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        "], $CellContext`SIGMA = {{1, 0.9}, {0.9, 1}}, $CellContext`U[
         Pattern[$CellContext`x, 
          Blank[]], 
         Pattern[$CellContext`y, 
          Blank[]]] = (
         5.263157894736843 $CellContext`x^2 - (
          9.473684210526319 $CellContext`x) $CellContext`y + 
         5.263157894736844 $CellContext`y^2)/2, $CellContext`dU[
         Pattern[$CellContext`x, 
          Blank[]], 
         Pattern[$CellContext`y, 
          Blank[]]] = {(10.526315789473687` $CellContext`x - 
          9.473684210526319 $CellContext`y)/
         2, ((-9.473684210526319) $CellContext`x + 
          10.526315789473689` $CellContext`y)/2}, $CellContext`ddU[
         Pattern[$CellContext`x, 
          Blank[]], 
         Pattern[$CellContext`y, 
          Blank[]]] = {{
        5.263157894736843, -4.736842105263159}, {-4.736842105263159, 
        5.263157894736844}}, $CellContext`GradientG[
         Pattern[$CellContext`f, 
          Blank[]], 
         PatternTest[
          Pattern[$CellContext`x, 
           Blank[List]], VectorQ]] := 
       D[$CellContext`f, {$CellContext`x, 1}], $CellContext`HessianH[
         Pattern[$CellContext`f, 
          Blank[]], 
         PatternTest[
          Pattern[$CellContext`x, 
           Blank[List]], VectorQ]] := 
       D[$CellContext`f, {$CellContext`x, 2}], $CellContext`hmc[
         Pattern[$CellContext`U, 
          Blank[]], 
         Pattern[$CellContext`dU, 
          Blank[]], 
         Pattern[$CellContext`ddU, 
          Blank[]], 
         Pattern[$CellContext`Dim, 
          Blank[]], 
         Pattern[$CellContext`BURNIN, 
          Blank[]], 
         Pattern[$CellContext`EPISODE, 
          Blank[]], 
         Pattern[$CellContext`vanilla0, 
          Blank[]], 
         Pattern[$CellContext`switch, 
          Blank[]]] := 
       Module[{$CellContext`CHAINS = 5, $CellContext`STEPS = 
          5, $CellContext`qAll, $CellContext`pAll, $CellContext`Utotal, \
$CellContext`Ktotal, $CellContext`Htotal, $CellContext`s, $CellContext`S, \
$CellContext`AS, $CellContext`ES, $CellContext`KtotalNew, $CellContext`dt, \
$CellContext`p, $CellContext`q0, $CellContext`UE, $CellContext`\[Alpha], \
$CellContext`q, $CellContext`j, $CellContext`Htotal1, $CellContext`Htotal2, \
$CellContext`i, $CellContext`ND = NormalDistribution[0, 1], $CellContext`UD = 
          UniformDistribution[], $CellContext`QS = {}, $CellContext`vanilla = \
$CellContext`vanilla0, $CellContext`dt1 = 1.*^-9, $CellContext`dt2 = 
          0.0001, $CellContext`ACS = {}}, $CellContext`pAll = 
          RandomVariate[$CellContext`ND, {$CellContext`CHAINS, \
$CellContext`Dim}]; $CellContext`qAll = 
          RandomVariate[$CellContext`ND, {$CellContext`CHAINS, \
$CellContext`Dim}]; $CellContext`Utotal = Sum[
            Apply[$CellContext`U, 
             Part[$CellContext`qAll, $CellContext`i]], {$CellContext`i, 
             1, $CellContext`CHAINS}]; $CellContext`Htotal1 = \
$CellContext`Utotal; $CellContext`Htotal2 = $CellContext`Utotal; 
         For[$CellContext`j = 1, $CellContext`j <= $CellContext`EPISODE, 
           
           Increment[$CellContext`j], $CellContext`pAll = 
            RandomVariate[$CellContext`ND, {$CellContext`CHAINS, \
$CellContext`Dim}]; $CellContext`KtotalNew = Sum[If[$CellContext`vanilla, 
                Dot[
                 Part[$CellContext`pAll, $CellContext`i], 
                 Part[$CellContext`pAll, $CellContext`i]], 
                Dot[
                 Part[$CellContext`pAll, $CellContext`i], 
                 LinearSolve[
                  Apply[$CellContext`ddU, 
                   Part[$CellContext`qAll, $CellContext`i]], 
                  Part[$CellContext`pAll, $CellContext`i]]]]/
              2, {$CellContext`i, 
               1, $CellContext`CHAINS}]; $CellContext`Utotal = Sum[
              Apply[$CellContext`U, 
               Part[$CellContext`qAll, $CellContext`i]], {$CellContext`i, 
               1, $CellContext`CHAINS}]; $CellContext`Htotal = 
            If[$CellContext`vanilla, $CellContext`Htotal1, \
$CellContext`Htotal2]; $CellContext`dt = 
            If[$CellContext`vanilla, $CellContext`dt1, $CellContext`dt2]; \
$CellContext`Ktotal = $CellContext`Htotal - $CellContext`Utotal; \
$CellContext`pAll = $CellContext`pAll Sqrt[
               
               Abs[$CellContext`Ktotal/$CellContext`KtotalNew]]; \
$CellContext`ES = {}; $CellContext`AS = {}; 
           For[$CellContext`i = 1, $CellContext`i <= $CellContext`CHAINS, 
             
             Increment[$CellContext`i], $CellContext`p = 
              Part[$CellContext`pAll, $CellContext`i]; $CellContext`q = 
              Part[$CellContext`qAll, $CellContext`i]; $CellContext`UE = {
                
                Apply[$CellContext`U, $CellContext`q]}; $CellContext`q0 = \
$CellContext`q; 
             For[$CellContext`s = 1; 
               Null, $CellContext`s <= $CellContext`STEPS, 
               
               Increment[$CellContext`s], $CellContext`p = $CellContext`p - \
$CellContext`dt 
                 Apply[$CellContext`dU, $CellContext`q]; $CellContext`q = \
$CellContext`q + $CellContext`dt If[$CellContext`vanilla, $CellContext`p, 
                    LinearSolve[
                    
                    Apply[$CellContext`ddU, $CellContext`q], $CellContext`p]]; \
$CellContext`UE = Append[$CellContext`UE, 
                  Apply[$CellContext`U, $CellContext`q]]]; $CellContext`ES = 
              Append[$CellContext`ES, $CellContext`UE]; $CellContext`\[Alpha] = 
              Exp[
                Clip[
                Apply[$CellContext`U, $CellContext`q0] - 
                 Apply[$CellContext`U, $CellContext`q], {-20, 
                 0}]]; $CellContext`AS = Append[$CellContext`AS, 
                N[$CellContext`\[Alpha]]]; 
             If[$CellContext`\[Alpha] < 
               RandomVariate[$CellContext`UD], $CellContext`q = \
$CellContext`q0]; Part[$CellContext`qAll, $CellContext`i] = $CellContext`q; 
             If[$CellContext`j > $CellContext`BURNIN, $CellContext`QS = 
               Append[$CellContext`QS, $CellContext`q]]]; $CellContext`s = 
            Union[
              Flatten[
               Table[
                Ordering[
                 Part[$CellContext`ES, $CellContext`i], 1], {$CellContext`i, 
                 1, $CellContext`CHAINS}]]]; $CellContext`S = Union[
              Flatten[
               Table[
                Ordering[
                 Part[$CellContext`ES, $CellContext`i], -1], {$CellContext`i, 
                 1, $CellContext`CHAINS}]]]; 
           If[$CellContext`j < $CellContext`BURNIN, If[
               Or[
                
                And[$CellContext`s == {
                  1, $CellContext`STEPS + 1}, $CellContext`S == {
                  1, $CellContext`STEPS + 1}], 
                
                And[$CellContext`s == {$CellContext`STEPS + 
                   1}, $CellContext`S == {
                  1}]], $CellContext`dt = $CellContext`dt 1.1]; If[
               
               Or[$CellContext`s == {
                 1}, $CellContext`S == {$CellContext`STEPS + 
                  1}], $CellContext`dt = $CellContext`dt/1.1]; If[
               And[$CellContext`Ktotal > 0, $CellContext`KtotalNew > 0], 
               If[
                
                And[$CellContext`Utotal < 
                 0, $CellContext`Htotal > $CellContext`Utotal/
                  100], $CellContext`Htotal = $CellContext`Utotal/100, 
                If[
                 
