The idea of eternal inflation might cut against this. Under eternal inflation new universes are always being created at an exponentially increasing rate so there are always far more young than old universes. So under this theory if you are uncertain of whether you are at a relatively early (pre-singularity) or relatively late (post-singularity) point in the universe you are almost certainly in the relatively early state because there are so many more universes in this state.

Note: Eliezer and Robin object to this idea for reasons I don't understand.