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Improvements to Poisson entropy computation

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File Statistics/Distribution/Poisson/Internal.hs

 
 import Data.List(unfoldr)
 import Numeric.MathFunctions.Constants (m_sqrt_2_pi, m_tiny, m_epsilon)
-import Numeric.SpecFunctions           (logGamma, stirlingError, choose)
+import Numeric.SpecFunctions (logGamma, stirlingError, choose, logFactorial)
 import Numeric.SpecFunctions.Extra     (bd0)
 
 -- | An unchecked, non-integer-valued version of Loader's saddle point
 {-# INLINE probability #-}
 
 -- | Compute entropy using Theorem 1 from "Sharp Bounds on the Entropy
--- of the Poisson Law" by José A. Adell, Alberto Lekuona, and Yaming
--- Yu.  This is highly robust for lambda <= 1.
+-- of the Poisson Law".  This function is unused because 'directEntorpy'
+-- is just as accurate and is faster by about a factor of 4.
 alyThm1 :: Double -> Double
 alyThm1 lambda =
   sum (takeWhile (\x -> abs x >= m_epsilon * lll) alySeries) + lll
   where lll = lambda * (1 - log lambda)
         alySeries =
-          [ alyc k * exp (fromIntegral k * log lambda 
-                          - logGamma (fromIntegral k))
+          [ alyc k * exp (fromIntegral k * log lambda - logFactorial k)
           | k <- [2..] ]
 
 alyc :: Int -> Double
           | even (k-j) = -1
           | otherwise  = 1
                          
-                         
--- | Compute entropy using Theorem 2 from the same paper. This is highly
--- accurate for large lambda.
-alyThm2 :: Double -> Double
-alyThm2 lambda =
+-- | Returns [x, x^2, x^3, x^4, ...]
+powers :: Double -> [Double]
+powers x = unfoldr (\y -> Just (y*x,y*x)) 1
+
+-- | Returns an upper bound according to theorem 2 of "Sharp Bounds on
+-- the Entropy of the Poisson Law"
+alyThm2Upper :: Double -> [Double] -> Double
+alyThm2Upper lambda coefficients =
   1.4189385332046727 + 0.5 * log lambda +
-  (sum $ map (uncurry (*)) (zip powers coefficients))
-  where powers = unfoldr (\x -> Just (x/lambda,x/lambda)) 1
-        coefficients =
-          [1/12, 1/24, 19/360, 9/80, 863/2520, -123365/1008, -3023561/2520,
-           -808157/720, -984451/5940, -1151/440, -1/1716]
+  zipCoefficients lambda coefficients
 
