-- |
-- Module      : Control.Monad.Bayes.Class
-- Description : Types for probabilistic modelling
-- Copyright   : (c) Adam Scibior, 2015-2020
-- License     : MIT
-- Maintainer  : leonhard.markert@tweag.io
-- Stability   : experimental
-- Portability : GHC
--
-- This module defines 'MonadInfer', which can be used to represent a simple model
-- like the following:
--
-- @
-- import Control.Monad (when)
-- import Control.Monad.Bayes.Class
--
-- model :: MonadInfer m => m Bool
-- model = do
--   rain <- bernoulli 0.3
--   sprinkler <-
--     bernoulli $
--     if rain
--       then 0.1
--       else 0.4
--   let wetProb =
--     case (rain, sprinkler) of
--       (True,  True)  -> 0.98
--       (True,  False) -> 0.80
--       (False, True)  -> 0.90
--       (False, False) -> 0.00
--   score wetProb
--   return rain
-- @
module Control.Monad.Bayes.Class
  ( MonadSample,
    random,
    uniform,
    normal,
    gamma,
    beta,
    bernoulli,
    categorical,
    logCategorical,
    uniformD,
    geometric,
    poisson,
    dirichlet,
    MonadCond,
    score,
    factor,
    condition,
    MonadInfer,
    discrete,
    normalPdf,
  )
where

import Control.Monad (when)
import Control.Monad.Trans.Class
import Control.Monad.Trans.Cont
import Control.Monad.Trans.Identity
import Control.Monad.Trans.List
import Control.Monad.Trans.Maybe
import Control.Monad.Trans.RWS hiding (tell)
import Control.Monad.Trans.Reader
import Control.Monad.Trans.State
import Control.Monad.Trans.Writer
import qualified Data.Vector as V
import Data.Vector.Generic as VG
import Numeric.Log
import Statistics.Distribution
import Statistics.Distribution.Beta (betaDistr)
import Statistics.Distribution.Gamma (gammaDistr)
import Statistics.Distribution.Geometric (geometric0)
import Statistics.Distribution.Normal (normalDistr)
import qualified Statistics.Distribution.Poisson as Poisson
import Statistics.Distribution.Uniform (uniformDistr)

-- | Monads that can draw random variables.
class Monad m => MonadSample m where
  -- | Draw from a uniform distribution.
  random ::
    -- | \(\sim \mathcal{U}(0, 1)\)
    m Double

  -- | Draw from a uniform distribution.
  uniform ::
    -- | lower bound a
    Double ->
    -- | upper bound b
    Double ->
    -- | \(\sim \mathcal{U}(a, b)\).
    m Double
  uniform a b = draw (uniformDistr a b)

  -- | Draw from a normal distribution.
  normal ::
    -- | mean μ
    Double ->
    -- | standard deviation σ
    Double ->
    -- | \(\sim \mathcal{N}(\mu, \sigma^2)\)
    m Double
  normal m s = draw (normalDistr m s)

  -- | Draw from a gamma distribution.
  gamma ::
    -- | shape k
    Double ->
    -- | scale θ
    Double ->
    -- | \(\sim \Gamma(k, \theta)\)
    m Double
  gamma shape scale = draw (gammaDistr shape scale)

  -- | Draw from a beta distribution.
  beta ::
    -- | shape α
    Double ->
    -- | shape β
    Double ->
    -- | \(\sim \mathrm{Beta}(\alpha, \beta)\)
    m Double
  beta a b = draw (betaDistr a b)

  -- | Draw from a Bernoulli distribution.
  bernoulli ::
    -- | probability p
    Double ->
    -- | \(\sim \mathrm{B}(1, p)\)
    m Bool
  bernoulli p = fmap (< p) random

  -- | Draw from a categorical distribution.
  categorical ::
    Vector v Double =>
    -- | event probabilities
    v Double ->
    -- | outcome category
    m Int
  categorical ps = fromPMF (ps !)

  -- | Draw from a categorical distribution in the log domain.
  logCategorical ::
    (Vector v (Log Double), Vector v Double) =>
    -- | event probabilities
    v (Log Double) ->
    -- | outcome category
    m Int
  logCategorical = categorical . VG.map (exp . ln)

  -- | Draw from a discrete uniform distribution.
  uniformD ::
    -- | observable outcomes @xs@
    [a] ->
    -- | \(\sim \mathcal{U}\{\mathrm{xs}\}\)
    m a
  uniformD xs = do
    let n = Prelude.length xs
    i <- categorical $ V.replicate n (1 / fromIntegral n)
    return (xs !! i)

  -- | Draw from a geometric distribution.
  geometric ::
    -- | success rate p
    Double ->
    -- | \(\sim\) number of failed Bernoulli trials with success probability p before first success
    m Int
  geometric = discrete . geometric0

