```module Data.HMM
(Prob, HMM, train, bestSequence, sequenceProb)
where

import qualified Data.Map as M
import Data.List (sort, groupBy, maximumBy)
import Data.Maybe (fromMaybe, fromJust)
import System.IO.Unsafe (unsafeInterleaveIO)
import System.Environment (getArgs)
import qualified Data.Foldable
import Debug.Trace
import Data.Lognum

type Prob = Lognum Double

-- | The type of Hidden Markov Models.
data HMM state observation = HMM [state] [Prob] [[Prob]] (observation -> [Prob])

-- | Perform a single step in the Viterbi algorithm.
--
--   Takes a list of path probabilities, and an observation, and returns the updated
--   list of (surviving) paths with probabilities.
viterbi :: Ord observation =>
HMM state observation
-> [(Prob, [state])]
-> observation
-> [(Prob, [state])]
viterbi (HMM states _ state_transitions observations) prev x =
[maximumBy (compare `on` fst)
[(transition_prob * prev_prob * observation_prob,
new_state:path)
| transition_prob <- transition_probs
| (prev_prob, path) <- prev
| observation_prob <- observation_probs]
| transition_probs <- state_transitions
| new_state <- states]
where
observation_probs = observations x

-- | The initial value for the Viterbi algorithm
viterbi_init :: HMM state observation -> [(Prob, [state])]
viterbi_init (HMM states state_probs _ _) = zip state_probs (map (:[]) states)

-- | Perform a single step of the forward algorithm
--
--   Each item in the input and output list is the probability that the system
--   ended in the respective state.
forward :: Ord observation =>
HMM state observation
-> [Prob]
-> observation
-> [Prob]
forward (HMM _ _ state_transitions observations) prev x =
[sum [transition_prob * prev_prob * observation_prob
| transition_prob <- transition_probs
| prev_prob <- prev
| observation_prob <- observation_probs]
| transition_probs <- state_transitions]
where
observation_probs = observations x

-- | The initial value for the forward algorithm
forward_init :: HMM state observation -> [Prob]
forward_init (HMM _ state_probs _ _) = state_probs

learn_states :: (Ord state, Fractional prob) => [(observation, state)] -> M.Map state prob
learn_states xs = histogram \$ map snd xs

learn_transitions :: (Ord state, Fractional prob) => [(observation, state)] -> M.Map (state, state) prob
learn_transitions xs = let xs' = map snd xs in
histogram \$ zip xs' (tail xs')

learn_observations ::  (Ord state, Ord observation, Fractional prob) =>
M.Map state prob
-> [(observation, state)]
-> M.Map (observation, state) prob
learn_observations state_prob = M.mapWithKey (\ (observation, state) prob -> prob / (fromJust \$ M.lookup state state_prob))
. histogram

histogram :: (Ord a, Fractional prob) => [a] -> M.Map a prob
histogram xs = let hist = foldr (flip (M.insertWith (+)) 1) M.empty xs in
M.map (/ M.fold (+) 0 hist) hist

readBrownFile :: FilePath -> IO [(String, String)]
where
split [] = ([], [])
split ('/':xs) = ([], xs)
split (x:xs) = let (first, snd) = split xs in
(x:first, snd)

-- | Calculate the parameters of an HMM from a list of observations
--   and the corresponding states.
train :: (Ord observation, Ord state) =>
[(observation, state)]
-> HMM state observation
train sample = model
where
states = learn_states sample
state_list = M.keys states

transitions = learn_transitions sample
trans_prob_mtx = [[fromMaybe 1e-10 \$ M.lookup (old_state, new_state) transitions
| old_state <- state_list]
| new_state <- state_list]

observations = learn_observations states sample
observation_probs = fromMaybe (fill state_list []) . (flip M.lookup \$
M.fromList \$ map (\ (e, xs) -> (e, fill state_list xs)) \$
map (\ xs -> (fst \$ head xs, map snd xs)) \$
groupBy     ((==) `on` fst)
[(observation, (state, prob))
| ((observation, state), prob) <- M.toAscList observations])

initial = map (\ state -> (fromJust \$ M.lookup state states, [state])) state_list

model = HMM state_list (fill state_list \$ M.toAscList states) trans_prob_mtx observation_probs

fill :: Eq state => [state] -> [(state, Prob)] -> [Prob]
fill states [] = map (const 1e-10) states
fill (s:states) xs@((s', p):xs') = if s /= s' then
1e-10 : fill states xs
else
p : fill states xs'

-- | Test Viterbi's algorithm on an HMM by comparing the predicted states
--   against known states for the observations.
testViterbi :: (Ord observation, Ord state) =>
HMM state observation
-> [(observation, state)]
-> Rational
testViterbi hmm testData = (fromIntegral \$ length \$ filter id \$ zipWith (==) (bestSequence hmm observations) states)
/ (fromIntegral \$ length testData)
where
observations = map fst testData
states = map snd testData

train2 :: FilePath -> IO (HMM String String)
train2 = (liftM train) . readBrownFile

-- | Calculate the most likely sequence of states for a given sequence of observations
--   using Viterbi's algorithm
bestSequence :: (Ord observation) => HMM state observation -> [observation] -> [state]
bestSequence hmm = (reverse . tail . snd . (maximumBy (compare `on` fst))) . (foldl (viterbi hmm) (viterbi_init hmm))

-- | Calculate the probability of a given sequence of observations
--    using the forward algorithm.
sequenceProb :: (Ord observation) => HMM state observation -> [observation] -> Prob
sequenceProb hmm = sum . (foldl (forward hmm) (forward_init hmm))

on f g a b = f (g a) (g b)
```