | Random number generation -- -- Random number generation based on modeling random variables in two -- complementary ways: first, by the parameters of standard mathematical -- distributions and, second, by an abstract type (RVar) which can -- be composed and manipulated monadically and sampled in either monadic -- or "pure" styles. -- -- The primary purpose of this library is to support defining and -- sampling a wide variety of high quality random variables. Quality is -- prioritized over speed, but performance is an important goal too. -- -- In my testing, I have found it capable of speed comparable to other -- Haskell libraries, but still a fair bit slower than straight C -- implementations of the same algorithms. @package random-fu @version 0.2.6.2 module Data.Random.Internal.Find findMax :: (Fractional a, Ord a) => (a -> Bool) -> a -- | Given an upward-closed predicate on an ordered Fractional type, find -- the smallest value satisfying the predicate. findMin :: (Fractional a, Ord a) => (a -> Bool) -> a -- | Given an upward-closed predicate on an ordered Fractional type, find -- the smallest value satisfying the predicate. Starts at the specified -- point with the specified stepsize, performs an exponential search out -- from there until it finds an interval bracketing the change-point of -- the predicate, and then performs a bisection search to isolate the -- change point. Note that infinitely-divisible domains such as -- Rational cannot be searched by this function because it does -- not terminate until it reaches a point where further subdivision of -- the interval has no effect. findMinFrom :: (Fractional a, Ord a) => a -> a -> (a -> Bool) -> a module Data.Random.Internal.Fixed resolutionOf :: HasResolution r => f r -> Integer resolutionOf2 :: HasResolution r => f (g r) -> Integer -- | The Fixed type doesn't expose its constructors, but I need a -- way to convert them to and from their raw representation in order to -- sample them. As long as Fixed is a newtype wrapping -- Integer, mkFixed and unMkFixed as defined here -- will work. Both are implemented using unsafeCoerce. mkFixed :: Integer -> Fixed r unMkFixed :: Fixed r -> Integer -- | Template Haskell utility code to replicate instance declarations to -- cover large numbers of types. I'm doing that rather than using class -- contexts because most Distribution instances need to cover multiple -- classes (such as Enum, Integral and Fractional) and that can't be done -- easily because of overlap. -- -- I experimented a bit with a convoluted type-level classification -- scheme, but I think this is simpler and easier to understand. It makes -- the haddock docs more cluttered because of the combinatorial explosion -- of instances, but overall I think it's just more sane than anything -- else I've come up with yet. module Data.Random.Internal.TH -- | replicateInstances standin types decls will take the -- template-haskell Decs in decls and substitute every -- instance of the Name standin with each Name in -- types, producing one copy of the Decs in -- decls for every Name in types. -- -- For example, Uniform has the following bit of TH code: -- -- 
--   $( replicateInstances ''Int integralTypes [d|
--
-- -- 
--   instance Distribution Uniform Int   where rvar (Uniform a b) = integralUniform a b
--
-- -- 
--   instance CDF Uniform Int            where cdf  (Uniform a b) = integralUniformCDF a b
--
-- -- 
--   |])
--
-- -- This code takes those 2 instance declarations and creates identical -- ones for every type named in integralTypes. replicateInstances :: (Monad m, Data t) => Name -> [Name] -> m [t] -> m [t] -- | Names of standard Integral types integralTypes :: [Name] -- | Names of standard RealFloat types realFloatTypes :: [Name] module Data.Random.Lift -- | A class for "liftable" data structures. Conceptually an extension of -- MonadTrans to allow deep lifting, but lifting need not be done -- between monads only. Eg lifting between Applicatives is -- allowed. -- -- For instances where m and n have 'return'/'pure' -- defined, these instances must satisfy lift (return x) == return -- x. -- -- This form of lift has an extremely general type and is used -- primarily to support sample. Its excessive generality is the -- main reason it's not exported from Data.Random. RVarT -- is, however, an instance of MonadTrans, which in most cases is -- the preferred way to do the lifting. class Lift m n lift :: Lift m n => m a -> n a -- | This instance is incoherent with the others. However, by the law -- lift (return x) == return x, the results must always be the -- same. -- | This instance is again incoherent with the others, but provides a -- more-specific instance to resolve the overlap between the Lift m -- (t m) and Lift Identity m instances. instance [incoherent] MonadTrans t => Lift Identity (t Identity) instance [incoherent] Lift (RVarT Identity) (RVarT m) instance [incoherent] Monad m => Lift Identity m instance [incoherent] Lift m m instance [incoherent] (Monad m, MonadTrans t) => Lift m (t m) module Data.Random.RVar -- | An opaque type modeling a "random variable" - a value which depends on -- the outcome of some random event. RVars 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 -- RVars: -- -- -- -- 
--   runRVar (uniform 1 100) DevRandom :: IO Int
--
-- -- -- -- 
--   sampleRVar (uniform 1 100) :: State PureMT Int
--
-- -- -- -- 
--   sampleState (uniform 1 100) :: StdGen -> (Int, StdGen)
--
-- -- (where sampleState = runState . sampleRVar) type RVar = RVarT Identity -- | "Run" an RVar - samples the random variable from the provided -- source of entropy. runRVar :: RandomSource m s => RVar a -> s -> 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. -- -- For example, a simple random walk can be implemented as an -- RVarT IO value: -- -- 
--   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
--
data RVarT (m :: * -> *) a :: (* -> *) -> * -> * -- | Like runRVarTWith, but using an implicit lifting (provided by -- the Lift class) runRVarT :: (Lift n m, RandomSource m s) => RVarT n a -> s -> m a -- | "Runs" an 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 -- IORef): -- -- 
--   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. runRVarTWith :: RandomSource m s => (forall t. n t -> m t) -> RVarT n a -> s -> m a module Data.Random.Distribution -- | A Distribution is a data representation of a random variable's -- probability structure. For example, in -- Data.Random.Distribution.Normal, the Normal -- distribution is defined as: -- -- 
--   data Normal a
--       = StdNormal
--       | Normal a a
--
-- -- Where the two parameters of the Normal data constructor are -- the mean and standard deviation of the random variable, respectively. -- To make use of the Normal type, one can convert it to an -- rvar and manipulate it or sample it directly: -- -- 
--   x <- sample (rvar (Normal 10 2))
--   x <- sample (Normal 10 2)
--