                 And[$CellContext`Utotal > 0, $CellContext`Htotal > 
                  100 $CellContext`Utotal], $CellContext`Htotal = 
                 100 $CellContext`Utotal, 
                 If[Mean[$CellContext`AS] > 
                   0.9, $CellContext`Ktotal = $CellContext`Ktotal 1.1, 
                   If[
                   Mean[$CellContext`AS] < 
                    0.1, $CellContext`Ktotal = $CellContext`Ktotal/
                    1.1]]; $CellContext`Htotal = $CellContext`Ktotal + \
$CellContext`Utotal]]]; 
             If[$CellContext`vanilla, $CellContext`Htotal1 = \
$CellContext`Htotal; $CellContext`dt1 = $CellContext`dt, $CellContext`Htotal2 = \
$CellContext`Htotal; $CellContext`dt2 = $CellContext`dt]]; 
           If[$CellContext`switch, $CellContext`vanilla = 
             Not[$CellContext`vanilla]]]; $CellContext`QS]}; 
     Typeset`initDone$$ = True),
    SynchronousInitialization->True,
    UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$},
    UnsavedVariables:>{Typeset`initDone$$},
    UntrackedVariables:>{Typeset`size$$}], "Manipulate",
   Deployed->True,
   StripOnInput->False],
  Manipulate`InterpretManipulate[1]]], "Output",
 CellChangeTimes->{3.817665256380991*^9, 3.817665344218412*^9, 
  3.817698848638335*^9, 3.8176989759101067`*^9, 3.8176990240059977`*^9},
 CellLabel->"(Debug) Out[10]=",
 CellID->1913502953,ExpressionUUID->"09e5c716-b11c-404c-9ee2-837c063d14fa"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["", "ManipulateCaptionSection",ExpressionUUID->"343ea5f6-a1e2-433e-a23a-ef37af0b05bb"],

Cell["\<\
Samples, the transformed samples using the covariance matrix, and the \
covariance matrix\
\>", "ManipulateCaption",
 CellChangeTimes->{{3.817699278669183*^9, 
  3.8176993250457697`*^9}},ExpressionUUID->"62c4c770-c515-46db-8779-\
e8362edab391"]
}, Open  ]],

Cell[CellGroupData[{

Cell["", "ThumbnailSection",ExpressionUUID->"8212c40d-46ef-4ecf-8636-6737412e60f3"],

Cell[BoxData[
 TagBox[
  StyleBox[
   DynamicModuleBox[{$CellContext`i$$ = 5, Typeset`show$$ = True, 
    Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", 
    Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = 
    "\"untitled\"", Typeset`specs$$ = {{
      Hold[$CellContext`i$$], 1, 8, 1}}, Typeset`size$$ = {
    468., {193., 197.}}, Typeset`update$$ = 0, Typeset`initDone$$, 
    Typeset`skipInitDone$$ = False, $CellContext`i$9439$$ = 0}, 
    DynamicBox[Manipulate`ManipulateBoxes[
     1, StandardForm, "Variables" :> {$CellContext`i$$ = 1}, 
      "ControllerVariables" :> {
        Hold[$CellContext`i$$, $CellContext`i$9439$$, 0]}, 
      "OtherVariables" :> {
       Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, 
        Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, 
        Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$,
         Typeset`skipInitDone$$}, "Body" :> 
      Module[{$CellContext`\[Rho]$, $CellContext`QS$, $CellContext`SIGMA$, \
$CellContext`U$, $CellContext`dU$, $CellContext`ddU$}, $CellContext`\[Rho]$ = 
         N[1 - 1/10^$CellContext`i$$]; $CellContext`SIGMA$ = {{
           1, $CellContext`\[Rho]$}, {$CellContext`\[Rho]$, 
            1}}; $CellContext`U$[
           Pattern[$CellContext`x$, 
            Blank[]], 
           Pattern[$CellContext`y$, 
            Blank[]]] = (1/2) Simplify[
            Dot[{$CellContext`x$, $CellContext`y$}, 
             
             LinearSolve[$CellContext`SIGMA$, {$CellContext`x$, \
$CellContext`y$}]]]; $CellContext`dU$[
           Pattern[$CellContext`x$, 
            Blank[]], 
           Pattern[$CellContext`y$, 
            Blank[]]] = $CellContext`GradientG[
           $CellContext`U$[$CellContext`x$, $CellContext`y$], \
{$CellContext`x$, $CellContext`y$}]; $CellContext`ddU$[
           Pattern[$CellContext`x$, 
            Blank[]], 
           Pattern[$CellContext`y$, 
            Blank[]]] = $CellContext`HessianH[
           $CellContext`U$[$CellContext`x$, $CellContext`y$], \
{$CellContext`x$, $CellContext`y$}]; $CellContext`QS$ = \
$CellContext`hmc[$CellContext`U$, $CellContext`dU$, $CellContext`ddU$, 2, 
           1000, 2000, True, True]; ListPlot[{$CellContext`QS$, 
           Dot[$CellContext`QS$, 
            MatrixPower[$CellContext`SIGMA$, -0.5]]}, PlotStyle -> Opacity[1],
           PlotLabel -> $CellContext`SIGMA$, 
          PlotLegends -> {"Samples", "Transformed"}, AspectRatio -> 1]], 
      "Specifications" :> {{$CellContext`i$$, 1, 8, 1}}, "Options" :> {}, 
      "DefaultOptions" :> {ControllerLinking -> True}],
     ImageSizeCache->{513., {237., 242.}},
     SingleEvaluation->True],
    Deinitialization:>None,
    DynamicModuleValues:>{},
    Initialization:>({$CellContext`\[Rho] = 
       0.9, $CellContext`QS = CompressedData["
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+o9PNz7Zxu00xxMBOR+0SuhSX9ButDvH/8oEKD0NWRL4ZBzkw6T21ip1IPfx
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tj2y3/K8ObYl+I09AF6w+hM=
        
        "], $CellContext`SIGMA = {{1, 0.9}, {0.9, 1}}, $CellContext`U[
         Pattern[$CellContext`x, 
          Blank[]], 
         Pattern[$CellContext`y, 
          Blank[]]] = (
         5.263157894736843 $CellContext`x^2 - (
          9.473684210526319 $CellContext`x) $CellContext`y + 
         5.263157894736844 $CellContext`y^2)/2, $CellContext`dU[
         Pattern[$CellContext`x, 
          Blank[]], 
         Pattern[$CellContext`y, 
          Blank[]]] = {(10.526315789473687` $CellContext`x - 
          9.473684210526319 $CellContext`y)/
         2, ((-9.473684210526319) $CellContext`x + 
          10.526315789473689` $CellContext`y)/2}, $CellContext`ddU[
         Pattern[$CellContext`x, 
          Blank[]], 
         Pattern[$CellContext`y, 
          Blank[]]] = {{
        5.263157894736843, -4.736842105263159}, {-4.736842105263159, 
        5.263157894736844}}, $CellContext`GradientG[
         Pattern[$CellContext`f, 
          Blank[]], 
         PatternTest[
          Pattern[$CellContext`x, 
           Blank[List]], VectorQ]] := 
       D[$CellContext`f, {$CellContext`x, 1}], $CellContext`HessianH[
         Pattern[$CellContext`f, 
          Blank[]], 
         PatternTest[
          Pattern[$CellContext`x, 
           Blank[List]], VectorQ]] := 
       D[$CellContext`f, {$CellContext`x, 2}], $CellContext`hmc[
         Pattern[$CellContext`U, 
          Blank[]], 
         Pattern[$CellContext`dU, 
          Blank[]], 
         Pattern[$CellContext`ddU, 
          Blank[]], 
         Pattern[$CellContext`Dim, 
          Blank[]], 
         Pattern[$CellContext`BURNIN, 
          Blank[]], 
         Pattern[$CellContext`EPISODE, 
          Blank[]], 
         Pattern[$CellContext`vanilla0, 
          Blank[]], 
         Pattern[$CellContext`switch, 
          Blank[]]] := 
       Module[{$CellContext`CHAINS = 5, $CellContext`STEPS = 
          5, $CellContext`qAll, $CellContext`pAll, $CellContext`Utotal, \
$CellContext`Ktotal, $CellContext`Htotal, $CellContext`s, $CellContext`S, \
$CellContext`AS, $CellContext`ES, $CellContext`KtotalNew, $CellContext`dt, \
$CellContext`p, $CellContext`q0, $CellContext`UE, $CellContext`\[Alpha], \
$CellContext`q, $CellContext`j, $CellContext`Htotal1, $CellContext`Htotal2, \
$CellContext`i, $CellContext`ND = NormalDistribution[0, 1], $CellContext`UD = 
          UniformDistribution[], $CellContext`QS = {}, $CellContext`vanilla = \
$CellContext`vanilla0, $CellContext`dt1 = 1.*^-9, $CellContext`dt2 = 
          0.0001, $CellContext`ACS = {}}, $CellContext`pAll = 
          RandomVariate[$CellContext`ND, {$CellContext`CHAINS, \
$CellContext`Dim}]; $CellContext`qAll = 
          RandomVariate[$CellContext`ND, {$CellContext`CHAINS, \
$CellContext`Dim}]; $CellContext`Utotal = Sum[
            Apply[$CellContext`U, 
             Part[$CellContext`qAll, $CellContext`i]], {$CellContext`i, 
             1, $CellContext`CHAINS}]; $CellContext`Htotal1 = \
$CellContext`Utotal; $CellContext`Htotal2 = $CellContext`Utotal; 
         For[$CellContext`j = 1, $CellContext`j <= $CellContext`EPISODE, 
           
