(*^
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(* This notebook explores some of the ideas presented in Cooper 
   and Herskovits [1992]. The following code assumes familiarity 
   with this paper. *)
:[font = input; preserveAspect]
Needs["Dynamics`Packages`CooperandHerskovits`"]
:[font = input; preserveAspect]
(* A database is a list of cases where a case is a list of assignments
   in which the ith element of the list is the assignment to the ith 
   variables in that particular case. Here is the database in Table 1
   of Cooper and Herskovits [1992]. *)
:[font = input; preserveAspect]
db1 = {{1, 0, 0},
       {1, 1, 1},
       {0, 0, 1},
       {1, 1, 1},
       {0, 0, 0},
       {0, 1, 1},
       {1, 1, 1},
       {0, 0, 0},
       {1, 1, 1},
       {0, 0, 0}} ;
:[font = input; preserveAspect; startGroup]
TableForm[db1]
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0}, {1, 1, 1}, {0, 0, 1}, {1, 1, 1}, {0, 0, 0}, 
 
   {0, 1, 1}, {1, 1, 1}, {0, 0, 0}, {1, 1, 1}, {0, 0, 0}}]
;[o]
1   0   0

1   1   1

0   0   1

1   1   1

0   0   0

0   1   1

1   1   1

0   0   0

1   1   1

0   0   0
:[font = input; preserveAspect]
(* A network structure is represented as a list of lists such that ith
   list is the list of the parents of the ith variable.  For example, 
   the network structure X_1 -> X_2 -> X_ 3 is represented by the list
   {{},{1},{2}} and the network structure X_3 <- X_1 -> X_2 is 
   represented by the list {{},{1},{1}}. The following correspond to
   B_{S1} and B_{S2} in Cooper and Herskovits [1992], respectively. *)
:[font = input; preserveAspect; startGroup]
bs1 = {{},{1},{2}} 
:[font = output; output; inactive; preserveAspect; endGroup]
{{}, {1}, {2}}
;[o]
{{}, {1}, {2}}
:[font = input; preserveAspect; startGroup]
bs2 = {{},{1},{1}} 
:[font = output; output; inactive; preserveAspect; endGroup]
{{}, {1}, {1}}
;[o]
{{}, {1}, {1}}
:[font = input; preserveAspect]
(* Compare the two Bayesian network structures given in 
   Figures 1 and 2 of Cooper and Herskovits [1992]. *)
:[font = input; preserveAspect]
(* First compute Pr(bs1,db1)/Pr(bs1). *)
:[font = input; preserveAspect; startGroup]
N[ComputeProbability[bs1,db1]] 
:[font = output; output; inactive; preserveAspect; endGroup]
2.226854078705930558*10^-9
;[o]
          -9
2.22685 10
:[font = input; preserveAspect]
(* Next compute Pr(bs2,db1)/Pr(bs2). *)
:[font = input; preserveAspect; startGroup]
N[ComputeProbability[bs2,db1]]
:[font = output; output; inactive; preserveAspect; endGroup]
2.226854078705930558*10^-10
;[o]
          -10
2.22685 10
:[font = input; preserveAspect]
Needs["Dynamics`Packages`MarkovExamples`"]
:[font = input; preserveAspect]
Needs["Dynamics`Packages`BayesianNetworks`"]
:[font = input; preserveAspect; startGroup]
network1 = MarkovExample[1]
:[font = output; output; inactive; preserveAspect; endGroup]
BayesianNetwork[9]
;[o]
BayesianNetwork[9]
:[font = input; preserveAspect; startGroup]
ShowGraph[DependencyGraph[network1], Directed] ;
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network2 = MarkovExample[2]
:[font = output; output; inactive; preserveAspect; endGroup]
BayesianNetwork[10]
;[o]
BayesianNetwork[10]
:[font = input; preserveAspect; startGroup]
ShowGraph[DependencyGraph[network2], Directed] ;
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:[font = input; preserveAspect; startGroup]
bs3 = Table[Parents[i,network1],{i,NumVars[network1]}]
:[font = output; output; inactive; preserveAspect; endGroup]
{{}, {}, {}, {}, {1, 6}, {2}, {2, 3}, {4}}
;[o]
{{}, {}, {}, {}, {1, 6}, {2}, {2, 3}, {4}}
:[font = input; preserveAspect; startGroup]
bs4 = Table[Parents[i,network2],{i,NumVars[network2]}]
:[font = output; output; inactive; preserveAspect; endGroup]
{{}, {}, {}, {}, {1, 6}, {2, 4}, {2, 3}, {2, 4}}
;[o]
{{}, {}, {}, {}, {1, 6}, {2, 4}, {2, 3}, {2, 4}}
:[font = input; preserveAspect]
(* The Cooper and Herskovits implementation requires that 
   Boolean True be represented as 1 and Boolean False as 0.
   Here we generate random samples in {True, False} format
   from the Bayesian network and convert them to {1,0} format. *)
:[font = input; preserveAspect]
db2 = Table[Map[(If[# == True,1,0])&,
                RandomSample[network1]],
            {10}] ;
:[font = input; preserveAspect; startGroup]
TableForm[db2]
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{0, 1, 0, 0, 1, 1, 1, 1}, {1, 0, 0, 0, 1, 1, 1, 1}, 
 