-- -- A Distribution is typically more transparent than an -- RVar but less composable (precisely because of that -- transparency). There are several practical uses for types implementing -- Distribution: -- -- -- -- On the other hand, most Distributions will not be closed under -- all the same operations as RVar (which, being a monad, has a -- fully turing-complete internal computational model). The sum of two -- uniformly-distributed variables, for example, is not uniformly -- distributed. To support general composition, the Distribution -- class defines a function rvar to construct the more-abstract -- and more-composable RVar representation of a random variable. class Distribution d t where rvar = rvarT rvarT d = lift (rvar d) rvar :: Distribution d t => d t -> RVar t rvarT :: Distribution d t => d t -> RVarT n t class Distribution d t => PDF d t where pdf d = exp . logPdf d logPdf d = log . pdf d pdf :: PDF d t => d t -> t -> Double logPdf :: PDF d t => d t -> t -> Double class Distribution d t => CDF d t cdf :: CDF d t => d t -> t -> Double module Data.Random.Distribution.Uniform -- | A definition of a uniform distribution over the type t. See -- also uniform. data Uniform t -- | A uniform distribution defined by a lower and upper range bound. For -- Integral and Enum types, the range is inclusive. For -- Fractional types the range includes the lower bound but not the -- upper. Uniform :: !t -> !t -> Uniform t uniform :: Distribution Uniform a => a -> a -> RVar a uniformT :: Distribution Uniform a => a -> a -> RVarT m a -- | A name for the "standard" uniform distribution over the type -- t, if one exists. See also stdUniform. -- -- For Integral and Enum types that are also -- Bounded, this is the uniform distribution over the full range -- of the type. For un-Bounded Integral types this is not -- defined. For Fractional types this is a random variable in the -- range [0,1) (that is, 0 to 1 including 0 but not including 1). data StdUniform t StdUniform :: StdUniform t -- | Get a "standard" uniformly distributed variable. For integral types, -- this means uniformly distributed over the full range of the type -- (there is no support for Integer). For fractional types, this -- means uniformly distributed on the interval [0,1). stdUniform :: Distribution StdUniform a => RVar a -- | Get a "standard" uniformly distributed process. For integral types, -- this means uniformly distributed over the full range of the type -- (there is no support for Integer). For fractional types, this -- means uniformly distributed on the interval [0,1). stdUniformT :: Distribution StdUniform a => RVarT m a -- | Like stdUniform but only returns positive values. stdUniformPos :: (Distribution StdUniform a, Num a, Eq a) => RVar a -- | Like stdUniform but only returns positive values. stdUniformPosT :: (Distribution StdUniform a, Num a, Eq a) => RVarT m a -- | Compute a random Integral value between the 2 values provided -- (inclusive). integralUniform :: Integral a => a -> a -> RVarT m a -- | realFloatUniform a b computes a uniform random value in the -- range [a,b) for any RealFloat type realFloatUniform :: RealFloat a => a -> a -> RVarT m a -- | floatUniform a b computes a uniform random Float value -- in the range [a,b) floatUniform :: Float -> Float -> RVarT m Float -- | doubleUniform a b computes a uniform random Double -- value in the range [a,b) doubleUniform :: Double -> Double -> RVarT m Double -- | fixedUniform a b computes a uniform random Fixed value -- in the range [a,b), with any desired precision. fixedUniform :: HasResolution r => Fixed r -> Fixed r -> RVarT m (Fixed r) -- | realFloatUniform a b computes a uniform random value in the -- range [a,b) for any Enum type enumUniform :: Enum a => a -> a -> RVarT m a -- | Compute a random value for a Bounded type, between -- minBound and maxBound (inclusive for Integral or -- Enum types, in [minBound, maxBound) for -- Fractional types.) boundedStdUniform :: (Distribution Uniform a, Bounded a) => RVar a -- | Compute a random value for a Bounded Enum type, between -- minBound and maxBound (inclusive) boundedEnumStdUniform :: (Enum a, Bounded a) => RVarT m a -- | Compute a uniform random value in the range [0,1) for any -- RealFloat type realFloatStdUniform :: RealFloat a => RVarT m a -- | Compute a uniform random Fixed value in the range [0,1), with -- any desired precision. fixedStdUniform :: HasResolution r => RVarT m (Fixed r) -- | Compute a uniform random Float value in the range [0,1) floatStdUniform :: RVarT m Float -- | Compute a uniform random Double value in the range [0,1) doubleStdUniform :: RVarT m Double boundedStdUniformCDF :: (CDF Uniform a, Bounded a) => a -> Double -- | The CDF of the random variable realFloatStdUniform. realStdUniformCDF :: Real a => a -> Double -- | realUniformCDF a b is the CDF of the random variable -- realFloatUniform a b. realUniformCDF :: RealFrac a => a -> a -> a -> Double enumUniformCDF :: (Enum a, Ord a) => a -> a -> a -> Double instance CDF StdUniform Ordering instance Distribution StdUniform Ordering instance CDF StdUniform Char instance Distribution StdUniform Char instance CDF StdUniform Bool instance Distribution StdUniform Bool instance CDF StdUniform () instance Distribution StdUniform () instance CDF Uniform Ordering instance Distribution Uniform Ordering instance CDF Uniform Bool instance Distribution Uniform Bool instance CDF Uniform Char instance Distribution Uniform Char instance CDF Uniform () instance Distribution Uniform () instance HasResolution r => CDF StdUniform (Fixed r) instance HasResolution r => Distribution StdUniform (Fixed r) instance HasResolution r => CDF Uniform (Fixed r) instance HasResolution r => Distribution Uniform (Fixed r) instance PDF StdUniform Double instance PDF StdUniform Float instance CDF StdUniform Double instance CDF StdUniform Float instance Distribution StdUniform Double instance Distribution StdUniform Float instance CDF Uniform Double instance CDF Uniform Float instance Distribution Uniform Double instance Distribution Uniform Float instance CDF StdUniform Int instance CDF StdUniform Int64 instance CDF StdUniform Int32 instance CDF StdUniform Int16 instance CDF StdUniform Int8 instance CDF StdUniform Word instance CDF StdUniform Word64 instance CDF StdUniform Word32 instance CDF StdUniform Word16 instance CDF StdUniform Word8 instance Distribution StdUniform Word instance Distribution StdUniform Int instance Distribution StdUniform Int64 instance Distribution StdUniform Int32 instance Distribution StdUniform Int16 instance Distribution StdUniform Int8 instance Distribution StdUniform Word64 instance Distribution StdUniform Word32 instance Distribution StdUniform Word16 instance Distribution StdUniform Word8 instance CDF Uniform Word64 instance Distribution Uniform Word64 instance CDF Uniform Word32 instance Distribution Uniform Word32 instance CDF Uniform Word16 instance Distribution Uniform Word16 instance CDF Uniform Word8 instance Distribution Uniform Word8 instance CDF Uniform Word instance Distribution Uniform Word instance CDF Uniform Int64 instance Distribution Uniform Int64 instance CDF Uniform Int32 instance Distribution Uniform Int32 instance CDF Uniform Int16 instance Distribution Uniform Int16 instance CDF Uniform Int8 instance Distribution Uniform Int8 instance CDF Uniform Int instance Distribution Uniform Int instance CDF Uniform Integer instance Distribution Uniform Integer module Data.Random.List -- | A random variable returning an arbitrary element of the given list. -- Every element has equal probability of being chosen. Because it is a -- pure RVar it has no memory - that is, it "draws with -- replacement." randomElement :: [a] -> RVar a randomElementT :: [a] -> RVarT m a -- | A random variable that returns the given list in an arbitrary