```{- | Tutorial explaining how to make infereces with the library.

Thus tutorial is using examples from the module "Bayes.Examples". Please,
refer to this module for documentation about how the example bayesian networks are
created or loaded.

/Inferences/

The function 'inferencesOnStandardNetwork' is showing how to use variable elimination
and factor elimination to make inferences.

First, the 'example' is loaded to make its variables and its bayesian network available:

@
let ([winter,sprinkler,rain,wet,road],exampleG) = 'example'
@

Then, we compute a prior marginal. Prior means that no evidence is used. A bayesian
network is a factorisation of a distribution P(A B C ...). If you want to know the
probability of only A, you need to sum out the other variables to eliminate them and get
P(A). To compute this prior marginal using variable elimnation, you need to give an elimination
order. The complexity of the computation is depending on the elimination order chosen.

For instance, if you want to compute the prior probability of rain, you can write:

@
'priorMarginal' exampleG [winter,sprinkler,wet,road] [rain]
@

Now, if you have observed that the grass is wet and want to take into account thios observation
to compute the posterior probability of rain (after observation):

@
'posteriorMarginal' exampleG [winter,sprinkler,wet,road] [rain]  [wet '=:' True]
@

If you want to combine several observations:

@
'posteriorMarginal' exampleG [winter,sprinkler,wet,road] [rain]  [wet '=:' True, sprinkler '=:' True]
@

There are several problems with variable elimination:

* You have to specify an elimination order

* If you want to compute another marginal (for instance probability of winter), you have
to recompute everything.

But, there exists another category of elimination algorithms based upon factor elimination.
They require the creation of an auxiliary data structure : the junction tree.

This tree is then used for computing all marginals (without having to recompute everything).
The junction tree is equivalent to giving an elimination order.

So, the previous examples can also be computed with factor elimination. First, the
junction tree must created:

@
let jt = 'createJunctionTree' 'nodeComparisonForTriangulation' exampleG
@

The junction tree being equivalent to an elimination order, the order chosen will
depend on a cost function. In the previous example, the cost function 'nodeComparisonForTriangulation'
is used. Other cost functions may be introduced in a futute version of this library.

Once the junction tree has been computd, it can be used to compute several marginals:

@
'posterior' jt rain
@

The function is called posterior and will compute posterior only when solme evidence has
been introduced into the tree. Otherwise it is computing a prior.

To set evidence, you need to update the junction tree with new evidence:

@
let jt' = 'updateEvidence' [wet '=:'' True] jt
'posterior' jt' rain
@

/Inferences with an imported network/

There is a slight additional difficulty with imported networks : you need
to create new data type to be able to set evidence.

For instance, in the cancer network there is a Coma variable with levels Present or Absent.
When imported, those levels are imported as number. But, the evidence API in this library is
requiring enumerations.

So, you need to create a 'Coma' type:

@
data Coma = Present | Absent deriving(Eq,Enum,Bounded)
@

and check that 'Present' is corresponding to the level 0 in the importd network.

Once this datatype is created, you can easily use the cancer network. First we load
the network and import the discrete variables of type 'DV' from the names of the nodes in the
network (not the label of the nodes)

@
print \"CANCER NETWORK\"
(varmap,cancer) <- 'exampleImport'
print cancer
let [varA,varB,varC,varD,varE] = fromJust \$ mapM (flip Map.lookup varmap) ["A","B","C","D","E"]
@

Once the variables are available, you can create the junction tree and start making inferences:

@
let jtcancer = 'createJunctionTree' 'nodeComparisonForTriangulation' cancer
--
mapM_ (\x -> putStrLn (show x) >> (print . 'posterior' jtcancer \$ x)) [varA,varB,varC,varD,varE]
--
print \"UPDATED EVIDENCE\"
let jtcancer' = 'updateEvidence' [varD '=:' Present] jtcancer
mapM_ (\x -> putStrLn (show x) >> (print . 'posterior' jtcancer' \$ x)) [varA,varB,varC,varD,varE]
@

-}
module Bayes.Examples.Tutorial(
-- * Tests with the standard network
inferencesOnStandardNetwork
-- * Tests with the cancer network
, inferencesOnCancerNetwork
, Coma(..)