  -- | Draw from a Poisson distribution.
  poisson ::
    -- | parameter λ
    Double ->
    -- | \(\sim \mathrm{Pois}(\lambda)\)
    m Int
  poisson = discrete . Poisson.poisson

  -- | Draw from a Dirichlet distribution.
  dirichlet ::
    Vector v Double =>
    -- | concentration parameters @as@
    v Double ->
    -- | \(\sim \mathrm{Dir}(\mathrm{as})\)
    m (v Double)
  dirichlet as = do
    xs <- VG.mapM (`gamma` 1) as
    let s = VG.sum xs
    let ys = VG.map (/ s) xs
    return ys

-- | Draw from a continuous distribution using the inverse cumulative density
-- function.
draw :: (ContDistr d, MonadSample m) => d -> m Double
draw d = fmap (quantile d) random

-- | Draw from a discrete distribution using a sequence of draws from
-- Bernoulli.
fromPMF :: MonadSample m => (Int -> Double) -> m Int
fromPMF p = f 0 1
  where
    f i r = do
      when (r < 0) $ error "fromPMF: total PMF above 1"
      let q = p i
      when (q < 0 || q > 1) $ error "fromPMF: invalid probability value"
      b <- bernoulli (q / r)
      if b then pure i else f (i + 1) (r - q)

-- | Draw from a discrete distributions using the probability mass function.
discrete :: (DiscreteDistr d, MonadSample m) => d -> m Int
discrete = fromPMF . probability

-- | Monads that can score different execution paths.
class Monad m => MonadCond m where
  -- | Record a likelihood.
  score ::
    -- | likelihood of the execution path
    Log Double ->
    m ()

-- | Synonym for 'score'.
factor ::
  MonadCond m =>
  -- | likelihood of the execution path
  Log Double ->
  m ()
factor = score

-- | Hard conditioning.
condition :: MonadCond m => Bool -> m ()
condition b = score $ if b then 1 else 0

-- | Monads that support both sampling and scoring.
class (MonadSample m, MonadCond m) => MonadInfer m

-- | Probability density function of the normal distribution.
normalPdf ::
  -- | mean μ
  Double ->
  -- | standard deviation σ
  Double ->
  -- | sample x
  Double ->
  -- | relative likelihood of observing sample x in \(\mathcal{N}(\mu, \sigma^2)\)
  Log Double
normalPdf mu sigma x = Exp $ logDensity (normalDistr mu sigma) x

----------------------------------------------------------------------------
-- Instances that lift probabilistic effects to standard tranformers.

instance MonadSample m => MonadSample (IdentityT m) where
  random = lift random
  bernoulli = lift . bernoulli

instance MonadCond m => MonadCond (IdentityT m) where
  score = lift . score

instance MonadInfer m => MonadInfer (IdentityT m)

instance MonadSample m => MonadSample (MaybeT m) where
  random = lift random

instance MonadCond m => MonadCond (MaybeT m) where
  score = lift . score

instance MonadInfer m => MonadInfer (MaybeT m)

instance MonadSample m => MonadSample (ReaderT r m) where
  random = lift random
  bernoulli = lift . bernoulli

instance MonadCond m => MonadCond (ReaderT r m) where
  score = lift . score

instance MonadInfer m => MonadInfer (ReaderT r m)

instance (Monoid w, MonadSample m) => MonadSample (WriterT w m) where
  random = lift random
  bernoulli = lift . bernoulli
  categorical = lift . categorical

instance (Monoid w, MonadCond m) => MonadCond (WriterT w m) where
  score = lift . score

instance (Monoid w, MonadInfer m) => MonadInfer (WriterT w m)

instance MonadSample m => MonadSample (StateT s m) where
  random = lift random
  bernoulli = lift . bernoulli
  categorical = lift . categorical

instance MonadCond m => MonadCond (StateT s m) where
  score = lift . score

instance MonadInfer m => MonadInfer (StateT s m)

instance (MonadSample m, Monoid w) => MonadSample (RWST r w s m) where
  random = lift random

instance (MonadCond m, Monoid w) => MonadCond (RWST r w s m) where
  score = lift . score

instance (MonadInfer m, Monoid w) => MonadInfer (RWST r w s m)

instance MonadSample m => MonadSample (ListT m) where
  random = lift random
  bernoulli = lift . bernoulli
  categorical = lift . categorical

instance MonadCond m => MonadCond (ListT m) where
  score = lift . score

instance MonadInfer m => MonadInfer (ListT m)

instance MonadSample m => MonadSample (ContT r m) where
  random = lift random

instance MonadCond m => MonadCond (ContT r m) where
  score = lift . score

instance MonadInfer m => MonadInfer (ContT r m)