--- | Compute entropy by brute force. Limits on floating point precision
--- cause this to underestimate by quite a bit, but for intermediate values
--- of lambda it's the best we can do.
-bruteforce :: Double -> Double
-bruteforce lambda =   
+-- | Returns the average of the upper and lower bounds accounding to
+-- theorem 2.
+alyThm2 :: Double -> [Double] -> [Double] -> Double
+alyThm2 lambda upper lower =
+  alyThm2Upper lambda upper + 0.5 * (zipCoefficients lambda lower)
+
+zipCoefficients :: Double -> [Double] -> Double 
+zipCoefficients lambda coefficients =
+  (sum $ map (uncurry (*)) (zip (powers $ recip lambda) coefficients))
+
+-- Mathematica code deriving the coefficients below:
+--
+-- poissonMoment[0, s_] := 1
+-- poissonMoment[1, s_] := 0
+-- poissonMoment[k_, s_] := 
+--   Sum[s * Binomial[k - 1, j] * poissonMoment[j, s], {j, 0, k - 2}]
+--
+-- upperSeries[m_]  :=
+--  Distribute[Integrate[
+--    Sum[(-1)^(j - 1) * 
+--      poissonMoment[j, \[Lambda]] / (j * (j - 1)* \[Lambda]^j),
+--     {j, 3, 2 m - 1}],
+--    \[Lambda]]]
+--
+-- lowerSeries[m_] :=
+--  Distribute[Integrate[
+--    poissonMoment[
+--      2 m + 2, \[Lambda]] / ((2 m + 
+--         1)*\[Lambda]^(2 m + 2)), \[Lambda]]]
+--
+-- upperBound[m_] := upperSeries[m] + (Log[2*Pi*\[Lambda]] + 1)/2 
+--
+-- lowerBound[m_] := upperBound[m] + lowerSeries[m]
+
+upperCoefficients4 :: [Double]
+upperCoefficients4 = [1/12, 1/24, -103/180, -13/40, -1/210]
+
+lowerCoefficients4 :: [Double]
+lowerCoefficients4 = [0,0,0, -105/4, -210, -2275/18, -167/21, -1/72]
+
+upperCoefficients6 :: [Double]
+upperCoefficients6 = [1/12, 1/24, 19/360, 9/80, -38827/2520,
+                      -74855/1008, -73061/2520, -827/720, -1/990]
+
+lowerCoefficients6 :: [Double]
+lowerCoefficients6 = [0,0,0,0,0, -3465/2, -45045, -466235/4, -531916/9,
+                      -56287/10, -629/11, -1/156]
+
+upperCoefficients8 :: [Double]
+upperCoefficients8 = [1/12, 1/24, 19/360, 9/80, 863/2520, 1375/1008,
+                      -3023561/2520, -15174047/720, -231835511/5940,
+                      -18927611/1320, -58315591/60060, -23641/3640,
+                      -1/2730]
+
+lowerCoefficients8 :: [Double]
+lowerCoefficients8 = [0,0,0,0,0,0,0, -2027025/8, -15315300, -105252147,
+                      -178127950, -343908565/4, -10929270, -3721149/14,
+                      -7709/15, -1/272]
+  
+upperCoefficients10 :: [Double]
+upperCoefficients10 = [1/12, 1/24, 19/360, 9,80, 863/2520, 1375/1008,
+                       33953/5040, 57281/1440, -2271071617/11880,
+                       -1483674219/176, -31714406276557/720720,
+                       -7531072742237/131040, -1405507544003/65520,
+                       -21001919627/10080, -1365808297/36720,
+                       -26059/544, -1/5814]
+                      
+lowerCoefficients10 :: [Double]
+lowerCoefficients10 = [0,0,0,0,0,0,0,0,0,-130945815/2, -7638505875,
+                       -438256243425/4, -435477637540, -3552526473925/6,
+                       -857611717105/3, -545654955967/12, -5794690528/3,
+                       -578334559/42, -699043/133, -1/420]
+                 
+upperCoefficients12 :: [Double]
+upperCoefficients12 = [1/12, 1/24, 19/360, 863/2520, 1375/1008,
+                       33953/5040, 57281/1440, 3250433/11880,
+                       378351/176, -37521922090657/720720,
+                       -612415657466657/131040, -3476857538815223/65520,
+                       -243882174660761/1440, -34160796727900637/183600,
+                       -39453820646687/544, -750984629069237/81396,
+                       -2934056300989/9576, -20394527513/12540,
+                       -3829559/9240, -1/10626]
+
+lowerCoefficients12 :: [Double]
+lowerCoefficients12 = [0,0,0,0,0,0,0,0,0,0,0,
+                       -105411381075/4, -5270569053750, -272908057767345/2,
+                       -1051953238104769, -24557168490009155/8,
+                       -3683261873403112, -5461918738302026/3,
+                       -347362037754732, -2205885452434521/100,
+                       -12237195698286/35, -16926981721/22,
+                       -6710881/155, -1/600]
+                      
+-- | Compute entropy directly from its definition. This is just as accurate
+-- as 'alyThm1' for lambda <= 1 and is faster, but is slow for large lambda,
+-- and produces some underestimation due to accumulation of floating point
+-- error.
+directEntropy :: Double -> Double
+directEntropy lambda =   
   negate . sum $
   takeWhile (< negate m_epsilon * lambda) $
   dropWhile (not . (< negate m_epsilon * lambda)) $
   [ let x = probability lambda k in x * log x | k <- [0..]]
 
--- | Compute the entropy of a poisson distribution. Use 'alyThm1' for
--- small lambda, 'alyThm2' for large lambda, and 'bruteforce' plus a
--- linear spline for intermediate lambda.
+-- | Compute the entropy of a poisson distribution using the best available
+-- method.
 poissonEntropy :: Double -> Double
 poissonEntropy lambda
   | lambda == 0 = 0
-  | lambda <= 1 = alyThm1 lambda
-  | lambda >= 4.671691395381868 = alyThm2 lambda
-  | otherwise = bruteforce lambda 
-                - 7.314355647771469e-2 * lambda
-                + 0.3417041234245674
+  | lambda <= 10 = directEntropy lambda
+  | lambda <= 12 = alyThm2 lambda upperCoefficients4 lowerCoefficients4
+  | lambda <= 18 = alyThm2 lambda upperCoefficients6 lowerCoefficients6
+  | lambda <= 24 = alyThm2 lambda upperCoefficients8 lowerCoefficients8
+  | lambda <= 30 = alyThm2 lambda upperCoefficients10 lowerCoefficients10
+  | otherwise = alyThm2 lambda upperCoefficients12 lowerCoefficients12