, miscTest
) where

import Bayes.Factor
import Bayes
import Bayes.VariableElimination
import Bayes.Examples(example, exampleJunction,exampleImport,exampleDiabete, exampleAsia, examplePoker, exampleFarm,examplePerso,anyExample)
import Bayes.FactorElimination
import Data.Function(on)
import qualified Data.Map as Map
import Data.Maybe(fromJust,mapMaybe)
import System.Exit(exitSuccess)
import qualified Data.List as L((\\))

miscDiabete = do
(varmap,perso) <- exampleDiabete
let jtperso = createJunctionTree nodeComparisonForTriangulation perso
cho0 = fromJust . Map.lookup "cho_0" \$ varmap
print \$ posterior jtperso cho0

miscTest s = do
(varmap,perso) <- anyExample s
let names = Map.keys varmap
l =  mapMaybe (flip Map.lookup varmap) names
jtperso = createJunctionTree nodeComparisonForTriangulation perso
print perso
print jtperso
print "FACTOR ELIMINATION"
let post (v,name) = do
putStrLn name
print \$ posterior jtperso v
mapM_ post  (zip l names)

print "VARIABLE ELIMINATION"
let prior (v,name) = do
putStrLn name
print \$ priorMarginal perso (l L.\\ [v]) [v]
mapM_ prior (zip l names)

-- | Type defined to set the evidence on the Coma variable
-- from the cancer network.
data Coma = Present | Absent deriving(Eq,Enum,Bounded)

-- | Inferences with the cancer network
inferencesOnCancerNetwork = do
print "CANCER NETWORK"
(varmap,cancer) <- exampleImport
print cancer
let [varA,varB,varC,varD,varE] = fromJust \$ mapM (flip Map.lookup varmap) ["A","B","C","D","E"]
let jtcancer = createJunctionTree nodeComparisonForTriangulation cancer

mapM_ (\x -> putStrLn (show x) >> (print . posterior jtcancer \$ x)) [varA,varB,varC,varD,varE]

print "UPDATED EVIDENCE : Coma present"
let jtcancer' = updateEvidence [varD =: Present] jtcancer
mapM_ (\x -> putStrLn (show x) >> (print . posterior jtcancer' \$ x)) [varA,varB,varC,varD,varE]

print "UPDATED EVIDENCE : Coma absent"
let jtcancer' = updateEvidence [varD =: Absent] jtcancer
mapM_ (\x -> putStrLn (show x) >> (print . posterior jtcancer' \$ x)) [varA,varB,varC,varD,varE]

-- | Inferences with the standard network
inferencesOnStandardNetwork = do
let ([winter,sprinkler,rain,wet,road],exampleG) = example

putStrLn ""
print "VARIABLE ELIMINATION"
putStrLn ""
print "Prior Marginal : probability of rain"
let m = priorMarginal exampleG [winter,sprinkler,wet,road] [rain]
print m
putStrLn ""

print "Posterior Marginal : probability of rain if grass wet"
let m = posteriorMarginal exampleG [winter,sprinkler,wet,road] [rain]  [wet =: True]
print m
putStrLn ""

print "Posterior Marginal : probability of rain if grass wet and sprinkler used"
let m = posteriorMarginal exampleG [winter,sprinkler,wet,road] [rain]  [wet =: True, sprinkler =: True]
print m
putStrLn ""

let jt = createJunctionTree nodeComparisonForTriangulation exampleG

putStrLn ""
print "FACTOR ELIMINATION"
putStrLn ""
print "Prior Marginal : probability of rain"
let m = posterior jt rain
print m
putStrLn ""

let jt' = updateEvidence [wet =: True] jt

print "Posterior Marginal : probability of rain if grass wet"
let m = posterior jt' rain
print m
putStrLn ""

let jt'' = clearEvidence jt'
print "Prior Marginal : probability of rain"
let m = posterior jt rain
print m
putStrLn ""

let jt3 = updateEvidence [wet =: True, sprinkler =: True] jt'

print "Posterior Marginal : probability of rain if grass wet and sprinkler used"
let m = posterior jt3 rain
print m
putStrLn ""

return ()
```