```{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE DeriveDataTypeable, DeriveGeneric #-}
-- |
-- Module    : Statistics.Distribution.Binomial
-- Copyright : (c) 2009 Bryan O'Sullivan
--
-- Maintainer  : bos@serpentine.com
-- Stability   : experimental
-- Portability : portable
--
-- The binomial distribution.  This is the discrete probability
-- distribution of the number of successes in a sequence of /n/
-- independent yes\/no experiments, each of which yields success with
-- probability /p/.

module Statistics.Distribution.Binomial
(
BinomialDistribution
-- * Constructors
, binomial
, binomialE
-- * Accessors
, bdTrials
, bdProbability
) where

import Control.Applicative
import Data.Aeson            (FromJSON(..), ToJSON, Value(..), (.:))
import Data.Binary           (Binary(..))
import Data.Data             (Data, Typeable)
import GHC.Generics          (Generic)
import Numeric.SpecFunctions           (choose,logChoose,incompleteBeta,log1p)
import Numeric.MathFunctions.Constants (m_epsilon)

import qualified Statistics.Distribution as D
import qualified Statistics.Distribution.Poisson.Internal as I
import Statistics.Internal

-- | The binomial distribution.
data BinomialDistribution = BD {
bdTrials      :: {-# UNPACK #-} !Int
-- ^ Number of trials.
, bdProbability :: {-# UNPACK #-} !Double
-- ^ Probability.
} deriving (Eq, Typeable, Data, Generic)

instance Show BinomialDistribution where
showsPrec i (BD n p) = defaultShow2 "binomial" n p i

instance ToJSON BinomialDistribution
instance FromJSON BinomialDistribution where
parseJSON (Object v) = do
n <- v .: "bdTrials"
p <- v .: "bdProbability"
maybe (fail \$ errMsg n p) return \$ binomialE n p
parseJSON _ = empty

instance Binary BinomialDistribution where
put (BD x y) = put x >> put y
get = do
n <- get
p <- get
maybe (fail \$ errMsg n p) return \$ binomialE n p

instance D.Distribution BinomialDistribution where
cumulative = cumulative

instance D.DiscreteDistr BinomialDistribution where
probability    = probability
logProbability = logProbability

instance D.Mean BinomialDistribution where
mean = mean

instance D.Variance BinomialDistribution where
variance = variance

instance D.MaybeMean BinomialDistribution where
maybeMean = Just . D.mean

instance D.MaybeVariance BinomialDistribution where
maybeStdDev   = Just . D.stdDev
maybeVariance = Just . D.variance

instance D.Entropy BinomialDistribution where
entropy (BD n p)
| n == 0 = 0
| n <= 100 = directEntropy (BD n p)
| otherwise = I.poissonEntropy (fromIntegral n * p)

instance D.MaybeEntropy BinomialDistribution where
maybeEntropy = Just . D.entropy

-- This could be slow for big n
probability :: BinomialDistribution -> Int -> Double
probability (BD n p) k
| k < 0 || k > n = 0
| n == 0         = 1
-- choose could overflow Double for n >= 1030 so we switch to
-- log-domain to calculate probability
| n < 1000       = choose n k * p^k * (1-p)^(n-k)
| otherwise      = exp \$ logChoose n k + log p * k' + log1p (-p) * nk'
where
k'  = fromIntegral k
nk' = fromIntegral \$ n - k

logProbability :: BinomialDistribution -> Int -> Double
logProbability (BD n p) k
| k < 0 || k > n          = (-1)/0
| n == 0                  = 0
| otherwise               = logChoose n k + log p * k' + log1p (-p) * nk'
where
k'  = fromIntegral   k
nk' = fromIntegral \$ n - k

-- Summation from different sides required to reduce roundoff errors
cumulative :: BinomialDistribution -> Double -> Double
cumulative (BD n p) x
| isNaN x      = error "Statistics.Distribution.Binomial.cumulative: NaN input"
| isInfinite x = if x > 0 then 1 else 0
| k <  0       = 0
| k >= n       = 1
| otherwise    = incompleteBeta (fromIntegral (n-k)) (fromIntegral (k+1)) (1 - p)
where
k = floor x

mean :: BinomialDistribution -> Double
mean (BD n p) = fromIntegral n * p

variance :: BinomialDistribution -> Double
variance (BD n p) = fromIntegral n * p * (1 - p)

directEntropy :: BinomialDistribution -> Double
directEntropy d@(BD n _) =
negate . sum \$
takeWhile (< negate m_epsilon) \$
dropWhile (not . (< negate m_epsilon)) \$
[ let x = probability d k in x * log x | k <- [0..n]]

-- | Construct binomial distribution. Number of trials must be
--   non-negative and probability must be in [0,1] range
binomial :: Int                 -- ^ Number of trials.
-> Double              -- ^ Probability.
-> BinomialDistribution
binomial n p = maybe (error \$ errMsg n p) id \$ binomialE n p

-- | Construct binomial distribution. Number of trials must be
--   non-negative and probability must be in [0,1] range
binomialE :: Int                 -- ^ Number of trials.
-> Double              -- ^ Probability.
-> Maybe BinomialDistribution
binomialE n p
| n < 0            = Nothing
| p >= 0 || p <= 1 = Just (BD n p)
| otherwise        = Nothing

errMsg :: Int -> Double -> String
errMsg n p
= "Statistics.Distribution.Binomial.binomial: n=" ++ show n
++ " p=" ++ show p ++ "but n>=0 and p in [0,1]"
```