Categorical Distribution

Question

Suppose I want to do calculations with a random variable $X$ that has a simple categorical distribution (a.k.a generalised Bernoulli distribution or discrete distribution).

I was expecting to be able to work with a distribution "symbolically". That is, I was expecting that there would be a function like CategoricalDistribution, that we could use like this.

distX = CategoricalDistribution[{1/3, 1/2, 1/6}];

But no distributions in tutorial/DiscreteDistributions looked like they were suitable. An example of how the distribution would be used would be

{Probability[x == 0, x \[Distributed] dist],
 Probability[x == 1, x \[Distributed] dist],
 Probability[x == 2, x \[Distributed] dist]}

would output

{1/3, 1/2, 1/6}

In this particular case, we can do a workaround, by doing

dist = TransformedDistribution[
  x1 + x2, {x1 \[Distributed] BernoulliDistribution[1/2], 
   x2 \[Distributed] BernoulliDistribution[1/3]}]

Unfortunately, this is not always possible, even with only 3 possible outcomes. For example, we cannot simulate CategoricalDistribution[{1/2, 0, 1/2}] like this, as we cannot have independent Bernoulli random variables that behave like this.

I was hoping the categorical distribution might be a special case of another distribution. Especially the MultinomialDistribution with 1 as its first argument looks useful, but I cannot transform that distribution into a distribution that gives an integer. Again it seems like I am missing a function like CategoricalDistribution to do so. To give an example

Position[RandomVariate@MultinomialDistribution[1, {1/2, 0, 1/2}], 
  1][[1, 1]]

"has the right distribution", but unfortunately I cannot use Position inside TransformedDistribution.

I am aware of functions like RandomChoice, but really I only want to do manipulations on distributions. Does anybody know how to do this?

Answer 1

Update: CategoricalDistribution is a built-in symbol as of Version 12.1 in 2020. Much of the requested functionality is incorporated. However since categories are inherently non-numeric, functions such as Mean are not implemented for categorical distributions.

https://reference.wolfram.com/language/ref/CategoricalDistribution.html

Answer 2

EmpiricalDistribution can assign probabilities to each element in a set of discrete values:

Here's an example:

In[1]:= d = EmpiricalDistribution[{1/3, 1/2, 1/6} -> {1, 2, 3}];

In[2]:= Mean[d]
Out[2]= 11/6

In[3]:= PDF[d, x]
Out[3]= 1/3 Boole[1 == x] + 1/2 Boole[2 == x] + 1/6 Boole[3 == x]

In[4]:= CDF[d, x]
Out[4]= 1/3 Boole[1 <= x] + 1/2 Boole[2 <= x] + 1/6 Boole[3 <= x]

In[5]:= RandomVariate[d, 10]
Out[5]= {1, 3, 1, 2, 1, 2, 2, 1, 2, 1}

In[6]:= Probability[x == 1, x \[Distributed] d]
Out[6]= 1/3