Does the following concept has a name ?
Starting point : "expected value"
What we call "expected value" for a random variable $X$ requires a vector space structure on its outcome set. Say for some discrete RV $X$, we have
$$ \mathbb{E}X = \sum_{x\in \Omega}x \mathbb{P}[X=x] $$
this clearly requires addition and multiplication by a scalar to be defined. Symbols sets $\{\circ,\diamond,\square\}$ or even natural numbers $\mathbb{N}$ don't have such structure. This results for instance in the expected outcome of a 6-face dice being 3.5 which is not in $\mathbb{N}$. And there is no "expected shape" when possible outcomes are $\{\circ,\diamond,\square\}$. So the natural definition of "expected value" is not always meaningful
Alternative definition
Alternatively, we could think of "expected value" as the fixed value $x\in \Omega$ that minimizes the expected "distance" to $X$'s outcome. Here we would just need to define a distance (aka metric) $d:\Omega \times \Omega \mapsto \mathbb{R}$ over the outcomes set, rather than a vector space structure. For RVs whose "expectation" means something, this in indeed the case that $x=\mathbb{E}X$ minimizes the expected squared deviation $\mathbb{E}[(x-X)^2]$. In the other cases however, this minimum doesn't coincide, may not be unique or even may not exist at all depending on $d$ and the distribution $p(X)$ :
- For our fair 6-faces dice, taking $d(a,b)=|a-b|$ we get x = 3 or x = 4
- For $\Omega=\{\circ,\diamond,\square\}$ with $\mathbb{P}[\circ]=0.8,\mathbb{P}[\diamond]=0.1,\mathbb{P}[\square]=0.1$ and $d(a,b)=\left\{\begin{array} 00 \text{ if } a=b \\ 1 \text{ otherwise} \end{array}\right.$, we have $x=\circ$.
- Whenever $\Omega$ is a vector space, $x$ is nothing but the "real" expectation $\mathbb{E}X=\sum_{x\in\Omega} x\mathbb{P}[X=x]$. Therefore, this "fake" expectation is somehow a generalization of $\mathbb{E}X$ to non-vector space outcome sets...
Okay this definition sounds weird in usual settings. But it is really useful in estimation theory where one wants to substitute some deterministic object $\hat{X}\in \Omega$ to the RV $X$ under an expected "loss" minimization constraint. In machine learning field, it is in fact pretty usual to define distances on sets that are not vector spaces and then seek the equivalent of a least-squares estimator. Note that the problem doesn't happen with MAP or ML estimation, wherein we always stay in the support of $X$.
So, finally, the question
Does this alternatively-defined concept has a NAME ?
thanks and sorry for not being concise :-b !
I think the term "median" would be appropriate, though I do not know if anyone else uses this term in this context. Your concept is a generalization of the median of a random variable. This is because when $X$ is real-valued, then letting $d(x,y)=|x-y|$ be the usual distance on $\mathbb R$, then the value(s) of $x$ which minimizes $E[d(X,x)]$ is the median(s) of $X$. In your dice example, both $3$ and $4$ minimize the value of $E[d(X,x)]$, and these are both media of $X$.
This term has some precedent. A geometric median of a finite set $S$ of points in $\mathbb R^n$ is defined as a point $x$ which minimizes $\sum_{y\in S}d(y,s)$, which corresponds exactly to the previous paragraph when $X$ is uniformly distributed on $S$.