           Increment[$CellContext`j], $CellContext`pAll = 
            RandomVariate[$CellContext`ND, {$CellContext`CHAINS, \
$CellContext`Dim}]; $CellContext`KtotalNew = Sum[If[$CellContext`vanilla, 
                Dot[
                 Part[$CellContext`pAll, $CellContext`i], 
                 Part[$CellContext`pAll, $CellContext`i]], 
                Dot[
                 Part[$CellContext`pAll, $CellContext`i], 
                 LinearSolve[
                  Apply[$CellContext`ddU, 
                   Part[$CellContext`qAll, $CellContext`i]], 
                  Part[$CellContext`pAll, $CellContext`i]]]]/
              2, {$CellContext`i, 
               1, $CellContext`CHAINS}]; $CellContext`Utotal = Sum[
              Apply[$CellContext`U, 
               Part[$CellContext`qAll, $CellContext`i]], {$CellContext`i, 
               1, $CellContext`CHAINS}]; $CellContext`Htotal = 
            If[$CellContext`vanilla, $CellContext`Htotal1, \
$CellContext`Htotal2]; $CellContext`dt = 
            If[$CellContext`vanilla, $CellContext`dt1, $CellContext`dt2]; \
$CellContext`Ktotal = $CellContext`Htotal - $CellContext`Utotal; \
$CellContext`pAll = $CellContext`pAll Sqrt[
               
               Abs[$CellContext`Ktotal/$CellContext`KtotalNew]]; \
$CellContext`ES = {}; $CellContext`AS = {}; 
           For[$CellContext`i = 1, $CellContext`i <= $CellContext`CHAINS, 
             
             Increment[$CellContext`i], $CellContext`p = 
              Part[$CellContext`pAll, $CellContext`i]; $CellContext`q = 
              Part[$CellContext`qAll, $CellContext`i]; $CellContext`UE = {
                
                Apply[$CellContext`U, $CellContext`q]}; $CellContext`q0 = \
$CellContext`q; 
             For[$CellContext`s = 1; 
               Null, $CellContext`s <= $CellContext`STEPS, 
               
               Increment[$CellContext`s], $CellContext`p = $CellContext`p - \
$CellContext`dt 
                 Apply[$CellContext`dU, $CellContext`q]; $CellContext`q = \
$CellContext`q + $CellContext`dt If[$CellContext`vanilla, $CellContext`p, 
                    LinearSolve[
                    
                    Apply[$CellContext`ddU, $CellContext`q], $CellContext`p]]; \
$CellContext`UE = Append[$CellContext`UE, 
                  Apply[$CellContext`U, $CellContext`q]]]; $CellContext`ES = 
              Append[$CellContext`ES, $CellContext`UE]; $CellContext`\[Alpha] = 
              Exp[
                Clip[
                Apply[$CellContext`U, $CellContext`q0] - 
                 Apply[$CellContext`U, $CellContext`q], {-20, 
                 0}]]; $CellContext`AS = Append[$CellContext`AS, 
                N[$CellContext`\[Alpha]]]; 
             If[$CellContext`\[Alpha] < 
               RandomVariate[$CellContext`UD], $CellContext`q = \
$CellContext`q0]; Part[$CellContext`qAll, $CellContext`i] = $CellContext`q; 
             If[$CellContext`j > $CellContext`BURNIN, $CellContext`QS = 
               Append[$CellContext`QS, $CellContext`q]]]; $CellContext`s = 
            Union[
              Flatten[
               Table[
                Ordering[
                 Part[$CellContext`ES, $CellContext`i], 1], {$CellContext`i, 
                 1, $CellContext`CHAINS}]]]; $CellContext`S = Union[
              Flatten[
               Table[
                Ordering[
                 Part[$CellContext`ES, $CellContext`i], -1], {$CellContext`i, 
                 1, $CellContext`CHAINS}]]]; 
           If[$CellContext`j < $CellContext`BURNIN, If[
               Or[
                
                And[$CellContext`s == {
                  1, $CellContext`STEPS + 1}, $CellContext`S == {
                  1, $CellContext`STEPS + 1}], 
                
                And[$CellContext`s == {$CellContext`STEPS + 
                   1}, $CellContext`S == {
                  1}]], $CellContext`dt = $CellContext`dt 1.1]; If[
               
               Or[$CellContext`s == {
                 1}, $CellContext`S == {$CellContext`STEPS + 
                  1}], $CellContext`dt = $CellContext`dt/1.1]; If[
               And[$CellContext`Ktotal > 0, $CellContext`KtotalNew > 0], 
               If[
                
                And[$CellContext`Utotal < 
                 0, $CellContext`Htotal > $CellContext`Utotal/
                  100], $CellContext`Htotal = $CellContext`Utotal/100, 
                If[
                 
                 And[$CellContext`Utotal > 0, $CellContext`Htotal > 
                  100 $CellContext`Utotal], $CellContext`Htotal = 
                 100 $CellContext`Utotal, 
                 If[Mean[$CellContext`AS] > 
                   0.9, $CellContext`Ktotal = $CellContext`Ktotal 1.1, 
                   If[
                   Mean[$CellContext`AS] < 
                    0.1, $CellContext`Ktotal = $CellContext`Ktotal/
                    1.1]]; $CellContext`Htotal = $CellContext`Ktotal + \
$CellContext`Utotal]]]; 
             If[$CellContext`vanilla, $CellContext`Htotal1 = \
$CellContext`Htotal; $CellContext`dt1 = $CellContext`dt, $CellContext`Htotal2 = \
$CellContext`Htotal; $CellContext`dt2 = $CellContext`dt]]; 
           If[$CellContext`switch, $CellContext`vanilla = 
             Not[$CellContext`vanilla]]]; $CellContext`QS]}; 
     Typeset`initDone$$ = True),
    SynchronousInitialization->True,
    UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$},
    UnsavedVariables:>{Typeset`initDone$$},
    UntrackedVariables:>{Typeset`size$$}], "Manipulate",
   Deployed->True,
   StripOnInput->False],
  Manipulate`InterpretManipulate[1]]], "Output",
 CellChangeTimes->{3.817665256380991*^9, 3.817665344218412*^9, 
  3.817698848638335*^9, 3.8176989759101067`*^9, 3.8176990240059977`*^9},
 CellLabel->"(Debug) Out[10]=",
 CellID->299801647,ExpressionUUID->"41325725-c29d-4a4a-94c2-3b3f0bb13b9d"]
}, Open  ]],

Cell[CellGroupData[{

Cell["", "SnapshotsSection",ExpressionUUID->"c41a30ce-d578-468b-bdd1-e5c4cdcf666c"],

