How to use pattern matching to apply a function to specific elements of a matrix

Question

I have a 3x3 matrix

testweightedgraph = {{{}, {p, q, pq, pq}, {}}, {{p, q, pq, 
pq}, {q}, {}}, {{}, {}, {}}};

whose elements are (possibly empty) lists.

I would like to apply the pure function

1 - Apply[Times, 1 - #] &

to every nonempty element of the matrix, leaving the empty elements exactly as they are. The result I want is this:

{{{}, {1 - (1 - p) (1 - pq)^2 (1 - q)}, {}}, {{1 - (1 - p) (1 - 
   pq)^2 (1 - q)}, {q}, {}}, {{}, {}, {}}}

If I use

Map[{1 - Apply[Times, 1 - #]} &, testweightedgraph, {2}]

then I get

{{{0}, {1 - (1 - p) (1 - pq)^2 (1 - q)}, {0}}, {{1 - (1 - p) (1 - 
   pq)^2 (1 - q)}, {q}, {0}}, {{0}, {0}, {0}}}

which is perfect, except that all instances of {0} should be replaced by {}.

Rather than patching the result, by somehow replacing {0}s by {}s, I would like to get it right in the first place.

One option seems to be to use Position and feed its results to MapAt. This has two disadvantages for me: 1) Perhaps Position needs to be told exactly what to look for, and cannot be asked to look for elements satisfying a pattern (such as, in this case "not being the empty list"). 2) If we are going to have to use pattern matching anyway, then isn't there a more direct way to do it, without using Position or MapAt ?

I have studied the tutorials on Pattern Matching but I confess I find it very hard to understand.

Answer 1

Design a function that ignores empty lists:

f[{}] = {};
f[{a__}] := {1 - Times @@ (1 - {a})}
Map[f, testweightedgraph, {2}]

{
 {{}, {1 - (1 - p) (1 - pq)^2 (1 - q)}, {}}, 
 {{1 - (1 - p) (1 - pq)^2 (1 - q)}, {q}, {}},
 {{}, {}, {}}
}

Answer 2

This is the sort of thing If[] was meant for:

Map[If[# =!= {}, 1 - Apply[Times, 1 - #], #] &, testweightedgraph, {2}]

Answer 3

With respect to your list of disadvantages about using Position to resolve the problem

Perhaps Position needs to be told exactly what to look for, and cannot be asked to look for elements satisfying a pattern (such as, in this case "not being the empty list")

I think that you can use a general pattern to extract non empty lists

testweightedgraph = {{{}, {p, q, pq, pq}, {}}, {{p, q, pq, 
    pq}, {q}, {}}, {{}, {}, {}}}

Position[testweightedgraph, {x__}, {2}]
(* {{1, 2}, {2, 1}, {2, 2}} *)

and then as you suggested use MapAt

MapAt[{1 - Apply[Times, 1 - #]} &, testweightedgraph, 
 Position[testweightedgraph, {a__}, {2}]]