Terminology: The difference between $X$'s convention

Question

I am reading the paper, Classification in Networked Data: A Toolkit and a Univariate Case Study. And I have a question about the terminology of this paper, on page 938:

Also, see the following equation, on page 947:

Here we have,

1. $\mathbf{X}$
2. $X_i$
3. $x_i$
4. $\chi$ (not quite the same font with the paper)

I can understand that 1 is the vector of all class labels since it is big and bold. That makes totally sense.

And 2 is an item of (1) on the index $i$, if I understand correctly. However what is the difference between 2 and 3?

4 also makes sense. However, why do we have $c_1, c_2, ...$ etc. instead of basically having $\chi_1$, $\chi_2$, etc.

Is the second equation even correct? Instead of $P(x_i = c~|~N_i)$ don't we need big $X$ like $P(X_i = c~|~N_i)$?

Summary: Can you explain the difference between these 4 X's like explaining to a 10-year-old? (examples are strongly encouraged)

Answer 1

Typically in probability and statistics, capital letters such as $X_i$ are used to denote a random variable, while lower-case letters such as $x_i$ denote realizations of the random variable. Rigorously speaking, you are correct in pointing out that $P(x_i = c|\mathcal{N}_i)$ should have been written as $P(X_i = c|\mathcal{N}_i)$, although this distinction is often overlooked when the underlying random variable is clear from context. For example, the technically correct way of writing $P(x_i|\mathcal{N}_i)$ should be $P(X_i = x_i|\mathcal{N}_i)$, but this slight abuse of notation is often employed and perhaps even preferred for its simplicity.

As for $\mathcal{X}$, careful authors tend to prefer script-letters, such as $X\in\mathcal{X}, Y\in\mathcal{Y}, x\in\mathbb{R}$ for denoting domains/ranges of numbers, random variables, and functions. However, when referring to particular elements of the domain/range, one would still abide by regular typography (lower-case for real numbers and upper-case for random variables) since it would be too much hustle and slightly abusing to use a calligraphic character for a real number.

Keep in mind, however, that these are all typographical conventions used by the community, and different authors may have different preferences depending on the level of exposition and their personal style. For instance, introductory linear algebra textbooks often denote matrices and vectors in bold letters such as $\boldsymbol{A}$ and $\boldsymbol{\beta}$, but this convention is often neglected in higher-level texts and papers$-$it may be unnecessary, not to mention troublesome, to embolden every letter in a book on multivariate analysis or matrix analysis.

Answer 2

The paper is concerned with a graph with attributes. The attributes are $X_1,\ldots,X_n$, and are collectively known as $\mathbf{X}$, which is either the vector $X_1,\ldots,X_n$ or the set $\{X_1,\ldots,X_n\}$.

Each attribute $X_i$ can take a value in the set $\mathcal{X} = \{c_1,\ldots,c_m\}$.

The value of the attribute $X_i$ is denoted $x_i$. There are two possibilities to interpret this distinction:

1. $X_i$ is the name of the property, and $x_i$ is its value. This seems like a superfluous distinction, since usually we identify the name of a variable with its value.

2. $X_i$ is a random variable, and $x_i$ is its value in a specific instantiation. This distinction is useful so that we can write expressions of the form $\Pr[X_1 = x_1 | X_2 = x_2]$. There is a difference, however, between $X_i$ and $x_i$: while $X_i$ is the "official" name of the attribute, $x_i$ is just an arbitrary choice for the specific value of $X_i$ in some context.

The equation on page 947 suggests the first interpretation, and that the authors have background in some very formal disciple such as program verification, programming languages or philosophical logic, in which such distinctions are given importance.