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{-# LANGUAGE TypeApplications #-}
module PPL.Distr where
import Data.Bits (countLeadingZeros, shiftL, shiftR, (.&.))
import Data.Functor ((<&>))
import Data.Word (Word64)
import PPL.Internal
import Unsafe.Coerce
-- Acklam's approximation
-- https://web.archive.org/web/20151030215612/http://home.online.no/~pjacklam/notes/invnorm/
{-# INLINE probit #-}
probit :: Double -> Double
probit p
| p < lower =
let q = sqrt (-2 * log p)
in (((((c1 * q + c2) * q + c3) * q + c4) * q + c5) * q + c6)
/ ((((d1 * q + d2) * q + d3) * q + d4) * q + 1)
| p < 1 - lower =
let q = p - 0.5
r = q * q
in (((((a1 * r + a2) * r + a3) * r + a4) * r + a5) * r + a6)
* q
/ (((((b1 * r + b2) * r + b3) * r + b4) * r + b5) * r + 1)
| otherwise = -probit (1 - p)
where
a1 = -3.969683028665376e+01
a2 = 2.209460984245205e+02
a3 = -2.759285104469687e+02
a4 = 1.383577518672690e+02
a5 = -3.066479806614716e+01
a6 = 2.506628277459239e+00
b1 = -5.447609879822406e+01
b2 = 1.615858368580409e+02
b3 = -1.556989798598866e+02
b4 = 6.680131188771972e+01
b5 = -1.328068155288572e+01
c1 = -7.784894002430293e-03
c2 = -3.223964580411365e-01
c3 = -2.400758277161838e+00
c4 = -2.549732539343734e+00
c5 = 4.374664141464968e+00
c6 = 2.938163982698783e+00
d1 = 7.784695709041462e-03
d2 = 3.224671290700398e-01
d3 = 2.445134137142996e+00
d4 = 3.754408661907416e+00
lower = 0.02425
iid :: Prob a -> Prob [a]
iid = sequence . repeat
gauss :: Prob Double
gauss = probit <$> uniform
norm :: Double -> Double -> Prob Double
norm m s = (+ m) . (* s) <$> gauss
-- Marsaglia's fast gamma rejection sampling
gamma :: Double -> Prob Double
gamma a = do
x <- gauss
u <- uniform
if u < 1 - 0.03331 * x ** 4
then pure $ d * v x
else gamma a
where
d = a - 1 / 3
v x = (1 + x / sqrt (9 * d)) ** 3
beta :: Double -> Double -> Prob Double
beta a b = do
x <- gamma a
y <- gamma b
pure $ x / (x + y)
beta' :: Double -> Double -> Prob Double
beta' a b = do
p <- beta a b
pure $ p / (1 - p)
bern :: Double -> Prob Bool
bern p = (< p) <$> uniform
binom :: Int -> Double -> Prob Int
binom n = fmap (length . filter id . take n) . iid . bern
exponential :: Double -> Prob Double
exponential lambda = negate . (/ lambda) . log <$> uniform
geom :: Double -> Prob Int
geom p = first 0 <$> iid (bern p)
where
first n (True : _) = n
first n (_ : xs) = first (n + 1) xs
first _ _ = undefined
-- Uses idea from 10.1145/3503512
bounded :: Double -> Double -> Prob Double
bounded lower upper
| hi - 2 <= 0xfffffffffffff = do
(k1, k2) <- split <$> boundedWord52 1 (hi - 2)
pure $ if abs lower <= abs upper then 4 * (upper / 4 - k1 * g) - k2 * g else 4 * (lower / 4 + k1 * g) + k2 * g
| otherwise = uniform <&> \x -> 2 * (lower / 2 + (upper / 2 - lower / 2) * x)
where
hi = ceilint lower upper g
g = max (up lower - lower) (upper - down upper)
split x = (fromIntegral $ x `shiftR` 2, fromIntegral $ x .&. 0x3)
up x = unsafeCoerce @Word64 @Double (unsafeCoerce x + 1)
down x = unsafeCoerce @Word64 @Double (unsafeCoerce x - 1)
ceilint :: Double -> Double -> Double -> Word64
ceilint a b g = ceiling s + if s == fromIntegral (ceiling s) && eps > 0 then 1 else 0
where
s = b / g - a / g
eps = if abs a <= abs b then negate a / g - (s - b / g) else b / g - (s + a / g)
-- Random mantissa part of a uniform double
unboundedWord52 = (.&. 0xfffffffffffff) . (unsafeCoerce @Double @Word64) <$> uniform
boundedWord52 l r = do
x <- (`mod` nextPow2 r) <$> unboundedWord52
if x <= r then pure (x + l) else boundedWord52 l r
where
nextPow2 x = 1 `shiftL` (64 - countLeadingZeros x)
bounded' :: Integral a => a -> a -> Prob a
bounded' lower upper
| r <= 0xfffffffffffff = fromIntegral <$> boundedWord52 (fromIntegral lower) (fromIntegral r)
| otherwise = round <$> bounded (fromIntegral lower) (fromIntegral upper)
where
r = upper - lower
cat :: [Double] -> Prob Int
cat xs = search 0 (tail $ scanl (+) 0 xs) <$> uniform
where
search i [] _ = i
search i (x : xs) r
| x > r = i
| otherwise = search (i + 1) xs r
dirichletProcess :: Double -> Prob [Double]
dirichletProcess p = go 1
where
go rest = do
x <- beta 1 p
(x * rest :) <$> go (rest - x * rest)
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