Cell[BoxData[
 TagBox[
  StyleBox[
   DynamicModuleBox[{$CellContext`i$$ = 1, Typeset`show$$ = True, 
    Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", 
    Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = 
    "\"untitled\"", Typeset`specs$$ = {{
      Hold[$CellContext`i$$], 1, 8, 1}}, Typeset`size$$ = {
    468., {195., 200.}}, Typeset`update$$ = 0, Typeset`initDone$$, 
    Typeset`skipInitDone$$ = False, $CellContext`i$9439$$ = 0}, 
    DynamicBox[Manipulate`ManipulateBoxes[
     1, StandardForm, "Variables" :> {$CellContext`i$$ = 1}, 
      "ControllerVariables" :> {
        Hold[$CellContext`i$$, $CellContext`i$9439$$, 0]}, 
      "OtherVariables" :> {
       Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, 
        Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, 
        Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$,
         Typeset`skipInitDone$$}, "Body" :> 
      Module[{$CellContext`\[Rho]$, $CellContext`QS$, $CellContext`SIGMA$, \
$CellContext`U$, $CellContext`dU$, $CellContext`ddU$}, $CellContext`\[Rho]$ = 
         N[1 - 1/10^$CellContext`i$$]; $CellContext`SIGMA$ = {{
           1, $CellContext`\[Rho]$}, {$CellContext`\[Rho]$, 
            1}}; $CellContext`U$[
           Pattern[$CellContext`x$, 
            Blank[]], 
           Pattern[$CellContext`y$, 
            Blank[]]] = (1/2) Simplify[
            Dot[{$CellContext`x$, $CellContext`y$}, 
             
             LinearSolve[$CellContext`SIGMA$, {$CellContext`x$, \
$CellContext`y$}]]]; $CellContext`dU$[
           Pattern[$CellContext`x$, 
            Blank[]], 
           Pattern[$CellContext`y$, 
            Blank[]]] = $CellContext`GradientG[
           $CellContext`U$[$CellContext`x$, $CellContext`y$], \
{$CellContext`x$, $CellContext`y$}]; $CellContext`ddU$[
           Pattern[$CellContext`x$, 
            Blank[]], 
           Pattern[$CellContext`y$, 
            Blank[]]] = $CellContext`HessianH[
           $CellContext`U$[$CellContext`x$, $CellContext`y$], \
{$CellContext`x$, $CellContext`y$}]; $CellContext`QS$ = \
$CellContext`hmc[$CellContext`U$, $CellContext`dU$, $CellContext`ddU$, 2, 
           1000, 2000, True, True]; ListPlot[{$CellContext`QS$, 
           Dot[$CellContext`QS$, 
            MatrixPower[$CellContext`SIGMA$, -0.5]]}, PlotStyle -> Opacity[1],
           PlotLabel -> $CellContext`SIGMA$, 
          PlotLegends -> {"Samples", "Transformed"}, AspectRatio -> 1]], 
      "Specifications" :> {{$CellContext`i$$, 1, 8, 1}}, "Options" :> {}, 
      "DefaultOptions" :> {ControllerLinking -> True}],
     ImageSizeCache->{513., {239., 244.}},
     SingleEvaluation->True],
    Deinitialization:>None,
    DynamicModuleValues:>{},
    Initialization:>({$CellContext`\[Rho] = 
       0.9, $CellContext`QS = CompressedData["
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elziUHAv7HNIqLs52geLvbgH51e2dFJwLp1hKj+chtYsxY3+zKOo2MZCOfuR
jj4eSacCPuVDdNW99+WOC7lrJIa+83097N7Xrrbpby/BO4maL/ufjuLbjNuq
wZ/IuK+2J5iXpQPZ5GXo82v6UKWgXEtBohe7mcUZW3/+wcVewsrKcpflSZD5
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tj2y3/K8ObYl+I09AF6w+hM=
        
        "], $CellContext`SIGMA = {{1, 0.9}, {0.9, 1}}, $CellContext`U[
         Pattern[$CellContext`x, 
          Blank[]], 
         Pattern[$CellContext`y, 
          Blank[]]] = (
         5.263157894736843 $CellContext`x^2 - (
          9.473684210526319 $CellContext`x) $CellContext`y + 
         5.263157894736844 $CellContext`y^2)/2, $CellContext`dU[
         Pattern[$CellContext`x, 
          Blank[]], 
         Pattern[$CellContext`y, 
          Blank[]]] = {(10.526315789473687` $CellContext`x - 
          9.473684210526319 $CellContext`y)/
         2, ((-9.473684210526319) $CellContext`x + 
          10.526315789473689` $CellContext`y)/2}, $CellContext`ddU[
         Pattern[$CellContext`x, 
          Blank[]], 
         Pattern[$CellContext`y, 
          Blank[]]] = {{
        5.263157894736843, -4.736842105263159}, {-4.736842105263159, 
        5.263157894736844}}, $CellContext`GradientG[
         Pattern[$CellContext`f, 
          Blank[]], 
         PatternTest[
          Pattern[$CellContext`x, 
           Blank[List]], VectorQ]] := 
       D[$CellContext`f, {$CellContext`x, 1}], $CellContext`HessianH[
         Pattern[$CellContext`f, 
          Blank[]], 
         PatternTest[
          Pattern[$CellContext`x, 
           Blank[List]], VectorQ]] := 
       D[$CellContext`f, {$CellContext`x, 2}], $CellContext`hmc[
         Pattern[$CellContext`U, 
          Blank[]], 
         Pattern[$CellContext`dU, 
          Blank[]], 
         Pattern[$CellContext`ddU, 
          Blank[]], 
         Pattern[$CellContext`Dim, 
          Blank[]], 
         Pattern[$CellContext`BURNIN, 
          Blank[]], 
         Pattern[$CellContext`EPISODE, 
          Blank[]], 
         Pattern[$CellContext`vanilla0, 
          Blank[]], 
         Pattern[$CellContext`switch, 
          Blank[]]] := 
       Module[{$CellContext`CHAINS = 5, $CellContext`STEPS = 
          5, $CellContext`qAll, $CellContext`pAll, $CellContext`Utotal, \
$CellContext`Ktotal, $CellContext`Htotal, $CellContext`s, $CellContext`S, \
$CellContext`AS, $CellContext`ES, $CellContext`KtotalNew, $CellContext`dt, \
$CellContext`p, $CellContext`q0, $CellContext`UE, $CellContext`\[Alpha], \
$CellContext`q, $CellContext`j, $CellContext`Htotal1, $CellContext`Htotal2, \
$CellContext`i, $CellContext`ND = NormalDistribution[0, 1], $CellContext`UD = 
          UniformDistribution[], $CellContext`QS = {}, $CellContext`vanilla = \
$CellContext`vanilla0, $CellContext`dt1 = 1.*^-9, $CellContext`dt2 = 
          0.0001, $CellContext`ACS = {}}, $CellContext`pAll = 
          RandomVariate[$CellContext`ND, {$CellContext`CHAINS, \
$CellContext`Dim}]; $CellContext`qAll = 
          RandomVariate[$CellContext`ND, {$CellContext`CHAINS, \
$CellContext`Dim}]; $CellContext`Utotal = Sum[
            Apply[$CellContext`U, 
             Part[$CellContext`qAll, $CellContext`i]], {$CellContext`i, 
             1, $CellContext`CHAINS}]; $CellContext`Htotal1 = \
$CellContext`Utotal; $CellContext`Htotal2 = $CellContext`Utotal; 
         For[$CellContext`j = 1, $CellContext`j <= $CellContext`EPISODE, 
           
           Increment[$CellContext`j], $CellContext`pAll = 
            RandomVariate[$CellContext`ND, {$CellContext`CHAINS, \
$CellContext`Dim}]; $CellContext`KtotalNew = Sum[If[$CellContext`vanilla, 
                Dot[
                 Part[$CellContext`pAll, $CellContext`i], 
                 Part[$CellContext`pAll, $CellContext`i]], 
                Dot[
                 Part[$CellContext`pAll, $CellContext`i], 
                 LinearSolve[
                  Apply[$CellContext`ddU, 
                   Part[$CellContext`qAll, $CellContext`i]], 
                  Part[$CellContext`pAll, $CellContext`i]]]]/
              2, {$CellContext`i, 
               1, $CellContext`CHAINS}]; $CellContext`Utotal = Sum[
              Apply[$CellContext`U, 
               Part[$CellContext`qAll, $CellContext`i]], {$CellContext`i, 
               1, $CellContext`CHAINS}]; $CellContext`Htotal = 
            If[$CellContext`vanilla, $CellContext`Htotal1, \
$CellContext`Htotal2]; $CellContext`dt = 
            If[$CellContext`vanilla, $CellContext`dt1, $CellContext`dt2]; \
$CellContext`Ktotal = $CellContext`Htotal - $CellContext`Utotal; \
$CellContext`pAll = $CellContext`pAll Sqrt[
               