shuffled -- order. Every ordering of the list has equal probability. shuffle :: [a] -> RVar [a] shuffleT :: [a] -> RVarT m [a] -- | A random variable that shuffles a list of a known length (or a list -- prefix of the specified length). Useful for shuffling large lists when -- the length is known in advance. Avoids needing to traverse the list to -- discover its length. Each ordering has equal probability. shuffleN :: Int -> [a] -> RVar [a] shuffleNT :: Int -> [a] -> RVarT m [a] -- | A random variable that selects N arbitrary elements of a list of known -- length M. shuffleNofM :: Int -> Int -> [a] -> RVar [a] shuffleNofMT :: Int -> Int -> [a] -> RVarT m [a] module Data.Random.Vector -- | Take a random element of a vector. randomElement :: Vector a -> RVar a module Data.Random.Distribution.Bernoulli -- | Generate a Bernoulli variate with the given probability. For -- Bool results, bernoulli p will return True (p*100)% -- of the time and False otherwise. For numerical types, True is replaced -- by 1 and False by 0. bernoulli :: Distribution (Bernoulli b) a => b -> RVar a -- | Generate a Bernoulli process with the given probability. For -- Bool results, bernoulli p will return True (p*100)% -- of the time and False otherwise. For numerical types, True is replaced -- by 1 and False by 0. bernoulliT :: Distribution (Bernoulli b) a => b -> RVarT m a -- | A random variable whose value is True the given fraction of the -- time and False the rest. boolBernoulli :: (Fractional a, Ord a, Distribution StdUniform a) => a -> RVarT m Bool boolBernoulliCDF :: Real a => a -> Bool -> Double -- | generalBernoulli t f p generates a random variable whose -- value is t with probability p and f with -- probability 1-p. generalBernoulli :: Distribution (Bernoulli b) Bool => a -> a -> b -> RVarT m a generalBernoulliCDF :: CDF (Bernoulli b) Bool => (a -> a -> Bool) -> a -> a -> b -> a -> Double newtype Bernoulli b a Bernoulli :: b -> Bernoulli b a instance (CDF (Bernoulli b) Bool, RealFloat a) => CDF (Bernoulli b) (Complex a) instance (Distribution (Bernoulli b) Bool, RealFloat a) => Distribution (Bernoulli b) (Complex a) instance (CDF (Bernoulli b) Bool, Integral a) => CDF (Bernoulli b) (Ratio a) instance (Distribution (Bernoulli b) Bool, Integral a) => Distribution (Bernoulli b) (Ratio a) instance CDF (Bernoulli b0) Bool => CDF (Bernoulli b0) Double instance Distribution (Bernoulli b0) Bool => Distribution (Bernoulli b0) Double instance CDF (Bernoulli b0) Bool => CDF (Bernoulli b0) Float instance Distribution (Bernoulli b0) Bool => Distribution (Bernoulli b0) Float instance CDF (Bernoulli b0) Bool => CDF (Bernoulli b0) Word64 instance Distribution (Bernoulli b0) Bool => Distribution (Bernoulli b0) Word64 instance CDF (Bernoulli b0) Bool => CDF (Bernoulli b0) Word32 instance Distribution (Bernoulli b0) Bool => Distribution (Bernoulli b0) Word32 instance CDF (Bernoulli b0) Bool => CDF (Bernoulli b0) Word16 instance Distribution (Bernoulli b0) Bool => Distribution (Bernoulli b0) Word16 instance CDF (Bernoulli b0) Bool => CDF (Bernoulli b0) Word8 instance Distribution (Bernoulli b0) Bool => Distribution (Bernoulli b0) Word8 instance CDF (Bernoulli b0) Bool => CDF (Bernoulli b0) Word instance Distribution (Bernoulli b0) Bool => Distribution (Bernoulli b0) Word instance CDF (Bernoulli b0) Bool => CDF (Bernoulli b0) Int64 instance Distribution (Bernoulli b0) Bool => Distribution (Bernoulli b0) Int64 instance CDF (Bernoulli b0) Bool => CDF (Bernoulli b0) Int32 instance Distribution (Bernoulli b0) Bool => Distribution (Bernoulli b0) Int32 instance CDF (Bernoulli b0) Bool => CDF (Bernoulli b0) Int16 instance Distribution (Bernoulli b0) Bool => Distribution (Bernoulli b0) Int16 instance CDF (Bernoulli b0) Bool => CDF (Bernoulli b0) Int8 instance Distribution (Bernoulli b0) Bool => Distribution (Bernoulli b0) Int8 instance CDF (Bernoulli b0) Bool => CDF (Bernoulli b0) Int instance Distribution (Bernoulli b0) Bool => Distribution (Bernoulli b0) Int instance CDF (Bernoulli b0) Bool => CDF (Bernoulli b0) Integer instance Distribution (Bernoulli b0) Bool => Distribution (Bernoulli b0) Integer instance (Distribution (Bernoulli b) Bool, Real b) => CDF (Bernoulli b) Bool instance (Fractional b, Ord b, Distribution StdUniform b) => Distribution (Bernoulli b) Bool module Data.Random.Distribution.Categorical -- | Categorical distribution; a list of events with corresponding -- probabilities. The sum of the probabilities must be 1, and no event -- should have a zero or negative probability (at least, at time of -- sampling; very clever users can do what they want with the numbers -- before sampling, just make sure that if you're one of those clever -- ones, you at least eliminate negative weights before sampling). data Categorical p a -- | Construct a Categorical random variable from a list of -- probabilities and categories, where the probabilities all sum to 1. categorical :: (Num p, Distribution (Categorical p) a) => [(p, a)] -> RVar a -- | Construct a Categorical random process from a list of -- probabilities and categories, where the probabilities all sum to 1. categoricalT :: (Num p, Distribution (Categorical p) a) => [(p, a)] -> RVarT m a -- | Construct a Categorical random variable from a list of -- probabilities and categories, where the probabilities all sum to 1. weightedCategorical :: (Fractional p, Eq p, Distribution (Categorical p) a) => [(p, a)] -> RVar a -- | Construct a Categorical random process from a list of -- probabilities and categories, where the probabilities all sum to 1. weightedCategoricalT :: (Fractional p, Eq p, Distribution (Categorical p) a) => [(p, a)] -> RVarT m a -- | Construct a Categorical distribution from a list of weighted -- categories. fromList :: Num p => [(p, a)] -> Categorical p a toList :: Num p => Categorical p a -> [(p, a)] totalWeight :: Num p => Categorical p a -> p numEvents :: Categorical p a -> Int -- | Construct a Categorical distribution from a list of weighted -- categories, where the weights do not necessarily sum to 1. fromWeightedList :: (Fractional p, Eq p) => [(p, a)] -> Categorical p a -- | Construct a Categorical distribution from a list of observed -- outcomes. Equivalent events will be grouped and counted, and the -- probabilities of each event in the returned distribution will be -- proportional to the number of occurrences of that event. fromObservations :: (Fractional p, Eq p, Ord a) => [a] -> Categorical p a -- | Like fmap, but for the probabilities of a categorical -- distribution. mapCategoricalPs :: (Num p, Num q) => (p -> q) -> Categorical p e -> Categorical q e -- | Adjust all the weights of a categorical distribution so that they sum -- to unity and remove all events whose probability is zero. normalizeCategoricalPs :: (Fractional p, Eq p) => Categorical p e -> Categorical p e -- | Simplify a categorical distribution by combining equivalent events -- (the new event will have a probability equal to the sum of all the -- originals). collectEvents :: (Ord e, Num p, Ord p) => Categorical p e -> Categorical p e -- | Simplify a categorical distribution by combining equivalent events -- (the new event will have a weight equal to the sum of all the -- originals). The comparator function is used to identify events to -- combine. Once chosen, the events and their weights are combined by the -- provided probability and