If you need a 4" deep frame, I suggest considering an acrylic box instead. The difference might amaze you. I mean the acrylic box might be cheaper, as well as better.
Because the deep box of wood would cast long shadows in most lighting conditions, you would probably have to increase its size in order to show off whatever is in there.
An acrylic box would show off the framed item(s) better and because light shines through, it probably could be built with narrower perimeter margins. And you can use any standard-rabbet-depth moulding, which opens up your design options.