symbolic - About Symbolize and Variables

I do believe this sort of similar questions had been asked many times, so I have read a LOT. For example this and this as well as many other related posts.

But a glance at what my main question is this:

Needs["Notation`"]

Symbolize[ParsedBoxWrapper[SubscriptBox["x", "_"]]]
Symbolize[ParsedBoxWrapper[SubscriptBox["y", "_"]]]
var1 = Table[ToExpression["Subscript[x, " <> ToString[i] <> "]"], {i, 10}]
var2 = Table[ToExpression["Subscript[y, " <> ToString[i] <> "]"], {i, 10}]

1/(1 + Exp[var1 + var2])
(* Does not work *)
Variables[%]


1/(1 + Exp[var1 + var2])
(* Does work *)
Variables[Level[%,4]]

Now the real challenge is: I need expression for TWO sets of variables (symbols) $p_{jc}$ and $\phi_{jc}$, like this

$$p_{jc} = \frac{1}{1+\exp{(-\mu_p-\tau_{pj}-\eta_{pc}})}$$ and $$\phi_{jc} = \frac{1}{1+\exp{(-\mu_\phi-\tau_{\phi j}-\eta_{\phi c}})}$$ for different $j$s and $c$s.

Ideally, I want to define variables like $\mu_{\phi},\mu_p,\tau_{pj}$, ect. They need to be treated like a variable by the function Variables, not when I Level out the expression.

I coded the whole thing in Maple with out any problem, but in Mathematica, this seen to be very difficult to do so. The similar command for Variables in Maple is indets.

I now wonder:

1 - Is that possible to define things like $\mu_{\phi},\tau_{\phi j},\eta_{\phi c}$. How?

2 - When I use these symbols, are they going to be recognized without any problem by Variables?

This is particularly important because I write functions that take $j$ and $c$ to create the variables above, then a complicated expression, involving exponential. I want to determine the "number of parameters".

3 - A bit vague, but more to do. I will use these symbols to create function, and do optimizations, and replace /. using rules. Would that have difficulties? See the PDF . Page 1, sskappa(ss,kappa); transforms expression (1) to expression (4) using rules (3).

4 - Just as a note to the Symbolize command, why does Variables do not treat the Symbolized variables as variables? (get me {} as shown above in the code)

Here is a PDF file, produced by Maple, on briefly what I am trying to get in Mathematica. Don't worry to much about the expression "kappa", but what I can do with "kappa", for example, using Maple command indets.

Any advise on how to approach this would be appreciated!

Answer

For the moment leaving aside the wisdom of using Subscripts as Symbols in Mathematica it would appear that your immediate problem has nothing to do with Subscript at all but rather the behavior of Variables.

Recall its definition:

Variables[poly] gives a list of all independent variables in a polynomial.

And observe that even when using true Symbols you do not get the result you desire:

var1 = Table[Symbol["x" <> ToString[i]], {i, 10}];
var2 = Table[Symbol["y" <> ToString[i]], {i, 10}];

expr = 1/(1 + Exp[var1 + var2]);

Variables[expr]

{}