event aggregation function. collectEventsBy :: Num p => (e -> e -> Ordering) -> ([(p, e)] -> (p, e)) -> Categorical p e -> Categorical p e instance (Eq p, Eq a) => Eq (Categorical p a) instance Fractional p => Applicative (Categorical p) instance Fractional p => Monad (Categorical p) instance Traversable (Categorical p) instance Foldable (Categorical p) instance Functor (Categorical p) instance (Fractional p, Ord p, Distribution Uniform p) => Distribution (Categorical p) a instance (Num p, Read p, Read a) => Read (Categorical p a) instance (Num p, Show p, Show a) => Show (Categorical p a) module Data.Random.Distribution.Exponential newtype Exponential a Exp :: a -> Exponential a floatingExponential :: (Floating a, Distribution StdUniform a) => a -> RVarT m a floatingExponentialCDF :: Real a => a -> a -> Double exponential :: Distribution Exponential a => a -> RVar a exponentialT :: Distribution Exponential a => a -> RVarT m a instance (Real a, Distribution Exponential a) => CDF Exponential a instance (Floating a, Distribution StdUniform a) => Distribution Exponential a module Data.Random.Distribution.StretchedExponential newtype StretchedExponential a StretchedExp :: (a, a) -> StretchedExponential a floatingStretchedExponential :: (Floating a, Distribution StdUniform a) => a -> a -> RVarT m a floatingStretchedExponentialCDF :: Real a => a -> a -> a -> Double stretchedExponential :: Distribution StretchedExponential a => a -> a -> RVar a stretchedExponentialT :: Distribution StretchedExponential a => a -> a -> RVarT m a instance (Real a, Distribution StretchedExponential a) => CDF StretchedExponential a instance (Floating a, Distribution StdUniform a) => Distribution StretchedExponential a -- | A generic "ziggurat algorithm" implementation. Fairly rough right now. -- -- There is a lot of room for improvement in findBin0 especially. -- It needs a fair amount of cleanup and elimination of redundant -- calculation, as well as either a justification for using the simple -- findMinFrom or a proper root-finding algorithm. -- -- It would also be nice to add (preferably by pulling in an external -- package) support for numerical integration and differentiation, so -- that tables can be derived from only a PDF (if the end user is willing -- to take the performance and accuracy hit for the convenience). module Data.Random.Distribution.Ziggurat -- | A data structure containing all the data that is needed to implement -- Marsaglia & Tang's "ziggurat" algorithm for sampling certain kinds -- of random distributions. -- -- The documentation here is probably not sufficient to tell a user -- exactly how to build one of these from scratch, but it is not really -- intended to be. There are several helper functions that will build -- Ziggurats. The pathologically curious may wish to read the -- runZiggurat source. That is the ultimate specification of the -- semantics of all these fields. data Ziggurat v t Ziggurat :: !(v t) -> !(v t) -> !(v t) -> !(forall m. RVarT m (Int, t)) -> (forall m. RVarT m t) -> !(forall m. t -> t -> RVarT m t) -> !(t -> t) -> !Bool -> Ziggurat v t -- | The X locations of each bin in the distribution. Bin 0 is the -- infinite one. -- -- In the case of bin 0, the value given is sort of magical - x[0] is -- defined to be V/f(R). It's not actually the location of any bin, but a -- value computed to make the algorithm more concise and slightly faster -- by not needing to specially-handle bin 0 quite as often. If you really -- need to know why it works, see the runZiggurat source or "the -- literature" - it's a fairly standard setup. zTable_xs :: Ziggurat v t -> !(v t) -- | The ratio of each bin's Y value to the next bin's Y value zTable_y_ratios :: Ziggurat v t -> !(v t) -- | The Y value (zFunc x) of each bin zTable_ys :: Ziggurat v t -> !(v t) -- | An RVar providing a random tuple consisting of: -- -- -- -- This is provided as a single RVar because it can be implemented -- more efficiently than naively sampling 2 separate values - a single -- random word (64 bits) can be efficiently converted to a double (using -- 52 bits) and a bin number (using up to 12 bits), for example. zGetIU :: Ziggurat v t -> !(forall m. RVarT m (Int, t)) -- | The distribution for the final "virtual" bin (the ziggurat algorithm -- does not handle distributions that wander off to infinity, so another -- distribution is needed to handle the last "bin" that stretches to -- infinity) zTailDist :: Ziggurat v t -> (forall m. RVarT m t) -- | A copy of the uniform RVar generator for the base type, so that -- Distribution Uniform t is not needed when sampling from a -- Ziggurat (makes it a bit more self-contained). zUniform :: Ziggurat v t -> !(forall m. t -> t -> RVarT m t) -- | The (one-sided antitone) PDF, not necessarily normalized zFunc :: Ziggurat v t -> !(t -> t) -- | A flag indicating whether the distribution should be mirrored about -- the origin (the ziggurat algorithm in its native form only samples -- from one-sided distributions. By mirroring, we can extend it to -- symmetric distributions such as the normal distribution) zMirror :: Ziggurat v t -> !Bool -- | Build a lazy recursive ziggurat. Uses a lazily-constructed ziggurat as -- its tail distribution (with another as its tail, ad nauseam). -- -- Arguments: -- -- mkZigguratRec :: (RealFloat t, Vector v t, Distribution Uniform t) => Bool -> (t -> t) -> (t -> t) -> (t -> t) -> t -> Int -> (forall m. RVarT m (Int, t)) -> Ziggurat v t -- | Build the tables to implement the "ziggurat algorithm" devised by -- Marsaglia & Tang, attempting to automatically compute the R and V -- values. -- -- Arguments are the same as for mkZigguratRec, with an additional -- argument for the tail distribution as a function of the selected R -- value. mkZiggurat :: (RealFloat t, Vector v t, Distribution Uniform t) => Bool -> (t -> t) -> (t -> t) -> (t -> t) -> t -> Int -> (forall m. RVarT m (Int, t)) -> (forall m. t -> RVarT m t) -> Ziggurat v t -- | Build the tables to implement the "ziggurat algorithm" devised by -- Marsaglia & Tang, attempting to automatically compute the R and V -- values. -- -- Arguments: -- -- mkZiggurat_ :: (RealFloat t, Vector v t, Distribution Uniform t) => Bool -> (t -> t) -> (t -> t) -> Int -> t -> t -> (forall m. RVarT m (Int, t)) -> (forall m. RVarT m t) -> Ziggurat v t -- | I suspect this isn't completely right, but it works well so far. -- Search the distribution for an appropriate R and V. -- -- Arguments: -- -- -- -- Result: (R,V) findBin0 :: RealFloat b => Int -> (b -> b) -> (b -> b) -> (b -> b) -> b -> (b, b) -- | Sample from the distribution encoded in a Ziggurat data -- structure. runZiggurat :: (Num a, Ord a, Vector v a) => Ziggurat v a -> RVarT m a instance (Num t, Ord t, Vector v t) => Distribution (Ziggurat v) t module Data.Random.Distribution.Normal -- | A specification of a normal distribution over the type a. data Normal a -- | The "standard" normal distribution - mean 0, stddev 1 StdNormal :: Normal a -- | Normal m s is a normal distribution with mean m and -- stddev sd. Normal :: a -> a -> Normal a -- | normal m s is a random variable with distribution -- Normal m s. normal :: Distribution Normal a => a -> a -> RVar a -- | normalT m s is a random process with distribution -- Normal m s. normalT :: Distribution Normal a => a -> a -> RVarT m a -- | stdNormal is a normal variable with