Okay, so I just tested everything thoroughly and I think I now have a pretty good handle on how it works. I also now understand why almost nobody ever sees Siege. It's a REALLY hard mode to get to! For starters, I didn't even realize there were two separate stages to Prison Multiball ...
During the first stage, you must shoot the prison walker to score a jackpot. You need to do this six times to complete the stage. Shooting any of the lit arrows or the closed prison doors will open the doors. Prison bombs can be used to score a jackpot, but only if the doors are already open. If they are closed, a bomb will open them, so if you have more than one prison bomb saved up, you can essentially use two in a row to get a jackpot. Once six jackpots have been collected, the Riot shot will start to strobe. You must shoot it to score the super jackpot and begin stage two.
During stage two, the prison doors will remain closed the entire time. You must shoot one of the two ramps and then shoot the prison in order to score a jackpot. To make things even more difficult, prison bombs can no longer be used to score a jackpot. They can be used to score a hit to one of the ramps, but the jackpots must be collected by physically shooting the closed prison doors. Once you score six jackpots, it will say "Siege is lit" on the DMD. This is a bit confusing, because the Siege insert will not actually light until the multiball is over. Perhaps "Siege is ready" would have made more sense.
When the multiball ends, the Siege insert will start to strobe and the next shot to the prison will start it. Once you start Siege, the mode continues until you either complete it or the game ends. The good news is that your progress carries over from ball to ball. I'm not going to spoil Siege, but the rules for it are pretty straightforward.
I hope this information helps. Prison Multiball is actually quite confusing, especially the second stage because both ramps as well as the prison are lit the entire time making it difficult to know what shot you are supposed to go for next. When you add in the fact that you are trying to accomplish all of this while juggling 2 or more balls, it makes Siege an extremely difficult mini-wizard mode to get to. Without the bombs, it must be even harder on the pro. Fortunately, Siege is amazingly cool, making it that much sweeter when you do manage to start it.