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module Numeric.Prob (Prob (..),
complement,
logOdds,
toFloating,
LogDouble
)
where
import Numeric.Log
import Numeric.LogDouble
import Data.Ratio
import Data.Floating.Types
{- We need to handle numbers that are > 1 in the following cases:
1/2 (2 > 1)
1-3*p (3 > 1)
So, we assume that IOdds z = 1 / Odds z -}
-- We could keep track of HOW MANY zeros!
-- ... | Zero Int | ...
-- Then we could penalize multiplying by an additional zero.
data Prob = Zero | Odds Double | One | IOdds Double | Infinity
complement Zero = One
complement (Odds y) = (Odds (-y))
complement One = Zero
complement _ = error "complement: not a probability"
fromProb :: (Real a, Pow a) => Prob -> a
fromProb Zero = 0
fromProb (Odds y) | y < 0 = let e = expTo y in e / (1 + e)
fromProb (Odds y) | otherwise = 1 / (1 + expTo(-y))
fromProb One = 1
fromProb (IOdds y)| y < 0 = let e = expTo y in (1 + e)/e
| otherwise = (1 + expTo(-y))
fromProb Infinity = 1 / 0
toProb :: Double -> Prob
toProb p | p < 0 = error "Negative Probability!"
| p == 0 = Zero
| p < 1 = Odds $ log $ p / (1-p)
| p == 1 = One
| p > 1 = recip $ toProb (1/p)
| isInfinite p = Infinity
| isNaN p = error "toProb: NaN"
| otherwise = error ("toProb: unknown number: " ++ show p)
-- Only defined on non-zero probabilities.
logProb :: Prob -> Double
logProb Zero = error "Probability: log(0)"
logProb (Odds y) | y < 0 = y - log1p (exp y)
| otherwise = -log1p (exp (-y))
logProb One = 0
logProb (IOdds y) = -logProb (Odds y)
logProb Infinity = 1/0
expToProb :: Double -> Prob
expToProb z | z < 0 = Odds $ z - log1p (-exp z)
| z == 0 = One
| z > 0 = let Odds z2 = expToProb (-z) in IOdds z2
-- If y12 is only slightly more than zero, we should be able to do better
plus (Odds y1) (Odds y2) | y12 > 0 = toProb (fromProb (Odds y1) + fromProb (Odds y2))
| y12 == 0 = One
| y12 > (-1) = One - toProb (expm1 (y12) / (1 + exp y1) / (1 + exp y2) )
| otherwise = toProb $ fromProb (Odds y1) + fromProb (Odds y2)
where y12 = y1 + y2
plus (Odds y) One | y < 0 = IOdds $ log1p(exp(y)) - y
| otherwise = IOdds $ log1p(exp(-y))
plus (IOdds y1) (Odds y2) = plus (Odds y2) (IOdds y1)
plus One (Odds y) = plus (Odds y) One
plus Zero x = x
plus x Zero = x
plus Infinity x = Infinity
plus x Infinity = Infinity
plus x1 x2 = toProb (fromProb x1 + fromProb x2)
sub x Zero = x
sub Infinity Infinity = error "Inf - Inf is undefined"
sub Infinity _ = Infinity
sub One One = Zero
sub One (Odds y) = Odds (-y)
sub p1@(Odds y1) p2@(Odds y2) | y1 >= y2 = p1 * (One - (p2/p1))
sub (IOdds y1) (Odds y2) = toProb (fromProb (IOdds y1) - fromProb (Odds y2))
sub (IOdds y1) (IOdds y2) | y1 == y2 = Zero
| y1 < y2 = toProb (fromProb (IOdds y1) - fromProb (IOdds y2))
sub _ _ = error "Negative probability"
mul (Odds y1) (Odds y2) | y1 > y2 = Odds $ y2 - log1p( exp(y2-y1) + exp(-y1) )
| otherwise = Odds $ y1 - log1p( exp(y1-y2) + exp(-y2) )
-- Note that as y1 and y2 increase, both tend towards 1, so whichever is SMALLER dominates.