               Abs[$CellContext`Ktotal/$CellContext`KtotalNew]]; \
$CellContext`ES = {}; $CellContext`AS = {}; 
           For[$CellContext`i = 1, $CellContext`i <= $CellContext`CHAINS, 
             
             Increment[$CellContext`i], $CellContext`p = 
              Part[$CellContext`pAll, $CellContext`i]; $CellContext`q = 
              Part[$CellContext`qAll, $CellContext`i]; $CellContext`UE = {
                
                Apply[$CellContext`U, $CellContext`q]}; $CellContext`q0 = \
$CellContext`q; 
             For[$CellContext`s = 1; 
               Null, $CellContext`s <= $CellContext`STEPS, 
               
               Increment[$CellContext`s], $CellContext`p = $CellContext`p - \
$CellContext`dt 
                 Apply[$CellContext`dU, $CellContext`q]; $CellContext`q = \
$CellContext`q + $CellContext`dt If[$CellContext`vanilla, $CellContext`p, 
                    LinearSolve[
                    
                    Apply[$CellContext`ddU, $CellContext`q], $CellContext`p]]; \
$CellContext`UE = Append[$CellContext`UE, 
                  Apply[$CellContext`U, $CellContext`q]]]; $CellContext`ES = 
              Append[$CellContext`ES, $CellContext`UE]; $CellContext`\[Alpha] = 
              Exp[
                Clip[
                Apply[$CellContext`U, $CellContext`q0] - 
                 Apply[$CellContext`U, $CellContext`q], {-20, 
                 0}]]; $CellContext`AS = Append[$CellContext`AS, 
                N[$CellContext`\[Alpha]]]; 
             If[$CellContext`\[Alpha] < 
               RandomVariate[$CellContext`UD], $CellContext`q = \
$CellContext`q0]; Part[$CellContext`qAll, $CellContext`i] = $CellContext`q; 
             If[$CellContext`j > $CellContext`BURNIN, $CellContext`QS = 
               Append[$CellContext`QS, $CellContext`q]]]; $CellContext`s = 
            Union[
              Flatten[
               Table[
                Ordering[
                 Part[$CellContext`ES, $CellContext`i], 1], {$CellContext`i, 
                 1, $CellContext`CHAINS}]]]; $CellContext`S = Union[
              Flatten[
               Table[
                Ordering[
                 Part[$CellContext`ES, $CellContext`i], -1], {$CellContext`i, 
                 1, $CellContext`CHAINS}]]]; 
           If[$CellContext`j < $CellContext`BURNIN, If[
               Or[
                
                And[$CellContext`s == {
                  1, $CellContext`STEPS + 1}, $CellContext`S == {
                  1, $CellContext`STEPS + 1}], 
                
                And[$CellContext`s == {$CellContext`STEPS + 
                   1}, $CellContext`S == {
                  1}]], $CellContext`dt = $CellContext`dt 1.1]; If[
               
               Or[$CellContext`s == {
                 1}, $CellContext`S == {$CellContext`STEPS + 
                  1}], $CellContext`dt = $CellContext`dt/1.1]; If[
               And[$CellContext`Ktotal > 0, $CellContext`KtotalNew > 0], 
               If[
                
                And[$CellContext`Utotal < 
                 0, $CellContext`Htotal > $CellContext`Utotal/
                  100], $CellContext`Htotal = $CellContext`Utotal/100, 
                If[
                 
                 And[$CellContext`Utotal > 0, $CellContext`Htotal > 
                  100 $CellContext`Utotal], $CellContext`Htotal = 
                 100 $CellContext`Utotal, 
                 If[Mean[$CellContext`AS] > 
                   0.9, $CellContext`Ktotal = $CellContext`Ktotal 1.1, 
                   If[
                   Mean[$CellContext`AS] < 
                    0.1, $CellContext`Ktotal = $CellContext`Ktotal/
                    1.1]]; $CellContext`Htotal = $CellContext`Ktotal + \
$CellContext`Utotal]]]; 
             If[$CellContext`vanilla, $CellContext`Htotal1 = \
$CellContext`Htotal; $CellContext`dt1 = $CellContext`dt, $CellContext`Htotal2 = \
$CellContext`Htotal; $CellContext`dt2 = $CellContext`dt]]; 
           If[$CellContext`switch, $CellContext`vanilla = 
             Not[$CellContext`vanilla]]]; $CellContext`QS]}; 
     Typeset`initDone$$ = True),
    SynchronousInitialization->True,
    UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$},
    UnsavedVariables:>{Typeset`initDone$$},
    UntrackedVariables:>{Typeset`size$$}], "Manipulate",
   Deployed->True,
   StripOnInput->False],
  Manipulate`InterpretManipulate[1]]], "Output",
 CellChangeTimes->{3.817665256380991*^9, 3.817665344218412*^9, 
  3.817698848638335*^9, 3.8176989759101067`*^9, 3.8176990240059977`*^9},
 CellID->316109030,ExpressionUUID->"d92329b3-1268-438c-85ac-aa59d3285fde"],

Cell[BoxData[
 TagBox[
  StyleBox[
   DynamicModuleBox[{$CellContext`i$$ = 3, Typeset`show$$ = True, 
    Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", 
    Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = 
    "\"untitled\"", Typeset`specs$$ = {{
      Hold[$CellContext`i$$], 1, 8, 1}}, Typeset`size$$ = {
    468., {195., 200.}}, Typeset`update$$ = 0, Typeset`initDone$$, 
    Typeset`skipInitDone$$ = False, $CellContext`i$9439$$ = 0}, 
    DynamicBox[Manipulate`ManipulateBoxes[
     1, StandardForm, "Variables" :> {$CellContext`i$$ = 1}, 
      "ControllerVariables" :> {
        Hold[$CellContext`i$$, $CellContext`i$9439$$, 0]}, 
      "OtherVariables" :> {
       Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, 
        Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, 
        Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$,
         Typeset`skipInitDone$$}, "Body" :> 
      Module[{$CellContext`\[Rho]$, $CellContext`QS$, $CellContext`SIGMA$, \
$CellContext`U$, $CellContext`dU$, $CellContext`ddU$}, $CellContext`\[Rho]$ = 
         N[1 - 1/10^$CellContext`i$$]; $CellContext`SIGMA$ = {{
           1, $CellContext`\[Rho]$}, {$CellContext`\[Rho]$, 
            1}}; $CellContext`U$[
           Pattern[$CellContext`x$, 
            Blank[]], 
           Pattern[$CellContext`y$, 
            Blank[]]] = (1/2) Simplify[
            Dot[{$CellContext`x$, $CellContext`y$}, 
             
             LinearSolve[$CellContext`SIGMA$, {$CellContext`x$, \
$CellContext`y$}]]]; $CellContext`dU$[
           Pattern[$CellContext`x$, 
            Blank[]], 
           Pattern[$CellContext`y$, 
            Blank[]]] = $CellContext`GradientG[
           $CellContext`U$[$CellContext`x$, $CellContext`y$], \
{$CellContext`x$, $CellContext`y$}]; $CellContext`ddU$[
           Pattern[$CellContext`x$, 
            Blank[]], 
           Pattern[$CellContext`y$, 
            Blank[]]] = $CellContext`HessianH[
           $CellContext`U$[$CellContext`x$, $CellContext`y$], \
{$CellContext`x$, $CellContext`y$}]; $CellContext`QS$ = \
$CellContext`hmc[$CellContext`U$, $CellContext`dU$, $CellContext`ddU$, 2, 
           1000, 2000, True, True]; ListPlot[{$CellContext`QS$, 
           Dot[$CellContext`QS$, 
            MatrixPower[$CellContext`SIGMA$, -0.5]]}, PlotStyle -> Opacity[1],
           PlotLabel -> $CellContext`SIGMA$, 
          PlotLegends -> {"Samples", "Transformed"}, AspectRatio -> 1]], 
      "Specifications" :> {{$CellContext`i$$, 1, 8, 1}}, "Options" :> {}, 
      "DefaultOptions" :> {ControllerLinking -> True}],
     ImageSizeCache->{513., {239., 244.}},
     SingleEvaluation->True],
    Deinitialization:>None,
    DynamicModuleValues:>{},
    Initialization:>({$CellContext`\[Rho] = 
       0.9, $CellContext`QS = CompressedData["
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eShhjOANHbPSO7ayAJSdKh49d8oleG2rJh9dsLXEwN1l6f0nMwjeCl9DeeHb
o7h5xdUnjIzjBG+nXfzyM2x1KHxmjnVnXQWSB4emP0jlgMSnj+659Y3/7H29
elziUHAv7HNIqLs52geLvbgH51e2dFJwLp1hKj+chtYsxY3+zKOo2MZCOfuR
jj4eSacCPuVDdNW99+WOC7lrJIa+83097N7Xrrbpby/BO4maL/ufjuLbjNuq
wZ/IuK+2J5iXpQPZ5GXo82v6UKWgXEtBohe7mcUZW3/+wcVewsrKcpflSZD5
PvH9tGb7kp7aT4mGlZfa8Hq/caOmfiZ8aLPROB7ZAbwCEY/lN/WD0iPX1+Sh
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rlYKGY4SvfSj1K/ru6BvOMzgW1sH8JW37ayomMTty3OCwvupuNhr/9TipOM9
BCXIqrzzextwuPs+masYgoq9hgXeTGT4MF+twchTD3P8s3ferSiD/7c2882v
k4z/7rcVXDDvqdZPDPMaXll0Ttq/eT/j44N/elff3e9+8Xh2aeaT/ReYeotT
yu/tRzevVJ37N6fFt/19//VNOCM+YZgXOW2D4bwnh/YnqXJPbU57t/9lRoqH
+YTX+9dyPZwm9vQzhnlTeuPkJhd83W+d+KQplPXdfusZc0oKvz3cX74hZcOf
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tj2y3/K8ObYl+I09AF6w+hM=
        