distribution -- StdNormal. stdNormal :: Distribution Normal a => RVar a -- | stdNormalT is a normal process with distribution -- StdNormal. stdNormalT :: Distribution Normal a => RVarT m a -- | A random variable sampling from the standard normal distribution over -- the Double type. doubleStdNormal :: RVarT m Double -- | A random variable sampling from the standard normal distribution over -- the Float type. floatStdNormal :: RVarT m Float -- | A random variable sampling from the standard normal distribution over -- any RealFloat type (subject to the rest of the constraints - it -- builds and uses a Ziggurat internally, which requires the -- Erf class). -- -- Because it computes a Ziggurat, it is very expensive to use for -- just one evaluation, or even for multiple evaluations if not used and -- reused monomorphically (to enable the ziggurat table to be let-floated -- out). If you don't know whether your use case fits this description -- then you're probably better off using a different algorithm, such as -- boxMullerNormalPair or knuthPolarNormalPair. And of -- course if you don't need the full generality of this definition then -- you're much better off using doubleStdNormal or -- floatStdNormal. -- -- As far as I know, this should be safe to use in any monomorphic -- Distribution Normal instance declaration. realFloatStdNormal :: (RealFloat a, Erf a, Distribution Uniform a) => RVarT m a -- | Draw from the tail of a normal distribution (the region beyond the -- provided value) normalTail :: (Distribution StdUniform a, Floating a, Ord a) => a -> RVarT m a -- | A random variable that produces a pair of independent -- normally-distributed values. normalPair :: (Floating a, Distribution StdUniform a) => RVar (a, a) -- | A random variable that produces a pair of independent -- normally-distributed values, computed using the Box-Muller method. -- This algorithm is slightly slower than Knuth's method but using a -- constant amount of entropy (Knuth's method is a rejection method). It -- is also slightly more general (Knuth's method require an Ord -- instance). boxMullerNormalPair :: (Floating a, Distribution StdUniform a) => RVar (a, a) -- | A random variable that produces a pair of independent -- normally-distributed values, computed using Knuth's polar method. -- Slightly faster than boxMullerNormalPair when it accepts on the -- first try, but does not always do so. knuthPolarNormalPair :: (Floating a, Ord a, Distribution Uniform a) => RVar (a, a) instance (Real a, Floating a, Distribution Normal a) => PDF Normal a instance (Real a, Distribution Normal a) => CDF Normal a instance Distribution Normal Float instance Distribution Normal Double module Data.Random.Distribution.Gamma data Gamma a Gamma :: a -> a -> Gamma a gamma :: Distribution Gamma a => a -> a -> RVar a gammaT :: Distribution Gamma a => a -> a -> RVarT m a newtype Erlang a b Erlang :: a -> Erlang a b erlang :: Distribution (Erlang a) b => a -> RVar b erlangT :: Distribution (Erlang a) b => a -> RVarT m b -- | derived from Marsaglia & Tang, "A Simple Method for generating -- gamma variables", ACM Transactions on Mathematical Software, Vol 26, -- No 3 (2000), p363-372. mtGamma :: (Floating a, Ord a, Distribution StdUniform a, Distribution Normal a) => a -> a -> RVarT m a instance (Integral a, Real b, Distribution (Erlang a) b) => CDF (Erlang a) b instance (Integral a, Floating b, Ord b, Distribution Normal b, Distribution StdUniform b) => Distribution (Erlang a) b instance (Real a, Distribution Gamma a) => CDF Gamma a instance (Floating a, Ord a, Distribution Normal a, Distribution StdUniform a) => Distribution Gamma a module Data.Random.Distribution.Beta fractionalBeta :: (Fractional a, Eq a, Distribution Gamma a, Distribution StdUniform a) => a -> a -> RVarT m a beta :: Distribution Beta a => a -> a -> RVar a betaT :: Distribution Beta a => a -> a -> RVarT m a data Beta a Beta :: a -> a -> Beta a logBetaPdf :: Double -> Double -> Double -> Double instance Distribution Beta Double instance Distribution Beta Float instance PDF Beta Float instance PDF Beta Double module Data.Random.Distribution.Binomial integralBinomial :: (Integral a, Floating b, Ord b, Distribution Beta b, Distribution StdUniform b) => a -> b -> RVarT m a integralBinomialCDF :: (Integral a, Real b) => a -> b -> a -> Double integralBinomialPDF :: (Integral a, Real b) => a -> b -> a -> Double integralBinomialLogPdf :: (Integral a, Real b) => a -> b -> a -> Double floatingBinomial :: (RealFrac a, Distribution (Binomial b) Integer) => a -> b -> RVar a floatingBinomialCDF :: (CDF (Binomial b) Integer, RealFrac a) => a -> b -> a -> Double floatingBinomialPDF :: (PDF (Binomial b) Integer, RealFrac a) => a -> b -> a -> Double floatingBinomialLogPDF :: (PDF (Binomial b) Integer, RealFrac a) => a -> b -> a -> Double binomial :: Distribution (Binomial b) a => a -> b -> RVar a binomialT :: Distribution (Binomial b) a => a -> b -> RVarT m a data Binomial b a Binomial :: a -> b -> Binomial b a instance PDF (Binomial b0) Integer => PDF (Binomial b0) Double instance CDF (Binomial b0) Integer => CDF (Binomial b0) Double instance Distribution (Binomial b0) Integer => Distribution (Binomial b0) Double instance PDF (Binomial b0) Integer => PDF (Binomial b0) Float instance CDF (Binomial b0) Integer => CDF (Binomial b0) Float instance Distribution (Binomial b0) Integer => Distribution (Binomial b0) Float instance (Real b0, Distribution (Binomial b0) Word64) => PDF (Binomial b0) Word64 instance (Real b0, Distribution (Binomial b0) Word64) => CDF (Binomial b0) Word64 instance (Floating b0, Ord b0, Distribution Beta b0, Distribution StdUniform b0) => Distribution (Binomial b0) Word64 instance (Real b0, Distribution (Binomial b0) Word32) => PDF (Binomial b0) Word32 instance (Real b0, Distribution (Binomial b0) Word32) => CDF (Binomial b0) Word32 instance (Floating b0, Ord b0, Distribution Beta b0, Distribution StdUniform b0) => Distribution (Binomial b0) Word32 instance (Real b0, Distribution (Binomial b0) Word16) => PDF (Binomial b0) Word16 instance (Real b0, Distribution (Binomial b0) Word16) => CDF (Binomial b0) Word16 instance (Floating b0, Ord b0, Distribution Beta b0, Distribution StdUniform b0) => Distribution (Binomial b0) Word16 instance (Real b0, Distribution (Binomial b0) Word8) => PDF (Binomial b0) Word8 instance (Real b0, Distribution (Binomial b0) Word8) => CDF (Binomial b0) Word8 instance (Floating b0, Ord b0, Distribution Beta b0, Distribution StdUniform b0) => Distribution (Binomial b0) Word8 instance (Real b0, Distribution (Binomial b0) Word) => PDF (Binomial b0) Word instance (Real b0, Distribution (Binomial b0) Word) => CDF (Binomial b0) Word instance (Floating b0, Ord b0, Distribution Beta b0, Distribution StdUniform b0) => Distribution (Binomial b0) Word instance (Real b0, Distribution (Binomial b0) Int64) => PDF (Binomial b0) Int64 instance (Real b0, Distribution (Binomial b0) Int64) => CDF (Binomial b0) Int64 instance (Floating b0, Ord b0, Distribution Beta b0, Distribution StdUniform b0) => Distribution (Binomial b0) Int64 instance (Real b0, Distribution (Binomial b0) Int32) => PDF (Binomial b0) Int32 instance (Real b0, Distribution (Binomial b0) Int32) => CDF (Binomial b0) Int32 instance (Floating b0, Ord b0, Distribution Beta b0, Distribution StdUniform b0) => Distribution (Binomial b0) Int32 instance (Real b0, Distribution (Binomial b0) Int16) => PDF (Binomial b0) Int16 instance (Real b0, Distribution (Binomial b0) Int16) => CDF (Binomial b0) Int16 instance (Floating b0, Ord b0, Distribution Beta b0, Distribution StdUniform b0) => Distribution (Binomial b0) Int16 instance (Real b0, Distribution (Binomial b0) Int8) => PDF (Binomial b0) Int8 instance (Real b0, Distribution (Binomial b0) Int8) => CDF (Binomial b0) Int8 instance (Floating b0, Ord b0, Distribution Beta b0, Distribution StdUniform b0) => Distribution (Binomial b0) Int8 instance (Real b0, Distribution (Binomial b0) Int) => PDF (Binomial b0) Int instance (Real b0, Distribution (Binomial b0) Int) => CDF (Binomial b0) Int instance (Floating b0, Ord b0, Distribution Beta b0, Distribution StdUniform b0) => Distribution (Binomial b0) Int instance (Real b0, Distribution (Binomial b0) Integer) => PDF (Binomial b0) Integer instance (Real b0, Distribution (Binomial b0) Integer) => CDF (Binomial b0) Integer instance (Floating b0, Ord b0, Distribution Beta b0, Distribution StdUniform b0) => Distribution (Binomial b0) Integer module Data.Random.Distribution.Multinomial multinomial :: Distribution (Multinomial p) [a] => [p] -> a -> RVar [a] multinomialT :: Distribution (Multinomial p) [a] => [p] -> a -> RVarT m [a] data Multinomial p a Multinomial :: [p] -> a -> Multinomial p [a] instance (Num a, Eq a, Fractional p, Distribution (Binomial p) a) => Distribution (Multinomial p) [a] module Data.Random.Distribution.ChiSquare chiSquare :: Distribution ChiSquare t => Integer -> RVar t chiSquareT :: Distribution ChiSquare t => Integer -> RVarT m t newtype ChiSquare b ChiSquare :: Integer -> ChiSquare b instance (Real t, Distribution ChiSquare t) => CDF ChiSquare t instance (Fractional t, Distribution Gamma t) => Distribution ChiSquare t module Data.Random.Distribution.Dirichlet fractionalDirichlet :: (Fractional a, Distribution Gamma a) => [a] -> RVarT m [a] dirichlet :: Distribution Dirichlet [a] => [a] -> RVar [a] dirichletT :: Distribution Dirichlet [a] => [a] -> RVarT m [a] newtype Dirichlet a Dirichlet :: a -> Dirichlet a instance Eq a => Eq (Dirichlet a) instance Show a => Show (Dirichlet a) instance (Fractional a, Distribution Gamma a) => Distribution Dirichlet [a] module Data.Random.Distribution.Poisson integralPoisson :: (Integral a, RealFloat b, Distribution StdUniform b, Distribution (Erlang a) b, Distribution (Binomial b) a) => b -> RVarT m a integralPoissonCDF :: (Integral a, Real b) => b -> a -> Double fractionalPoisson :: (Num a, Distribution (Poisson b) Integer) => b -> RVarT m a fractionalPoissonCDF :: (CDF (Poisson b) Integer, RealFrac a) => b -> a -> Double poisson :: Distribution (Poisson b) a => b -> RVar a poissonT :: Distribution (Poisson b) a => b -> RVarT m a newtype Poisson b a Poisson :: b -> Poisson b a instance CDF (Poisson b0) Integer => CDF (Poisson b0) Double instance Distribution (Poisson b0) Integer => Distribution (Poisson b0) Double instance CDF (Poisson b0) Integer => CDF (Poisson b0) Float instance Distribution (Poisson b0) Integer => Distribution (Poisson b0) Float instance (Real b0, Distribution (Poisson b0) Word64) => CDF (Poisson b0) Word64 instance (RealFloat b0, Distribution StdUniform b0, Distribution (Erlang Word64) b0, Distribution (Binomial b0) Word64) => Distribution (Poisson b0) Word64 instance (Real b0, Distribution (Poisson b0) Word32) => CDF (Poisson b0) Word32 instance (RealFloat b0, Distribution StdUniform b0, Distribution (Erlang Word32) b0, Distribution (Binomial b0) Word32) => Distribution (Poisson b0) Word32 instance (Real b0, Distribution (Poisson b0) Word16) => CDF (Poisson b0) Word16 instance (RealFloat b0, Distribution StdUniform b0, Distribution (Erlang Word16) b0, Distribution (Binomial b0) Word16) => Distribution (Poisson b0) Word16 instance (Real b0, Distribution (Poisson b0) Word8) => CDF (Poisson b0) Word8 instance (RealFloat b0, Distribution StdUniform b0, Distribution (Erlang Word8) b0, Distribution (Binomial b0) Word8) => Distribution (Poisson b0) Word8 instance (Real b0, Distribution (Poisson b0) Word) => CDF (Poisson b0) Word instance (RealFloat b0, Distribution StdUniform b0, Distribution (Erlang Word) b0, Distribution (Binomial b0) Word) => Distribution (Poisson b0) Word instance (Real b0, Distribution (Poisson b0) Int64) => CDF (Poisson b0) Int64 instance (RealFloat b0, Distribution StdUniform b0, Distribution (Erlang Int64) b0, Distribution (Binomial b0) Int64) => Distribution (Poisson b0) Int64 instance (Real b0, Distribution (Poisson b0) Int32) => CDF (Poisson b0) Int32 instance (RealFloat b0, Distribution StdUniform b0, Distribution (Erlang Int32) b0, Distribution (Binomial b0) Int32) => Distribution (Poisson b0) Int32 instance (Real b0, Distribution (Poisson b0) Int16) => CDF (Poisson b0) Int16 instance (RealFloat b0, Distribution StdUniform b0, Distribution (Erlang Int16) b0, Distribution (Binomial b0) Int16) => Distribution (Poisson b0) Int16 instance (Real b0, Distribution (Poisson b0) Int8) => CDF (Poisson b0) Int8 instance (RealFloat b0, Distribution StdUniform b0, Distribution (Erlang Int8) b0, Distribution (Binomial b0) Int8) => Distribution (Poisson b0) Int8 instance (Real b0, Distribution (Poisson b0) Int) => CDF (Poisson b0) Int instance (RealFloat b0, Distribution StdUniform b0, Distribution (Erlang Int) b0, Distribution (Binomial b0) Int) => Distribution (Poisson b0) Int instance (Real b0, Distribution (Poisson b0) Integer) => CDF (Poisson b0) Integer instance (RealFloat b0, Distribution StdUniform b0, Distribution (Erlang Integer) b0, Distribution (Binomial b0) Integer) => Distribution (Poisson b0) Integer module Data.Random.Distribution.Rayleigh floatingRayleigh :: (Floating a, Eq a, Distribution StdUniform a) => a -> RVarT m a -- | The rayleigh distribution with a specified mode ("sigma") parameter. -- Its mean will be sigma*sqrt(pi/2) and its variance will be -- sigma^2*(4-pi)/2 -- -- (therefore if you want one with a particular mean m, -- sigma should be m*sqrt(2/pi)) newtype Rayleigh a Rayleigh :: a -> Rayleigh a rayleigh :: Distribution Rayleigh a => a -> RVar a rayleighT :: Distribution Rayleigh a => a -> RVarT m a rayleighCDF :: Real a => a -> a -> Double instance (Real a, Distribution Rayleigh a) => CDF Rayleigh a instance (RealFloat a, Distribution StdUniform a) => Distribution Rayleigh a module Data.Random.Distribution.T t :: Distribution T a => Integer -> RVar a tT :: Distribution T a => Integer -> RVarT m a newtype T a T :: Integer -> T a instance Eq (T a) instance Ord (T a) instance Show (T a) instance (Real a, Distribution T a) => CDF T a instance (Floating a, Distribution Normal a, Distribution ChiSquare a) => Distribution T a module Data.Random.Distribution.Triangular -- | A description of a triangular distribution - a distribution whose PDF -- is a triangle ramping up from a lower bound to a specified midpoint -- and back down to the upper bound. This is a very simple distribution -- that does not generally occur naturally but is used sometimes as an -- estimate of a true distribution when only the range of the values and -- an approximate mode of the true distribution are known. data Triangular a Triangular :: a -> a -> a -> Triangular a -- | The lower bound of the triangle in the PDF (the smallest number the -- distribution can generate) triLower :: Triangular a -> a -- | The midpoint of the triangle (also the mode of the distribution) triMid :: Triangular a -> a -- | The upper bound of the triangle (and the largest number the -- distribution can generate) triUpper :: Triangular a -> a -- | Compute a triangular distribution for a Floating type. floatingTriangular :: (Floating a, Ord a, Distribution StdUniform a) => a -> a -> a -> RVarT m a -- | triangularCDF a b c is the CDF of realFloatTriangular a b -- c. triangularCDF :: RealFrac a => a -> a -> a -> a -> Double instance Eq a => Eq (Triangular a) instance Show a => Show (Triangular a) instance (RealFrac a, Distribution Triangular a) => CDF Triangular a instance (RealFloat a, Ord a, Distribution StdUniform a) => Distribution Triangular a module Data.Random.Distribution.Weibull data Weibull a Weibull :: !a -> !a -> Weibull a weibullLambda :: Weibull a -> !a weibullK :: Weibull a -> !a instance Eq a => Eq (Weibull a) instance Show a => Show (Weibull a) instance (Real a, Distribution Weibull a) => CDF Weibull a instance (Floating a, Distribution StdUniform a) => Distribution Weibull a module Data.Random.Sample -- | A typeclass allowing Distributions and RVars to be -- sampled. Both may also be sampled via runRVar or -- runRVarT, but I find it psychologically pleasing to be able to -- sample both using this function, as they are two separate abstractions -- for one base concept: a random variable. class Sampleable d m t sampleFrom :: (Sampleable d m t, RandomSource m s) => s -> d t -> m t -- | Sample a random variable using the default source of entropy for the -- monad in which the sampling occurs. sample :: (Sampleable d m t, MonadRandom m) => d t -> m t -- | Sample a random variable in a "functional" style. Typical -- instantiations of s are System.Random.StdGen or -- System.Random.Mersenne.Pure64.PureMT. sampleState :: (Sampleable d (State s) t, MonadRandom (State s)) => d t -> s -> (t, s) -- | Sample a random variable in a "semi-functional" style. Typical -- instantiations of s are System.Random.StdGen or -- System.Random.Mersenne.Pure64.PureMT. sampleStateT :: (Sampleable d (StateT s m) t, MonadRandom (StateT s m)) => d t -> s -> m (t, s) instance [incoherent] Lift m n => Sampleable (RVarT m) n t instance [incoherent] Distribution d t => Sampleable d m t -- | Flexible modeling and sampling of random variables. -- -- The central abstraction in this library is the concept of a random -- variable. It is not fully formalized in the standard measure-theoretic -- language, but rather is informally defined as a "thing you can get -- random values out of". Different random variables may have different -- types of values they can return or the same types but different -- probabilities for each value they can return. The random values you -- get out of them are traditionally called "random variates". -- -- Most imperative-language random number libraries are all about -- obtaining and manipulating random variates. This one is about -- defining, manipulating and sampling random variables. Computationally, -- the distinction is small and mostly just a matter of perspective, but -- from a program design perspective it provides both a powerfully -- composable abstraction and a very useful separation of concerns. -- -- Abstract random variables as implemented by RVar are -- composable. They can be defined in a monadic / "imperative" style that -- amounts to manipulating variates, but with strict type-level -- isolation. Concrete random variables are also provided, but they do -- not compose as generically. The Distribution type class allows -- concrete random variables to "forget" their concreteness so that they -- can be composed. For examples of both, see the documentation for -- RVar and Distribution, as well as the code for any of -- the concrete distributions such as Uniform, Gamma, etc. -- -- Both abstract and concrete random variables can be sampled (despite -- the types GHCi may list for the functions) by the functions in -- Data.Random.Sample. -- -- Random variable sampling is done with regard to a generic basis of -- primitive random variables defined in -- Data.Random.Internal.Primitives. This basis is very low-level -- and the actual set of primitives is still fairly experimental, which -- is why it is in the "Internal" sub-heirarchy. User-defined variables -- should use the existing high-level variables such as Uniform -- and Normal rather than these basis variables. -- Data.Random.Source defines classes for entropy sources that -- provide implementations of these primitive variables. Several -- implementations are available in the Data.Random.Source.* modules. module Data.Random -- | An opaque type modeling a "random variable" - a value which depends on -- the outcome of some random event. RVars 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 -- RVars: -- -- -- -- 
--   runRVar (uniform 1 100) DevRandom :: IO Int
--
-- -- -- -- 
--   sampleRVar (uniform 1 100) :: State PureMT Int
--
-- -- -- -- 
--   sampleState (uniform 1 100) :: StdGen -> (Int, StdGen)
--
-- -- (where sampleState = runState . sampleRVar) type RVar = RVarT Identity -- | 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. -- -- For example, a simple random walk can be implemented as an -- RVarT IO value: -- -- 
--   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
--
data RVarT (m :: * -> *) a :: (* -> *) -> * -> * -- | "Run" an RVar - samples the random variable from the provided -- source of entropy. runRVar :: RandomSource m s => RVar a -> s -> m a -- | Like runRVarTWith, but using an implicit lifting (provided by -- the Lift class) runRVarT :: (Lift n m, RandomSource m s) => RVarT n a -> s -> m a -- | "Runs" an 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 -- IORef): -- -- 
--   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. runRVarTWith :: RandomSource m s => (forall t. n t -> m t) -> RVarT n a -> s -> m a -- | A Distribution is a data representation of a random variable's -- probability structure. For example, in -- Data.Random.Distribution.Normal, the Normal -- distribution is defined as: -- -- 
--   data Normal a
--       = StdNormal
--       | Normal a a
--
-- -- Where the two parameters of the Normal data constructor are -- the mean and standard deviation of the random variable, respectively. -- To make use of the Normal type, one can convert it to an -- rvar and manipulate it or sample it directly: -- -- 
--   x <- sample (rvar (Normal 10 2))
--   x <- sample (Normal 10 2)
--