   {1, 1, 1, 0, 1, 1, 1, 1}, {1, 0, 1, 1, 1, 1, 1, 1}, 
 
   {0, 0, 0, 0, 1, 1, 0, 0}, {0, 1, 0, 0, 1, 1, 1, 1}, 
 
   {0, 0, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 1, 1, 1, 0, 1}, 
 
   {0, 0, 0, 1, 1, 1, 0, 1}, {1, 0, 1, 1, 1, 1, 0, 1}}]
;[o]
0   1   0   0   1   1   1   1

1   0   0   0   1   1   1   1

1   1   1   0   1   1   1   1

1   0   1   1   1   1   1   1

0   0   0   0   1   1   0   0

0   1   0   0   1   1   1   1

0   0   1   1   1   1   1   1

0   0   0   1   1   1   0   1

0   0   0   1   1   1   0   1

1   0   1   1   1   1   0   1
:[font = input; preserveAspect]
(* Compute Pr(bs3,db2)/Pr(bs3). *)
:[font = input; preserveAspect; startGroup]
N[ComputeProbability[bs3,db2]]
:[font = output; output; inactive; preserveAspect; endGroup]
1.764232682734354533*10^-22
;[o]
          -22
1.76423 10
:[font = input; preserveAspect]
(* Compare with Pr(bs4,db2)/Pr(bs4). *)
:[font = input; preserveAspect; startGroup]
N[ComputeProbability[bs4,db2]]
:[font = output; output; inactive; preserveAspect; endGroup]
9.801292681857525183*10^-23
;[o]
          -23
9.80129 10
:[font = input; preserveAspect]
Needs["Dynamics`Packages`TemporalBayesianNetworks`"]
:[font = input; preserveAspect; startGroup]
process1 = TemporalBayesianNetworkToMarkovProcess[network1]
:[font = output; output; inactive; preserveAspect; endGroup]
MarkovProcess[5]
;[o]
MarkovProcess[5]
:[font = input; preserveAspect]
Needs["Dynamics`Packages`MarkovProcesses`"]
:[font = input; preserveAspect; startGroup]
ShowGraph[TransitionGraph[process1]] ;
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup]
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(* Generate n independent random samples using the 
   Markov process representation. *)
:[font = input; preserveAspect]
Module[{u,v,s = {},n = 10}, 
  Do[u = Random[Integer,{1,NumStates[process1]}] ;
     v = RandomNextState[u,process1] ;
     s = Append[s,Join[IntegerToStateVector[u,process1],
                       IntegerToStateVector[v,process1]]],
     {n}] ;
  db3 = s] ;
:[font = input; preserveAspect; startGroup]
TableForm[db3]
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0, 1, 1, 1, 0, 1}, {0, 0, 1, 0, 1, 1, 0, 0}, 
 
   {1, 1, 1, 1, 1, 1, 1, 1}, {1, 0, 1, 0, 1, 1, 0, 0}, 
 
   {0, 0, 1, 0, 1, 1, 1, 0}, {1, 1, 1, 1, 1, 1, 1, 1}, 
 
   {0, 0, 0, 0, 1, 1, 0, 0}, {0, 0, 1, 0, 1, 1, 1, 0}, 
 
   {1, 1, 1, 0, 1, 1, 1, 0}, {0, 0, 1, 0, 1, 1, 1, 0}}]
;[o]
1   0   0   1   1   1   0   1

0   0   1   0   1   1   0   0

1   1   1   1   1   1   1   1

1   0   1   0   1   1   0   0

0   0   1   0   1   1   1   0

1   1   1   1   1   1   1   1

0   0   0   0   1   1   0   0

0   0   1   0   1   1   1   0

1   1   1   0   1   1   1   0

0   0   1   0   1   1   1   0
:[font = input; preserveAspect]
(* Generate n = h k dependent random samples from h trajectories of
   length k using the Markov process representation. *)
:[font = input; preserveAspect]
Module[{u,v,s = {},h = 2,k = 5}, 
  Do[u = Random[Integer,{1,NumStates[process1]}] ;
     Do[v = RandomNextState[u,process1] ;
        u = v ;
        s = Append[s,Join[IntegerToStateVector[u,process1],
                          IntegerToStateVector[v,process1]]], 
        {k}], 
     {h}] ; 
  db4 = s] ;
:[font = input; preserveAspect; startGroup]
TableForm[db4]
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 1, 0, 1, 0, 1, 0}, {1, 1, 0, 1, 1, 1, 0, 1}, 
 
   {1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1}, 
 
   {1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1}, 
 
   {1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1}, 
 
   {1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1}}]
;[o]
1   0   1   0   1   0   1   0

1   1   0   1   1   1   0   1

1   1   1   1   1   1   1   1

1   1   1   1   1   1   1   1

1   1   1   1   1   1   1   1

1   1   1   1   1   1   1   1

1   1   1   1   1   1   1   1

1   1   1   1   1   1   1   1

1   1   1   1   1   1   1   1

1   1   1   1   1   1   1   1
:[font = input; preserveAspect]
(* Compute Pr(bs3,db3)/Pr(bs3). *)
:[font = input; preserveAspect; startGroup]
N[ComputeProbability[bs3,db3]]
:[font = output; output; inactive; preserveAspect; endGroup]
1.575864089004905221*10^-20
;[o]
          -20
1.57586 10
:[font = input; preserveAspect]
(* Compare with Pr(bs3,db4)/Pr(bs3). *)
:[font = input; preserveAspect; startGroup]
N[ComputeProbability[bs4,db3]]
:[font = output; output; inactive; preserveAspect; endGroup]
2.286968292433422542*10^-21
;[o]
          -21
2.28697 10
^*)