        "], $CellContext`SIGMA = {{1, 0.9}, {0.9, 1}}, $CellContext`U[
         Pattern[$CellContext`x, 
          Blank[]], 
         Pattern[$CellContext`y, 
          Blank[]]] = (
         5.263157894736843 $CellContext`x^2 - (
          9.473684210526319 $CellContext`x) $CellContext`y + 
         5.263157894736844 $CellContext`y^2)/2, $CellContext`dU[
         Pattern[$CellContext`x, 
          Blank[]], 
         Pattern[$CellContext`y, 
          Blank[]]] = {(10.526315789473687` $CellContext`x - 
          9.473684210526319 $CellContext`y)/
         2, ((-9.473684210526319) $CellContext`x + 
          10.526315789473689` $CellContext`y)/2}, $CellContext`ddU[
         Pattern[$CellContext`x, 
          Blank[]], 
         Pattern[$CellContext`y, 
          Blank[]]] = {{
        5.263157894736843, -4.736842105263159}, {-4.736842105263159, 
        5.263157894736844}}, $CellContext`GradientG[
         Pattern[$CellContext`f, 
          Blank[]], 
         PatternTest[
          Pattern[$CellContext`x, 
           Blank[List]], VectorQ]] := 
       D[$CellContext`f, {$CellContext`x, 1}], $CellContext`HessianH[
         Pattern[$CellContext`f, 
          Blank[]], 
         PatternTest[
          Pattern[$CellContext`x, 
           Blank[List]], VectorQ]] := 
       D[$CellContext`f, {$CellContext`x, 2}], $CellContext`hmc[
         Pattern[$CellContext`U, 
          Blank[]], 
         Pattern[$CellContext`dU, 
          Blank[]], 
         Pattern[$CellContext`ddU, 
          Blank[]], 
         Pattern[$CellContext`Dim, 
          Blank[]], 
         Pattern[$CellContext`BURNIN, 
          Blank[]], 
         Pattern[$CellContext`EPISODE, 
          Blank[]], 
         Pattern[$CellContext`vanilla0, 
          Blank[]], 
         Pattern[$CellContext`switch, 
          Blank[]]] := 
       Module[{$CellContext`CHAINS = 5, $CellContext`STEPS = 
          5, $CellContext`qAll, $CellContext`pAll, $CellContext`Utotal, \
$CellContext`Ktotal, $CellContext`Htotal, $CellContext`s, $CellContext`S, \
$CellContext`AS, $CellContext`ES, $CellContext`KtotalNew, $CellContext`dt, \
$CellContext`p, $CellContext`q0, $CellContext`UE, $CellContext`\[Alpha], \
$CellContext`q, $CellContext`j, $CellContext`Htotal1, $CellContext`Htotal2, \
$CellContext`i, $CellContext`ND = NormalDistribution[0, 1], $CellContext`UD = 
          UniformDistribution[], $CellContext`QS = {}, $CellContext`vanilla = \
$CellContext`vanilla0, $CellContext`dt1 = 1.*^-9, $CellContext`dt2 = 
          0.0001, $CellContext`ACS = {}}, $CellContext`pAll = 
          RandomVariate[$CellContext`ND, {$CellContext`CHAINS, \
$CellContext`Dim}]; $CellContext`qAll = 
          RandomVariate[$CellContext`ND, {$CellContext`CHAINS, \
$CellContext`Dim}]; $CellContext`Utotal = Sum[
            Apply[$CellContext`U, 
             Part[$CellContext`qAll, $CellContext`i]], {$CellContext`i, 
             1, $CellContext`CHAINS}]; $CellContext`Htotal1 = \
$CellContext`Utotal; $CellContext`Htotal2 = $CellContext`Utotal; 
         For[$CellContext`j = 1, $CellContext`j <= $CellContext`EPISODE, 
           
           Increment[$CellContext`j], $CellContext`pAll = 
            RandomVariate[$CellContext`ND, {$CellContext`CHAINS, \
$CellContext`Dim}]; $CellContext`KtotalNew = Sum[If[$CellContext`vanilla, 
                Dot[
                 Part[$CellContext`pAll, $CellContext`i], 
                 Part[$CellContext`pAll, $CellContext`i]], 
                Dot[
                 Part[$CellContext`pAll, $CellContext`i], 
                 LinearSolve[
                  Apply[$CellContext`ddU, 
                   Part[$CellContext`qAll, $CellContext`i]], 
                  Part[$CellContext`pAll, $CellContext`i]]]]/
              2, {$CellContext`i, 
               1, $CellContext`CHAINS}]; $CellContext`Utotal = Sum[
              Apply[$CellContext`U, 
               Part[$CellContext`qAll, $CellContext`i]], {$CellContext`i, 
               1, $CellContext`CHAINS}]; $CellContext`Htotal = 
            If[$CellContext`vanilla, $CellContext`Htotal1, \
$CellContext`Htotal2]; $CellContext`dt = 
            If[$CellContext`vanilla, $CellContext`dt1, $CellContext`dt2]; \
$CellContext`Ktotal = $CellContext`Htotal - $CellContext`Utotal; \
$CellContext`pAll = $CellContext`pAll Sqrt[
               
               Abs[$CellContext`Ktotal/$CellContext`KtotalNew]]; \
$CellContext`ES = {}; $CellContext`AS = {}; 
           For[$CellContext`i = 1, $CellContext`i <= $CellContext`CHAINS, 
             
             Increment[$CellContext`i], $CellContext`p = 
              Part[$CellContext`pAll, $CellContext`i]; $CellContext`q = 
              Part[$CellContext`qAll, $CellContext`i]; $CellContext`UE = {
                
                Apply[$CellContext`U, $CellContext`q]}; $CellContext`q0 = \
$CellContext`q; 
             For[$CellContext`s = 1; 
               Null, $CellContext`s <= $CellContext`STEPS, 
               
               Increment[$CellContext`s], $CellContext`p = $CellContext`p - \
$CellContext`dt 
                 Apply[$CellContext`dU, $CellContext`q]; $CellContext`q = \
$CellContext`q + $CellContext`dt If[$CellContext`vanilla, $CellContext`p, 
                    LinearSolve[
                    
                    Apply[$CellContext`ddU, $CellContext`q], $CellContext`p]]; \
$CellContext`UE = Append[$CellContext`UE, 
                  Apply[$CellContext`U, $CellContext`q]]]; $CellContext`ES = 
              Append[$CellContext`ES, $CellContext`UE]; $CellContext`\[Alpha] = 
              Exp[
                Clip[
                Apply[$CellContext`U, $CellContext`q0] - 
                 Apply[$CellContext`U, $CellContext`q], {-20, 
                 0}]]; $CellContext`AS = Append[$CellContext`AS, 
                N[$CellContext`\[Alpha]]]; 
             If[$CellContext`\[Alpha] < 
               RandomVariate[$CellContext`UD], $CellContext`q = \
$CellContext`q0]; Part[$CellContext`qAll, $CellContext`i] = $CellContext`q; 
             If[$CellContext`j > $CellContext`BURNIN, $CellContext`QS = 
               Append[$CellContext`QS, $CellContext`q]]]; $CellContext`s = 
            Union[
              Flatten[
               Table[
                Ordering[
                 Part[$CellContext`ES, $CellContext`i], 1], {$CellContext`i, 
                 1, $CellContext`CHAINS}]]]; $CellContext`S = Union[
              Flatten[
               Table[
                Ordering[
                 Part[$CellContext`ES, $CellContext`i], -1], {$CellContext`i, 
                 1, $CellContext`CHAINS}]]]; 
           If[$CellContext`j < $CellContext`BURNIN, If[
               Or[
                
