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Cell[CellGroupData[{
Cell["BeliefMatrices.nb", "Title"],

Cell["\<\

Matrix calculus for belief functions\
\>", "Subtitle"],

Cell["Minh Ha-Duong, 2002, 2003", "Author"],

Cell["minh.ha.duong@cmu.edu", "Address"],

Cell[BoxData[
    \(Version : \ \(25/3\)/2003\)], "Input"],

Cell["\<\
This package implements   Matrix calculus for belief functions, as \
described by Philippe Smets (2001)
http://iridia.ulb.ac.be/~psmets
Licensed under GPL version 2 or later.\
\>", "Abstract"],

Cell[CellGroupData[{

Cell["TODO", "Subsubsection"],

Cell["\<\
Add {:Keywords:}
Replace b, pl, bel, q by english full words.
BeliefMatrices::usage
i defaults to Length[\[CapitalOmega]]\
\>", "Abstract"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Public interface", "Subsection",
  InitializationCell->True],

Cell[BoxData[
    \(BeginPackage["\<BeliefMatrices`\>"]\)], "Input",
  InitializationCell->True],

Cell[BoxData[{
    \(Belief::usage = \*"\"\<Belief[m,i] lists the   \!\(2\^i\)-1   values \
the belief function defined by the basic belief assignment  m  when the frame \
of reference has  i  elements. Belief of subset  k  is Belief[m,i]\
\[LeftDoubleBracket]k\[RightDoubleBracket], except for k=0 (the empty subset) \
in which case belief is 0.\>\""\n\), "\[IndentingNewLine]", 
    \(Plausibility::usage = \*"\"\<Plausibility[m,i] lists the   \!\(2\^i\)-1 \
 values of the plausibility function defined by the basic belief assignment  \
m  when the frame of reference has  i  elements. Plausibility of subset  k  \
is Plausibility[m,i]\[LeftDoubleBracket]k\[RightDoubleBracket], except for \
k=0 (the empty subset) in which case plausibility is 0.\>\""\
\[IndentingNewLine]\), "\n", 
    \(Commonality::usage = \*"\"\<Commonality[m,i] lists the  \!\(2\^i\)-1  \
values of the commonality function defined by the  basic belief assignment  m \
 when the frame of reference has  i  elements. Commonality of subset  k  is \
Commonality[m,i]\[LeftDoubleBracket]k\[RightDoubleBracket], except for k=0 \
(the empty subset) in which case commonality is 1.\>\""\[IndentingNewLine]\), \
"\n", 
    \(Implicability::usage = \*"\"\<Implicability[m,i]  lists the   \
\!\(2\^i\)-1  values of the commonality function defined by the  basic belief \
assignment  m  when the frame of reference has  i  elements. Implicability of \
subset  k  is  Implicability[m,i]\[LeftDoubleBracket]k\[RightDoubleBracket], \
except for k=0 (the empty subset) in which case implicability is  m[0].\>\""\n\
\), "\[IndentingNewLine]", 
    \(PignisticProbability::usage = "\<PignisticProbability[m,i]  lists the  \
i  values of the pignistic probability function defined by the  basic belief \
assignment  m  when the frame of reference has  i  elements. Pignistic \
probability of outcome  k  is PignisticProbability[m,i]\[LeftDoubleBracket]k\
\[RightDoubleBracket].\>"\)}], "Input",
  CellLabel->"In[193]:=",
  InitializationCell->True],

Cell[BoxData[{
    \(\(bba::usage = "\<Symbol  bba  represents the the basic belief \
assignment (also noted  m  in the litterature)\>";\)\), \
"\[IndentingNewLine]", 
    \(\(b::usage = "\<Symbol  b  represents the implicability \
function\>";\)\), "\[IndentingNewLine]", 
    \(\(q::usage = "\<Symbol  q  represents the commonality function\>";\)\), \
"\[IndentingNewLine]", 
    \(\(bel::usage = "\<Symbol  bel  represents the belief function\>";\)\), \
"\[IndentingNewLine]", 
    \(\(pl::usage = "\<Symbol  pl  represents the plausibility function\>";\)\
\)}], "Input",
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  InitializationCell->True],

Cell[BoxData[
    \(\(TransformMatrix::usage = "\<TransformMatrix[x,y,i] is the 2^i \
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x  and  y  are ordered according to usage in the belief function theory: In x\
\[LeftDoubleBracket]k\[RightDoubleBracket] , the argument  k  must be read in \
binary as a bitfield representing a subset of {1, ..., i}.\n
Acceptable x and y are:\n
 bba     the basic belief assignment (also noted  m  in the litterature),\n
 b       the implicability function\n
 q       the commonality function\n
 bel     the belief function (with the convention bel[0] = 1)\n
 pl      the plausibility function (with the convention pl[0]=1).\>";\)\)], \
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basic belief assignment  m. They form a commutative and associative \
familly.\>";\)\(\[IndentingNewLine]\)
    \)\), "\[IndentingNewLine]", 
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2^i  dimension square Depsterian generalization matrix associated with the \
basic belief assignment  m. They form a commutative and associative \
familly.\>";\)\(\[IndentingNewLine]\)
    \)\), "\[IndentingNewLine]", 
    \(\(ConditionatingMatrix::usage = "\<ConditionatingMatrix[k,i] is the  \
2^i  dimension square Dempsterian specialization matrix conditioning with \
respect to subset  k .\>";\)\)}], "Input",
  CellLabel->"In[19]:=",
  InitializationCell->True],

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m1[k]  , for 0 \[LessEqual] k \[LessEqual] 2^i - 1 ).\>";\)\)], "Input",
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\[LessEqual] k \[LessEqual] 2^i - 1 ), \[Alpha] is the junction parameter and \
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\[IndentingNewLine]\)
    \)\), "\[IndentingNewLine]", 
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\[LessEqual] 2^i - 1 ). Unspecificity is a generalization of cardinality.\>";\
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\[LessEqual] k \[LessEqual] 2^i - 1 ).\>";\)\)], "Input",
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        i > 0\  := \[IndentingNewLine]\(TransformMatrix[b, bba, 
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          TransformMatrix[j, i]\)\)\(\[IndentingNewLine]\)
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    \)\), "\[IndentingNewLine]", 
    \(\(\(TransformMatrix[bel, bba, i_] := \(TransformMatrix[bel, bba, i] = \ 
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    \(\(\(TransformMatrix[pl, bba, i_] := \(TransformMatrix[pl, bba, i] = 
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Cell["\<\
Compare with Smets (2001) Tables 4, 5 and 6 page 7:
Matrices for bel and pl are modified so that they are no more singular \
\>", \
"Text"],

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Check.
(note: need to do also the probability -> possibility transformation. See \
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TODO also the probability -> possibility transformation. See Smets \
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Example, conditioning wrt to {a,b} when \[CapitalOmega]={a,b,c} see \
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Check that \[Alpha]=1 corresponds to specialization\
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This unspecificity generalizes cardinality. As discussed in Rocha \
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Example: Smets (2000) QEPT seen as Hyper Cautious TBM tables 1-4
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End of Mathematica Notebook file.
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