Referencing defined symbols when simplifying equations

Question

Question: How can I tell Mathematica to replace a sub-expression (in a simplified expression) with a symbol if the symbol is defined to be that sub-expression?

Say you have defined:

f[x_, y_] := some_complicated_expr_using_x_and_y_1
g[x_, y_] := some_complicated_expr_using_x_and_y_2
h := some_complicated_expr_using_f_and_g

then you do D[h,x] // Simplify, you would get an expression that has sub-expressions some_complicated_expr_using_x_and_y_1 and some_complicated_expr_using_x_and_y_2.

How can I tell Mathematica to replace some_complicated_expr_using_x_and_y_1 with f[x,y]?

For a concrete example, consider:

h[μ_, r_] := Sum[x[μ, s]*w[s, r], {s, 1, n}]
yhat[μ_, r_] := σ[h[μ, r]]
e := (1/(2 m)) Sum[Sum[(y[μ, r] - yhat[μ, r])^2, {r, 1, p}], {μ, 1, m}]
h[μ, r]
yhat[μ, r]
e
D[e, w[i, j]]

I you evaluate the above you will get

How can I get it to produce:

(which equals to Out[97])

I have tried ReplaceRepeated but got no luck:

Answer 1

Here is the result of the operation D[e, w[i, j]]:

Clear[yhat];
expr=(1/(2 m) (\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(\[Mu] = 1\), \(m\)]\(
\*UnderoverscriptBox[\(\[Sum]\), \(r = 1\), \(p\)]\((\(-2\)\ \((
\*UnderoverscriptBox[\(\[Sum]\), \(s = 1\), \(n\)]\*
TemplateBox[{RowBox[{"i", ",", "s"}]},
"KroneckerDeltaSeq"]\ \*
TemplateBox[{RowBox[{"j", ",", "r"}]},
"KroneckerDeltaSeq"]\ x[\[Mu], s])\)\ y[\[Mu], r]\ \*
SuperscriptBox["\[Sigma]", "\[Prime]",
MultilineFunction->None][
\*UnderoverscriptBox[\(\[Sum]\), \(s = 1\), \(n\)]w[s, r]\ x[\[Mu], 
              s]] + 2\ \((
\*UnderoverscriptBox[\(\[Sum]\), \(s = 1\), \(n\)]\*
TemplateBox[{RowBox[{"i", ",", "s"}]},
"KroneckerDeltaSeq"]\ \*
TemplateBox[{RowBox[{"j", ",", "r"}]},
"KroneckerDeltaSeq"]\ x[\[Mu], s])\)\ \[Sigma][
\*UnderoverscriptBox[\(\[Sum]\), \(s = 1\), \(n\)]w[s, r]\ x[\[Mu], 
              s]]\ \*
SuperscriptBox["\[Sigma]", "\[Prime]",
MultilineFunction->None][
\*UnderoverscriptBox[\(\[Sum]\), \(s = 1\), \(n\)]w[s, r]\ x[\[Mu], 
              s]])\)\)\))) /. \[Sigma][a_] -> yhat[\[Mu], r] /. 
 Derivative[1][\[Sigma]][a_] -> yhat'[\[Mu], r]

Try this:

expr /. \[Sigma][a_] -> yhat[\[Mu], r] /.Derivative[1][\[Sigma]][a_] -> yhat'[\[Mu], r]

yielding thew following:

Have fun!