                And[$CellContext`s == {
                  1, $CellContext`STEPS + 1}, $CellContext`S == {
                  1, $CellContext`STEPS + 1}], 
                
                And[$CellContext`s == {$CellContext`STEPS + 
                   1}, $CellContext`S == {
                  1}]], $CellContext`dt = $CellContext`dt 1.1]; If[
               
               Or[$CellContext`s == {
                 1}, $CellContext`S == {$CellContext`STEPS + 
                  1}], $CellContext`dt = $CellContext`dt/1.1]; If[
               And[$CellContext`Ktotal > 0, $CellContext`KtotalNew > 0], 
               If[
                
                And[$CellContext`Utotal < 
                 0, $CellContext`Htotal > $CellContext`Utotal/
                  100], $CellContext`Htotal = $CellContext`Utotal/100, 
                If[
                 
                 And[$CellContext`Utotal > 0, $CellContext`Htotal > 
                  100 $CellContext`Utotal], $CellContext`Htotal = 
                 100 $CellContext`Utotal, 
                 If[Mean[$CellContext`AS] > 
                   0.9, $CellContext`Ktotal = $CellContext`Ktotal 1.1, 
                   If[
                   Mean[$CellContext`AS] < 
                    0.1, $CellContext`Ktotal = $CellContext`Ktotal/
                    1.1]]; $CellContext`Htotal = $CellContext`Ktotal + \
$CellContext`Utotal]]]; 
             If[$CellContext`vanilla, $CellContext`Htotal1 = \
$CellContext`Htotal; $CellContext`dt1 = $CellContext`dt, $CellContext`Htotal2 = \
$CellContext`Htotal; $CellContext`dt2 = $CellContext`dt]]; 
           If[$CellContext`switch, $CellContext`vanilla = 
             Not[$CellContext`vanilla]]]; $CellContext`QS]}; 
     Typeset`initDone$$ = True),
    SynchronousInitialization->True,
    UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$},
    UnsavedVariables:>{Typeset`initDone$$},
    UntrackedVariables:>{Typeset`size$$}], "Manipulate",
   Deployed->True,
   StripOnInput->False],
  Manipulate`InterpretManipulate[1]]], "Output",
 CellChangeTimes->{3.817665256380991*^9, 3.817665344218412*^9, 
  3.817698848638335*^9, 3.8176989759101067`*^9, 3.8176990240059977`*^9},
 CellID->1210265810,ExpressionUUID->"896a0515-f506-4626-a5d9-d3e37523c996"],

Cell[BoxData[
 TagBox[
  StyleBox[
   DynamicModuleBox[{$CellContext`i$$ = 6, Typeset`show$$ = True, 
    Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", 
    Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = 
    "\"untitled\"", Typeset`specs$$ = {{
      Hold[$CellContext`i$$], 1, 8, 1}}, Typeset`size$$ = {
    468., {198., 202.}}, Typeset`update$$ = 0, Typeset`initDone$$, 
    Typeset`skipInitDone$$ = False, $CellContext`i$9439$$ = 0}, 
    DynamicBox[Manipulate`ManipulateBoxes[
     1, StandardForm, "Variables" :> {$CellContext`i$$ = 1}, 
      "ControllerVariables" :> {
        Hold[$CellContext`i$$, $CellContext`i$9439$$, 0]}, 
      "OtherVariables" :> {
       Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, 
        Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, 
        Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$,
         Typeset`skipInitDone$$}, "Body" :> 
      Module[{$CellContext`\[Rho]$, $CellContext`QS$, $CellContext`SIGMA$, \
$CellContext`U$, $CellContext`dU$, $CellContext`ddU$}, $CellContext`\[Rho]$ = 
         N[1 - 1/10^$CellContext`i$$]; $CellContext`SIGMA$ = {{
           1, $CellContext`\[Rho]$}, {$CellContext`\[Rho]$, 
            1}}; $CellContext`U$[
           Pattern[$CellContext`x$, 
            Blank[]], 
           Pattern[$CellContext`y$, 
            Blank[]]] = (1/2) Simplify[
            Dot[{$CellContext`x$, $CellContext`y$}, 
             
             LinearSolve[$CellContext`SIGMA$, {$CellContext`x$, \
$CellContext`y$}]]]; $CellContext`dU$[
           Pattern[$CellContext`x$, 
            Blank[]], 
           Pattern[$CellContext`y$, 
            Blank[]]] = $CellContext`GradientG[
           $CellContext`U$[$CellContext`x$, $CellContext`y$], \
{$CellContext`x$, $CellContext`y$}]; $CellContext`ddU$[
           Pattern[$CellContext`x$, 
            Blank[]], 
           Pattern[$CellContext`y$, 
            Blank[]]] = $CellContext`HessianH[
           $CellContext`U$[$CellContext`x$, $CellContext`y$], \
{$CellContext`x$, $CellContext`y$}]; $CellContext`QS$ = \
$CellContext`hmc[$CellContext`U$, $CellContext`dU$, $CellContext`ddU$, 2, 
           1000, 2000, True, True]; ListPlot[{$CellContext`QS$, 
           Dot[$CellContext`QS$, 
            MatrixPower[$CellContext`SIGMA$, -0.5]]}, PlotStyle -> Opacity[1],
           PlotLabel -> $CellContext`SIGMA$, 
          PlotLegends -> {"Samples", "Transformed"}, AspectRatio -> 1]], 
      "Specifications" :> {{$CellContext`i$$, 1, 8, 1}}, "Options" :> {}, 
      "DefaultOptions" :> {ControllerLinking -> True}],
     ImageSizeCache->{513., {242., 247.}},
     SingleEvaluation->True],
    Deinitialization:>None,
    DynamicModuleValues:>{},
    Initialization:>({$CellContext`\[Rho] = 
       0.9, $CellContext`QS = CompressedData["
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BCXIqrzzextwuPs+masYgoq9hgXeTGT4MF+twchTD3P8s3ferSiD/7c2882v
k4z/7rcVXDDvqdZPDPMaXll0Ttq/eT/j44N/elff3e9+8Xh2aeaT/ReYeotT
yu/tRzevVJ37N6fFt/19//VNOCM+YZgXOW2D4bwnh/YnqXJPbU57t/9lRoqH
+YTX+9dyPZwm9vQzhnlTeuPkJhd83W+d+KQplPXdfusZc0oKvz3cX74hZcOf
8CcY5m1+bHhSwvvWfodbp/u4Tj3BMC8/dWHR3wUf9st8qpsuve7R/m1VZ4pn
tj2y3/K8ObYl+I09AF6w+hM=
        
