Background
I'm studying the basic theory of Random Finite Sets (RFS), which is the name that is used in my field to denote simple point processes. A simple point process is a random variable whose outcome is a finite set composed by distinguished elements. From a practical point of view, we can characterize an RFS in terms of:
- A discrete distribution that expresses how the cardinality, i.e., the number of elements, of the RFS is distributed;
- For each possible cardinality, there is a joint distribution that expresses how the elements of the RFS are distributed.
Since we are modeling sets, the joint distributions cannot be arbitrary but must satisfy some constraints:
- the distributions must be symmetric because sets are unordered. To get a simple example, consider the special case where the RFS is a doubleton with elements $x_1,x_2 \in \mathbb{R}^d$. In this case, symmetry requires that the distribution $\Pi(\cdot)$ satisfies the equation:\begin{equation*}\Pi(A_1,A_2)=\Pi(A_2,A_1)\end{equation*} for all events $A_1,A_2 \subseteq \mathbb{R}^d$. In other words, the distribution $\Pi(\cdot)$ has to be symmetric with respect to the diagonal $\{x_1=x_2\}$
- the distributions have to be null over the diagonal $\{x_1=x_2\}$ because we do not allow repeated elements. In the doubleton case, this means that two elements must be $x_1\neq x_2$.
Now, in RFS theory, it is of interest to compute the union of two or more independent RFSs. However, a technical problem arises: due to their independence, we cannot guarantee that the union will result in a set without distinguished elements. But why is it so important to exclude repeated elements? What's the problem with non-simple point processes? I would like to understand what happens in this non-simple case.
Question
I cannot understand why we have to restrict our attention to just simple point processes. Clearly, there is a good reason for considering only simple point processes, but I cannot discern what this reason is. In other words, I don't see any problems with considering repeated elements. The consequence, as far as I understand, is simply that the distribution $\Pi(\cdot)$ does not nullify over the diagonals $\{x_1=x_2\}$ (doubleton case), $\{x_1=x_2, x_1=x_3, x_2=x_3\}$ (tripleton case), etc... So, I have two specific questions
- What is the problem with non-simple point processes?
- Is it possible to provide a simple toy counter-example where the drawbacks of non-simple point processes are clearly highlighted?
References
My reference textbook is Advances in Statistical Multisource-Multitarget Information Fusion by Mahler. The source of confusion is the following proposition (page 51)
if there are repeated elements in $[y_1\dots y_n]$ (i..e if $y_i=y_j$ for some $i\neq j$), then $f_{\mathcal{P}}([y_1,\dots,y_n])=\infty$, which cannot be used in formulas of practical interest.
here $f_{\mathcal{P}}([y_1,\dots,y_n])$ is the probability density associated to the random set $\{y_1,\dots,y_n\}$. I can't see why this should be case if we have repeated elements and, moreover, I can't see why having a non-finite density is problematic.
The quoted proposition is a Mahler's reinterpretation of a proposition from the book An Introduction to the Theory of Point Processes Volume I: Elementary Theory and Methods by Daley and Vere-Jones. Such proposition is the following (page 138)
Proposition 5.4.V. (a) A necessary and sufficient condition for a point process to be simple is that, for all $n = 1, 2,...,$ the associated Janossy measure $J_n(·)$ allots zero mass to the ‘diagonals’ $\{x_i = x_j\}$. (b) When $\mathcal{X} = \mathbb{R}^d$, the process is simple if for all such $n$ the Janossy measures have densities $j_n(·)$ with respect to $(nd)$-dimensional Lebesgue measure.
moreover, Daley and Vere-Jones say the following
The existence of densities is closely linked to the concept of orderliness, or more properly, simplicity, in the sense of Chapter 3, that with probability 1 there are no coincidences amongst the points. Suppose on the contrary that, for some population size $n$, the probability that two points coincide is positive. In terms of the measure $J_n(·)$, the necessary and sufficient condition for this probability to be positive is that $J_n(·)$ should allot nonzero mass to at least one (and hence all) of the diagonal sets $\{x_i = x_j\}$, where $x_i$ is a point in the $i$th coordinate space.
So, it seems that the main problem with non-simple point processes lies in the fact that we cannot describe them in terms of densities (only measures). Is this true? If so, why do densities not exist? Is this fact related to the Radon-Nykodim theorem?
