The Wikipedia article about p-boxes only talks about cumulative probability density functions, which are meaningful for continuous sample spaces. <https://en.wikipedia.org/wiki/Probability_box>


Just out of curiosity, is it possible, and meaningful, to define p-boxes for discrete sample spaces? What axioms should they fulfil? Some quick google searches for "discrete p-box" and variants thereof did not turn up any results.


Take a three-element sample space $\{0,1,2\}$ for example. We can define a probability mass function (pmf) on it, like $\{0 \mapsto 0.5, 1 \mapsto 0.5, 2 \mapsto 0\}$, where 0 has probability 0.5, 1 has probability 0.5, and 2 has probability 0. Another one would be $\{0 \mapsto 0, 1 \mapsto 0.5, 2 \mapsto 0.5\}$.


My attempt at defining p-boxes on this would be to allow intervals for the probabilities, with the intention that it specifies all pmf's whose values lie in these intervals.


For example a p-box on the three-element set could be $\{0 \mapsto [0,0.5], 1 \mapsto [0,0.5], 2 \mapsto [0,0.5]\}$. This specifies all pmf's whose values lie between 0 and 0.5, like the two concrete ones above.


Not all of those would be meaningful, for example $\{0 \mapsto [0,0.2], 1 \mapsto [0,0.2], 2,[0 \mapsto 0.2]\}$ does not include any valid pmf, because no possible sum adds up to 1.


Another one that is not meaningful would be $\{0 \mapsto [0,\frac{1}{3}], 1 \mapsto [0,\frac{1}{3}], 2 \mapsto [0, \frac{1}{3}]\}$, because the lower bounds could never be used.


It seems that if one were to specify p-boxes in this way, there is some constraint between the upper and lower bounds.


Are p-boxes on discrete sample spaces meaningful? Are they described in some area of probability theory?



Are p-boxes for discrete sample spaces meaningful?
There is no such thing as a "cumulative probability density function". You seem to get confused between the notions of (i) the (cumulative) probability distribution function, (ii) the probability density function, and (iii) the probability mass function. I suggest you read an introductory textbook on probability or statistics, or just the corresponding Wikipedia articles. 


Concerning now p-boxes specifically: The Wikipedia article linked in your post defines a p-box in terms of a cumulative probability distribution function, which in turn is defined for arbitrary real-valued random variables -- discrete, continuous, and neither discrete nor continuous. So, there is no reason to worry about p-boxes in the discrete case. 


As for the usefulness/meaningfulness of p-boxes, even the Wikipedia article linked in your post has some [criticisms](https://en.wikipedia.org/wiki/Probability_box#Criticisms)  of the notion of a p-box. I have never before encountered this notion, and I don't believe it could be really useful. 


