What is a good way to display many discrete probability distributions on the same plot in Mathematica?

Question

For something like 2 to 30 distributions each with 2 to 30 values.

If we assume the distributions have unique values, a standard stacked bar chart is insufficient, but maybe something similar would be good? Unique values in the distributions means that not only do they have different probabilities, but their possible values are over the reals and are different in each distribution.

So, for example, if I were trying to represent the proportion of horses of various colors in different geographic regions(latitudes), then the standard stacked bar chart would work, one distribution for each region. But, if I were trying to represent various lotteries which each have their own system of payouts with various probabilities, then the chart would not be sufficient, because the payouts would be in the reals, and they wouldn't necessarily fall into neat categories like brown, black, white, grey. See my hack, each outcome has a different value in addition to its probability.

If color is used to display the value of each possibility within the discrete probability distribution, each probability could be represented by the height of the bar region, and each region could be vertically ordered by value.

So, check out this hack I made of a DensityPlot using piecewise functions to demonstrate the idea; there must be a better solution in Mathematica.

valseg[n_] := {Sort[RandomReal[1, n]], Sort[RandomReal[1, n - 1]]};
piece[{vals_, segs_}] := 
  Piecewise[
   Prepend[Append[
 Table[{vals[[i + 1]], segs[[i]] < y < segs[[i + 1]]}, {i, 
   Length[segs] - 1}], {vals[[1]], 
  y < segs[[1]]}], {vals[[Length[vals]]], 
 y > segs[[Length[segs]]]}]];
piecefunc1 = piece[valseg[5]];
piecefunc2 = piece[valseg[5]];
d1 = DensityPlot[piecefunc1, {x, 0, 1}, {y, 0, 1}, 
   ColorFunction -> "ThermometerColors" ];
d2 = DensityPlot[piecefunc2, {x, 1, 2}, {y, 0, 1} , 
   ColorFunction -> "ThermometerColors"];
Show[d1, d2, PlotRange -> {{0, 2}, {0, 1}}]

There is another example given with sample values: Distribution 1: 50% chance of getting a 10.4, 50% chance of getting a 20.5 Distribution 2: 80% chance of getting a 50.6, 20% chance of getting a 5.5. Distribution 3: 10% chance each for getting 5.1, 10.2, 15.3, 20.4, 15.5, 20.6, 25.7, 30.8, 35.9, 40. For the hack, it looks like this:

Answer 1

For the example given:

• Distribution 1: 50% chance of getting a 10.4, 50% chance of getting a 20.5
• Distribution 2: 80% chance of getting a 50.6, 20% chance of getting a 5.5.

• Distribution 3: 10% chance each for getting 5.1, 10.2, 15.3, 20.4, 15.5, 20.6, 25.7, 30.8, 35.9, 40

  Histogram3D[ {{{1, 10.4}, {1, 20.6}}, {{2, 50.6}, {2, 50.6}, {2, 50.6}, {2, 50.6}, {2, 5.5}}, {{3, 5.1}, {3, 10.2}, {3, 15.3}, {3, 20.4}, {3, 20.6}, {3, 25.7}, {3, 30.8}, {3, 35.9}, {3, 40}}}, {3, 10}, "Probability", AxesLabel -> {Text[Style["Distribution index", 18]], Text[Style["Value", 18]], Text[Style["Probability", 18]]}, ImageSize -> 700]

Alternatively:

    ListPointPlot3D[{
  {{1, 10.4, .5}, {1, 20.6, .5}}, 
  {{2, 50.6, .8}, {2, 5.5, .2}},
  {{3, 5.1, .1}, {3, 10.2, .1}, {3, 15.3, .1}, {3, 20.4, .1}, 
   {3, 20.6, .1}, {3, 25.7, .1}, {3, 30.8, .1}, {3, 35.9, .1}, 
   {3, 40, .1}}},
 Filling -> Bottom,
 PlotRange -> {0, 1},
 PlotStyle -> {{PointSize[0.02], Red}, {PointSize[0.02], 
    Green}, {PointSize[0.02], Blue}},
 FillingStyle -> Thick,
 Ticks -> {{1, 2, 3}, Automatic, Automatic},
 AxesLabel -> {
   Text[Style["Distribution index", 16]], 
   Text[Style["Value", 16]], 
   Text[Style["Probability", 16]]},
 ImageSize -> 500]

Answer 2

Perhaps

BarChart[RandomReal[1, {5, 5}], ChartLayout -> "Percentile", Joined -> True]

Answer 3

If you have the discrete probabilities in $\{value,prob\}$ pairs.

dist1 = {{10.4, .5}, {20.5, .5}};
dist2 = {{50.6, .8}, {5.5, .2}};
dist3 = {#, .1} & /@ {5.1, 10.2, 15.3, 20.4, 15.5, 20.6, 25.7, 30.8, 35.9, 40.};

Then a ListPlot could be used to plot the PDF.

ListPlot[{dist1, dist2, dist3}, Filling -> Axis, 
 PlotMarkers -> 
  Graphics[{EdgeForm[{Thin, Black}], Opacity[.65], Disk[{0, 0}, Scaled[.03]]}],
 FillingStyle -> Medium]

---

Update

For many distributions a ListPointPlot3D can be used.

ListPointPlot3D[
 MapIndexed[
  Function[{item, index}, {First@index, Sequence @@ #} & /@ item], 
  {dist1, dist2, dist3, dist1, dist2, dist3}, 1],
 Filling -> Bottom, 
 PlotRange -> Full,
 PlotStyle -> Directive[PointSize[Large], Opacity[.7]],
 FaceGrids -> {{{-1, 0, 0}, {None, Automatic}}, {{0, 1, 0}, {None, 
     Automatic}}},
 FaceGridsStyle -> Directive[LightGray],
 Boxed -> False,
 AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}}]

---

A BubbleChart would work. It is 2D but it displays 3D data. There is no overlap of the distributions. The LabelingFunction has been updated to show the probabilities.

BubbleChart[
 MapIndexed[
  Function[{item, index}, {First@index, Sequence @@ #} & /@ item], 
  {dist1, dist2, dist3, dist1, dist2, dist3}, 1],
 ChartBaseStyle -> 
  Directive[EdgeForm[{Black, Thin, Opacity[1]}], Opacity[0.5]],
 FrameTicks -> 
  {
   {Automatic, Automatic},
   {{#, StringTemplate["\!\(\*SubscriptBox[\(X\), \(`1`\)]\)"][#]} & /@ Range[6], Automatic}
   },
 LabelingFunction -> 
  Function[{value, index, lbls}, 
   StringTemplate[
     "\[DoubleStruckCapitalP]( \!\(\*SubscriptBox[\(X\), \(`1`\)]\) = `2`) = `3`"][First@index, value[[2]], value[[3]]]]]