mul (Odds y1) (IOdds y2) | y1 == y2 = One
| y1 < y2 = Odds $ y1 + log1pexp(-y2) - log1mexp(y1-y2)
| otherwise = IOdds $ y2 + log1pexp(-y1) - log1mexp(y2-y1)
mul (IOdds y1) (Odds y2) = mul (Odds y2) (IOdds y1)
mul (IOdds y1) (IOdds y2) = recip $ mul (Odds y1) (Odds y2)
mul Zero Infinity = error "0 * Inf is undefined"
mul Infinity Zero = error "Inf * 0 is undefined"
mul Zero x = Zero
mul x Zero = Zero
mul One x = x
mul x One = x
mul Infinity x = Infinity
mul x Infinity = Infinity
instance Eq Prob where
(Odds y1) == (Odds y2) = y1 == y2
(IOdds y1) == (IOdds y2) = y1 == y2
x == y = pord x == pord y
pord Zero = 0
pord (Odds _) = 1
pord One = 2
pord (IOdds _) = 3
pord Infinity = 4
instance Ord Prob where
(Odds y1) < (Odds y2) = y1 < y2
(IOdds y1) < (IOdds y2) = y1 > y2
x < y = pord x < pord y
instance Num Prob where
(+) = plus
(-) = sub
(*) = mul
abs = id
negate = error "Can't negate a probability"
signum Zero = 0
signum _ = 1
fromInteger 0 = Zero
fromInteger 1 = One
fromInteger x | x < 0 = error "Negative Probability!"
| otherwise = IOdds $ integerToInvLogOdds x
instance Real Prob where
toRational Zero = 0 % 1
toRational (Odds y) = toRational $ (fromProb (Odds y) :: Double)
toRational One = 1 % 1
toRational (IOdds y) = toRational $ (fromProb (IOdds y) :: Double)
toRational Infinity = 1 % 0
instance Fractional Prob where
recip Zero = Infinity
recip (Odds y) = IOdds y
recip One = One
recip (IOdds y) = Odds y
recip Infinity = Zero
instance Show Prob where
show Zero = "0"
show (Odds y) = show $ (fromProb (Odds y) :: Double)
show One = "1"
show (IOdds y) = show $ (fromProb (IOdds y) :: Double)
show Infinity = "Inf"
logOdds (Odds y) = y
logOdds Zero = -1/0
logOdds One = 1/0
logOdds (IOdds _) = error "logOdds(x) where x > 1"
logOdds Infinity = error "logOdds Infinity"
-- t regions: (-Inf,-1], (-1, 0), (0,1), [1,Inf)
-- y regions for t > 1: (-Inf,0),[0,30),[30,Inf)
-- maybe for t in (0,1) we need different y regions?
-- we need to avoid exp(t*y) when t*y is large and positive.
-- It would be nice if we could write y ** t, but t is not a Prob
powProb :: Prob -> Double -> Prob
powProb One t = One
powProb y t | t == 0 = One
powProb Zero t | t < 0 = Infinity
| otherwise = Zero -- t > 0
powProb Infinity t | t < 0 = Zero
| otherwise = Infinity -- t > 0
-- This probably works best for t > 1, when exp(tx) < exp(x).
powProb (Odds y) t | y > 30 = Odds $ y - log t
| y < 0 = Odds $ t*y - log ( (1+exp(y))**t - exp(t*y) )
| otherwise = Odds $ negate $ log $ expm1 ( t * log1p ( exp (-y)) )
powProb (IOdds y) t = recip $ pow (Odds y) t
instance Pow Prob where
ln = logProb
pow = powProb
expTo = expToProb
instance FloatConvert Prob Double where
toFloating = fromProb
instance FloatConvert Prob LogDouble where
toFloating = fromProb
instance FloatConvert LogDouble Prob where
toFloating = expTo . ln
instance FloatConvert Double Prob where
toFloating = toProb
-- Handle Int, Integer, and anything else that we can get a double from.
instance {-# OVERLAPPABLE #-} FloatConvert a Double => FloatConvert a Prob where
toFloating i = toFloating (toFloating i :: Double)
foreign import bpcall "Real:" integerToInvLogOdds :: Integer -> Double
|
