## Probability Trees

Posted on September 30, 2016
``````{-# language DeriveFunctor, DeriveFoldable #-}
{-# language PatternSynonyms, ViewPatterns #-}

module ProbTree where

import Data.Monoid
import qualified Data.Map.Strict as Map
import Data.Map.Strict (Map)
import Control.Arrow
import Data.Ratio
import Data.Foldable``````

Previously, I tried to figure out how to make the probability monad more “listy”. I read a little more about the topic (especially Erwig and Kollmansberger 2006; and Kidd 2007).

I then thought about what a probability monad would look like if it was based on other data structures. I feel like the standard version really wants to be:

``````newtype ProperProb a = ProperProb
{ yes :: Map a (Product Rational) }``````

But of course a monad instance isn’t allowed.

Similar to a map, though, is a binary tree:

``````data BinaryTree a = Leaf
| Node (BinaryTree a) a (BinaryTree a)``````

And it feels better for probability - flatter, somehow. Transmuting it into a probability-thing:

``````data Odds a = Certain a
| Choice (Odds a) Rational (Odds a)
deriving (Eq, Functor, Foldable, Show)``````

That looks good to me. A choice between two different branches feels more natural than a choice between a head and a tail.

The fold is similar to before, with an unfold for good measure:

``````foldOdds :: (b -> Rational -> b -> b) -> (a -> b) -> Odds a -> b
foldOdds f b = r where
r (Certain x) = b x
r (Choice xs p ys) = f (r xs) p (r ys)

unfoldOdds :: (b -> Either a (b,Rational,b)) -> b -> Odds a
unfoldOdds f = r where
r b = case f b of
Left a -> Certain a
Right (x,p,y) -> Choice (r x) p (r y)

fi :: Bool -> a -> a -> a
fi True  t _ = t
fi False _ f = f``````

I changed the pattern synonym a little:

``````
unRatio :: Num a => Rational -> (a,a)
unRatio = numerator   &&& denominator
>>> fromInteger *** fromInteger

pattern n :% d <- (unRatio -> (n,d))``````

Then, the `probOf` function:

``````probOf :: Eq a => a -> Odds a -> Rational
probOf e = foldOdds f b where
b x = fi (e == x) 1 0
f x (n:%d) y = (x * n + y * d) / (n + d)``````

This version doesn’t have the option for short-circuiting on the first value it finds.

For generating from lists, you can try to evenly divide the list among each branch.

``````fromListOdds :: (([b], Int) -> Integer) -> (b -> a) -> [b] -> Maybe (Odds a)
fromListOdds fr e = r where
r [] = Nothing
r xs = Just (unfoldOdds f (xs, length xs))
f ([x],_) = Left (e x)
f (xs ,n) = Right ((ys,l), fr (ys,l) % fr (zs,r), (zs,r)) where
l = n `div` 2
r = n - l
(ys,zs) = splitAt l xs

equalOdds :: [a] -> Maybe (Odds a)
equalOdds = fromListOdds (fromIntegral . snd) id

fromDistrib :: [(a,Integer)] -> Maybe (Odds a)
fromDistrib = fromListOdds (sum . map snd . fst) fst``````

What’s really nice about this version is the fact that the old `append` is just the `Choice` constructor, leaving the instances to be really nice:

``````flatten :: Odds (Odds a) -> Odds a
flatten = foldOdds Choice id

instance Applicative Odds where
pure = Certain
fs <*> xs = flatten (fmap (<\$> xs) fs)

instance Monad Odds where
x >>= f = flatten (f <\$> x)``````

Finally, as a bonus, to remove duplicates:

``````lcd :: Foldable f => f Rational -> Integer
lcd = foldl' (\a e -> lcm a (denominator e)) 1

toDistrib :: Odds a -> [(a,Integer)]
toDistrib = factorOut . foldOdds f b where
b x = [(x,1)]
f l p r = (map.fmap) (n%t*) l ++ (map.fmap) (d%t*) r where
n = numerator p
d = denominator p
t = n + d
factorOut xs = (map.fmap) (numerator . (lcd'*)) xs where
lcd' = fromIntegral . lcd . map snd \$ xs

counts :: (Ord a, Num n) => [(a,n)] -> [(a,n)]
counts =
Map.assocs .
Map.fromListWith (+)

compress :: Ord a => Odds a -> Odds a
compress xs = let Just ys = (fromDistrib . counts . toDistrib) xs in ys``````

After reading yet more on this, I found that the main issue with the monad is its performance. Two articles in particular: Larsen (2011), and Ścibior, Ghahramani, and Gordon (2015), refer to a GADT implementation of the monad which maximises laziness.

Erwig, Martin, and Steve Kollmansberger. 2006. “Functional pearls: Probabilistic functional programming in Haskell.” Journal of Functional Programming 16 (1): 21–34. doi:10.1017/S0956796805005721.

Kidd, Eric. 2007. “Build your own probability monads.”

Larsen, Ken Friis. 2011. “Memory Efficient Implementation of Probability Monads.”

Ścibior, Adam, Zoubin Ghahramani, and Andrew D. Gordon. 2015. “Practical Probabilistic Programming with Monads.” In Proceedings of the 2015 ACM SIGPLAN Symposium on Haskell, 50:165–176. Haskell ’15. New York, NY, USA: ACM. doi:10.1145/2804302.2804317.