ProbabilityDistribution only accepts dx=1 for discrete distributions

Question

Bug introduced in 10.4.1 or earlier and somewhat fixed in 12.0.0

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This question maybe related to (95436) but neither its answer nor the documentation for ProbabilityDistribution state whether we may build a discrete probability distribution using a step size $dx$ different from $1$.

Minimum Working Example

Let's say that we want to build a discrete probability distribution (say a prior distribution) for some random variable $X$ which can only take values in the range $[0,1]$. We do know that $Pr(X=\frac{1}{12}) = 0.25$, $Pr(X=\frac{1}{6})=0.5$, $Pr(X=\frac{1}{4})=0.25$ and the probability is zero for all other values.

From reading the documentation one might assume that we can use Piecewise within ProbabilityDistribution using a discrete range specification with $dx = \frac{1}{12}$ to build a discrete probability distribution matching our prior information:

priorDistribution = ProbabilityDistribution[
    Piecewise[
      {
        { 0.25, x == 1/12 },
        { 0.50, x == 1/6  },
        { 0.25, x == 1/4  }
      }
    ],
    { x, 0, 1, 1/12 }
]

But this surprisingly will not work in Version 11.0 :

PDF[priorDistribution, 1/4]

PDF[ProbabilityDistribution[[Piecewise] 0.25 [FormalX]==1/12 0.5 [FormalX]==1/6 0.25 [FormalX]==1/4 0 True ,{[FormalX],0,1,1/12}],0.25]

RandomVariate[ priorDistribution, 10 ]

RandomVariate::noimp: Sampling from ProbabilityDistribution[[Piecewise] 0.25 [FormalX]==1/12 0.5 [FormalX]==1/6 0.25 [FormalX]==1/4 0 True ,{[FormalX],0,1,1/12}] is not implemented.

Am I missing something? It seems that ProbabilityDistribution only accepts $dx = 1$ (I tried $dx = 2$ for a different range and it also did not work).

Update (Version 12.0.0):

While the above given example still does not work, the limitation with regard to dx has been fixed (at least with regard to dx > 1):

dist = ProbabilityDistribution[
      0.25 Boole[ x == 4]
    + 0.50 Boole[ x == 6]
    + 0.25 Boole[ x == 12 ]
    ,
    { x, 0, 12, 2}
]

PDF[ dist, 4 ] 

0.25

Answer 1

It does seem to be broken for increments other than one. The workaround is either to rescale or else use EmpiricalDistribution

priorDistribution =
  ProbabilityDistribution[
   Piecewise[{
     {0.25, x == 1},
     {0.50, x == 2},
     {0.25, x == 3}}],
   {x, 1, 12, 1}];

DiscretePlot[PDF[priorDistribution, 12 x], {x, 0, 1, 1/12}]

Alternatively,

wts = {0.25, 0.50, 0.25};

data = {1/12, 1/6, 1/4};

priorDistribution =
  EmpiricalDistribution[wts -> data];

DiscretePlot[PDF[priorDistribution, x], {x, 0, 1, 1/12}]

(* plot same as above *)