        "], $CellContext`SIGMA = {{1, 0.9}, {0.9, 1}}, $CellContext`U[
         Pattern[$CellContext`x, 
          Blank[]], 
         Pattern[$CellContext`y, 
          Blank[]]] = (
         5.263157894736843 $CellContext`x^2 - (
          9.473684210526319 $CellContext`x) $CellContext`y + 
         5.263157894736844 $CellContext`y^2)/2, $CellContext`dU[
         Pattern[$CellContext`x, 
          Blank[]], 
         Pattern[$CellContext`y, 
          Blank[]]] = {(10.526315789473687` $CellContext`x - 
          9.473684210526319 $CellContext`y)/
         2, ((-9.473684210526319) $CellContext`x + 
          10.526315789473689` $CellContext`y)/2}, $CellContext`ddU[
         Pattern[$CellContext`x, 
          Blank[]], 
         Pattern[$CellContext`y, 
          Blank[]]] = {{
        5.263157894736843, -4.736842105263159}, {-4.736842105263159, 
        5.263157894736844}}, $CellContext`GradientG[
         Pattern[$CellContext`f, 
          Blank[]], 
         PatternTest[
          Pattern[$CellContext`x, 
           Blank[List]], VectorQ]] := 
       D[$CellContext`f, {$CellContext`x, 1}], $CellContext`HessianH[
         Pattern[$CellContext`f, 
          Blank[]], 
         PatternTest[
          Pattern[$CellContext`x, 
           Blank[List]], VectorQ]] := 
       D[$CellContext`f, {$CellContext`x, 2}], $CellContext`hmc[
         Pattern[$CellContext`U, 
          Blank[]], 
         Pattern[$CellContext`dU, 
          Blank[]], 
         Pattern[$CellContext`ddU, 
          Blank[]], 
         Pattern[$CellContext`Dim, 
          Blank[]], 
         Pattern[$CellContext`BURNIN, 
          Blank[]], 
         Pattern[$CellContext`EPISODE, 
          Blank[]], 
         Pattern[$CellContext`vanilla0, 
          Blank[]], 
         Pattern[$CellContext`switch, 
          Blank[]]] := 
       Module[{$CellContext`CHAINS = 5, $CellContext`STEPS = 
          5, $CellContext`qAll, $CellContext`pAll, $CellContext`Utotal, \
$CellContext`Ktotal, $CellContext`Htotal, $CellContext`s, $CellContext`S, \
$CellContext`AS, $CellContext`ES, $CellContext`KtotalNew, $CellContext`dt, \
$CellContext`p, $CellContext`q0, $CellContext`UE, $CellContext`\[Alpha], \
$CellContext`q, $CellContext`j, $CellContext`Htotal1, $CellContext`Htotal2, \
$CellContext`i, $CellContext`ND = NormalDistribution[0, 1], $CellContext`UD = 
          UniformDistribution[], $CellContext`QS = {}, $CellContext`vanilla = \
$CellContext`vanilla0, $CellContext`dt1 = 1.*^-9, $CellContext`dt2 = 
          0.0001, $CellContext`ACS = {}}, $CellContext`pAll = 
          RandomVariate[$CellContext`ND, {$CellContext`CHAINS, \
$CellContext`Dim}]; $CellContext`qAll = 
          RandomVariate[$CellContext`ND, {$CellContext`CHAINS, \
$CellContext`Dim}]; $CellContext`Utotal = Sum[
            Apply[$CellContext`U, 
             Part[$CellContext`qAll, $CellContext`i]], {$CellContext`i, 
             1, $CellContext`CHAINS}]; $CellContext`Htotal1 = \
$CellContext`Utotal; $CellContext`Htotal2 = $CellContext`Utotal; 
         For[$CellContext`j = 1, $CellContext`j <= $CellContext`EPISODE, 
           
           Increment[$CellContext`j], $CellContext`pAll = 
            RandomVariate[$CellContext`ND, {$CellContext`CHAINS, \
$CellContext`Dim}]; $CellContext`KtotalNew = Sum[If[$CellContext`vanilla, 
                Dot[
                 Part[$CellContext`pAll, $CellContext`i], 
                 Part[$CellContext`pAll, $CellContext`i]], 
                Dot[
                 Part[$CellContext`pAll, $CellContext`i], 
                 LinearSolve[
                  Apply[$CellContext`ddU, 
                   Part[$CellContext`qAll, $CellContext`i]], 
                  Part[$CellContext`pAll, $CellContext`i]]]]/
              2, {$CellContext`i, 
               1, $CellContext`CHAINS}]; $CellContext`Utotal = Sum[
              Apply[$CellContext`U, 
               Part[$CellContext`qAll, $CellContext`i]], {$CellContext`i, 
               1, $CellContext`CHAINS}]; $CellContext`Htotal = 
            If[$CellContext`vanilla, $CellContext`Htotal1, \
$CellContext`Htotal2]; $CellContext`dt = 
            If[$CellContext`vanilla, $CellContext`dt1, $CellContext`dt2]; \
$CellContext`Ktotal = $CellContext`Htotal - $CellContext`Utotal; \
$CellContext`pAll = $CellContext`pAll Sqrt[
               
               Abs[$CellContext`Ktotal/$CellContext`KtotalNew]]; \
$CellContext`ES = {}; $CellContext`AS = {}; 
           For[$CellContext`i = 1, $CellContext`i <= $CellContext`CHAINS, 
             
             Increment[$CellContext`i], $CellContext`p = 
              Part[$CellContext`pAll, $CellContext`i]; $CellContext`q = 
              Part[$CellContext`qAll, $CellContext`i]; $CellContext`UE = {
                
                Apply[$CellContext`U, $CellContext`q]}; $CellContext`q0 = \
$CellContext`q; 
             For[$CellContext`s = 1; 
               Null, $CellContext`s <= $CellContext`STEPS, 
               
               Increment[$CellContext`s], $CellContext`p = $CellContext`p - \
$CellContext`dt 
                 Apply[$CellContext`dU, $CellContext`q]; $CellContext`q = \
$CellContext`q + $CellContext`dt If[$CellContext`vanilla, $CellContext`p, 
                    LinearSolve[
                    
                    Apply[$CellContext`ddU, $CellContext`q], $CellContext`p]]; \
$CellContext`UE = Append[$CellContext`UE, 
                  Apply[$CellContext`U, $CellContext`q]]]; $CellContext`ES = 
              Append[$CellContext`ES, $CellContext`UE]; $CellContext`\[Alpha] = 
              Exp[
                Clip[
                Apply[$CellContext`U, $CellContext`q0] - 
                 Apply[$CellContext`U, $CellContext`q], {-20, 
                 0}]]; $CellContext`AS = Append[$CellContext`AS, 
                N[$CellContext`\[Alpha]]]; 
             If[$CellContext`\[Alpha] < 
               RandomVariate[$CellContext`UD], $CellContext`q = \
$CellContext`q0]; Part[$CellContext`qAll, $CellContext`i] = $CellContext`q; 
             If[$CellContext`j > $CellContext`BURNIN, $CellContext`QS = 
               Append[$CellContext`QS, $CellContext`q]]]; $CellContext`s = 
            Union[
              Flatten[
               Table[
                Ordering[
                 Part[$CellContext`ES, $CellContext`i], 1], {$CellContext`i, 
                 1, $CellContext`CHAINS}]]]; $CellContext`S = Union[
              Flatten[
               Table[
                Ordering[
                 Part[$CellContext`ES, $CellContext`i], -1], {$CellContext`i, 
                 1, $CellContext`CHAINS}]]]; 
           If[$CellContext`j < $CellContext`BURNIN, If[
               Or[
                
                And[$CellContext`s == {
                  1, $CellContext`STEPS + 1}, $CellContext`S == {
                  1, $CellContext`STEPS + 1}], 
                
                And[$CellContext`s == {$CellContext`STEPS + 
                   1}, $CellContext`S == {
                  1}]], $CellContext`dt = $CellContext`dt 1.1]; If[
               
               Or[$CellContext`s == {
                 1}, $CellContext`S == {$CellContext`STEPS + 
                  1}], $CellContext`dt = $CellContext`dt/1.1]; If[
               And[$CellContext`Ktotal > 0, $CellContext`KtotalNew > 0], 
               If[
                
                And[$CellContext`Utotal < 
                 0, $CellContext`Htotal > $CellContext`Utotal/
                  100], $CellContext`Htotal = $CellContext`Utotal/100, 
                If[
                 
                 And[$CellContext`Utotal > 0, $CellContext`Htotal > 
                  100 $CellContext`Utotal], $CellContext`Htotal = 
                 100 $CellContext`Utotal, 
                 If[Mean[$CellContext`AS] > 
                   0.9, $CellContext`Ktotal = $CellContext`Ktotal 1.1, 
                   If[
                   Mean[$CellContext`AS] < 
                    0.1, $CellContext`Ktotal = $CellContext`Ktotal/
                    1.1]]; $CellContext`Htotal = $CellContext`Ktotal + \
$CellContext`Utotal]]]; 
             If[$CellContext`vanilla, $CellContext`Htotal1 = \
$CellContext`Htotal; $CellContext`dt1 = $CellContext`dt, $CellContext`Htotal2 = \
$CellContext`Htotal; $CellContext`dt2 = $CellContext`dt]]; 
           If[$CellContext`switch, $CellContext`vanilla = 
             Not[$CellContext`vanilla]]]; $CellContext`QS]}; 
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(* NotebookSignature Gw0x7rivJn7NKA1J5xg6ys7y *)