-- -- A Distribution is typically more transparent than an -- RVar but less composable (precisely because of that -- transparency). There are several practical uses for types implementing -- Distribution: -- -- -- -- On the other hand, most Distributions will not be closed under -- all the same operations as RVar (which, being a monad, has a -- fully turing-complete internal computational model). The sum of two -- uniformly-distributed variables, for example, is not uniformly -- distributed. To support general composition, the Distribution -- class defines a function rvar to construct the more-abstract -- and more-composable RVar representation of a random variable. class Distribution d t where rvar = rvarT rvarT d = lift (rvar d) rvar :: Distribution d t => d t -> RVar t rvarT :: Distribution d t => d t -> RVarT n t class Distribution d t => CDF d t cdf :: CDF d t => d t -> t -> Double class Distribution d t => PDF d t where pdf d = exp . logPdf d logPdf d = log . pdf d pdf :: PDF d t => d t -> t -> Double logPdf :: PDF d t => d t -> t -> Double -- | A typeclass allowing Distributions and RVars to be -- sampled. Both may also be sampled via runRVar or -- runRVarT, but I find it psychologically pleasing to be able to -- sample both using this function, as they are two separate abstractions -- for one base concept: a random variable. class Sampleable d m t sampleFrom :: (Sampleable d m t, RandomSource m s) => s -> d t -> m t -- | Sample a random variable using the default source of entropy for the -- monad in which the sampling occurs. sample :: (Sampleable d m t, MonadRandom m) => d t -> m t -- | Sample a random variable in a "functional" style. Typical -- instantiations of s are System.Random.StdGen or -- System.Random.Mersenne.Pure64.PureMT. sampleState :: (Sampleable d (State s) t, MonadRandom (State s)) => d t -> s -> (t, s) -- | Sample a random variable in a "semi-functional" style. Typical -- instantiations of s are System.Random.StdGen or -- System.Random.Mersenne.Pure64.PureMT. sampleStateT :: (Sampleable d (StateT s m) t, MonadRandom (StateT s m)) => d t -> s -> m (t, s) -- | A definition of a uniform distribution over the type t. See -- also uniform. data Uniform t -- | A uniform distribution defined by a lower and upper range bound. For -- Integral and Enum types, the range is inclusive. For -- Fractional types the range includes the lower bound but not the -- upper. Uniform :: !t -> !t -> Uniform t uniform :: Distribution Uniform a => a -> a -> RVar a uniformT :: Distribution Uniform a => a -> a -> RVarT m a -- | A name for the "standard" uniform distribution over the type -- t, if one exists. See also stdUniform. -- -- For Integral and Enum types that are also -- Bounded, this is the uniform distribution over the full range -- of the type. For un-Bounded Integral types this is not -- defined. For Fractional types this is a random variable in the -- range [0,1) (that is, 0 to 1 including 0 but not including 1). data StdUniform t StdUniform :: StdUniform t -- | Get a "standard" uniformly distributed variable. For integral types, -- this means uniformly distributed over the full range of the type -- (there is no support for Integer). For fractional types, this -- means uniformly distributed on the interval [0,1). stdUniform :: Distribution StdUniform a => RVar a -- | Get a "standard" uniformly distributed process. For integral types, -- this means uniformly distributed over the full range of the type -- (there is no support for Integer). For fractional types, this -- means uniformly distributed on the interval [0,1). stdUniformT :: Distribution StdUniform a => RVarT m a -- | A specification of a normal distribution over the type a. data Normal a -- | The "standard" normal distribution - mean 0, stddev 1 StdNormal :: Normal a -- | Normal m s is a normal distribution with mean m and -- stddev sd. Normal :: a -> a -> Normal a -- | normal m s is a random variable with distribution -- Normal m s. normal :: Distribution Normal a => a -> a -> RVar a -- | stdNormal is a normal variable with distribution -- StdNormal. stdNormal :: Distribution Normal a => RVar a -- | normalT m s is a random process with distribution -- Normal m s. normalT :: Distribution Normal a => a -> a -> RVarT m a -- | stdNormalT is a normal process with distribution -- StdNormal. stdNormalT :: Distribution Normal a => RVarT m a data Gamma a Gamma :: a -> a -> Gamma a gamma :: Distribution Gamma a => a -> a -> RVar a gammaT :: Distribution Gamma a => a -> a -> RVarT m a -- | A typeclass for monads with a chosen source of entropy. For example, -- RVar is such a monad - the source from which it is -- (eventually) sampled is the only source from which a random variable -- is permitted to draw, so when directly requesting entropy for a random -- variable these functions are used. -- -- Minimum implementation is either the internal getRandomPrim or -- all other functions. Additionally, this class's interface is subject -- to extension at any time, so it is very, very strongly recommended -- that the monadRandom Template Haskell function be used to -- implement this function rather than directly implementing it. That -- function takes care of choosing default implementations for any -- missing functions; as long as at least one function is implemented, it -- will derive sensible implementations of all others. -- -- To use monadRandom, just wrap your instance declaration as -- follows (and enable the TemplateHaskell and GADTs language -- extensions): -- -- 
--   $(monadRandom [d|
--           instance MonadRandom FooM where
--               getRandomDouble = return pi
--               getRandomWord16 = return 4
--               {- etc... -}
--       |])
--
class Monad m => MonadRandom (m :: * -> *) -- | A source of entropy which can be used in the given monad. -- -- See also MonadRandom. -- -- Minimum implementation is either the internal getRandomPrimFrom -- or all other functions. Additionally, this class's interface is -- subject to extension at any time, so it is very, very strongly -- recommended that the randomSource Template Haskell function -- be used to implement this function rather than directly implementing -- it. That function takes care of choosing default implementations for -- any missing functions; as long as at least one function is -- implemented, it will derive sensible implementations of all others. -- -- To use randomSource, just wrap your instance declaration as -- follows (and enable the TemplateHaskell, MultiParamTypeClasses and -- GADTs language extensions, as well as any others required by your -- instances, such as FlexibleInstances): -- -- 
--   $(randomSource [d|
--           instance RandomSource FooM Bar where
--               {- at least one RandomSource function... -}
--       |])
--
class Monad m => RandomSource (m :: * -> *) s -- | A token representing the "standard" entropy source in a -- MonadRandom monad. Its sole purpose is to make the following -- true (when the types check): -- -- 
--   runRVar x StdRandom === sampleRVar
--
data StdRandom :: * StdRandom :: StdRandom -- | A random variable returning an arbitrary element of the given list. -- Every element has equal probability of being chosen. Because it is a -- pure RVar it has no memory - that is, it "draws with -- replacement." randomElement :: [a] -> RVar a -- | A random variable that returns the given list in an arbitrary shuffled -- order. Every ordering of the list has equal probability. shuffle :: [a] -> RVar [a] -- | A random variable that shuffles a list of a known length (or a list -- prefix of the specified length). Useful for shuffling large lists when -- the length is known in advance. Avoids needing to traverse the list to -- discover its length. Each ordering has equal probability. shuffleN :: Int -> [a] -> RVar [a] -- | A random variable that selects N arbitrary elements of a list of known -- length M. shuffleNofM :: Int -> Int -> [a] -> RVar [a] module Data.Random.Distribution.Pareto pareto :: Distribution Pareto a => a -> a -> RVar a paretoT :: Distribution Pareto a => a -> a -> RVarT m a data Pareto a Pareto :: !a -> !a -> Pareto a instance (Real a, Distribution Pareto a) => CDF Pareto a instance (Floating a, Distribution StdUniform a) => Distribution Pareto a