Funkyderek,
If the evidence is only circumstantial, then he shouldn't have to prove his innocence. He should be assumed innocent until proven guilty.
He didn't have to prove his innocence, as he was presumed innocent, before trial. However, once a jury decides he is guilty based on the evidence presented to them, then the burden of proof shifts after trial. Now the onus is on Scott to sit in jail and if he can, prove his innocence.
Juries are allowed to convict on circumstantial evidence, though it is a hard case for a prosecutor to nail down. The problem is that Scott Peterson had a lot of circumstantial evidence against him, plus his attorney, Mark Gerrigos may not have properly defended him.
For example: Long before they found Laci's body, tt was well known in the media that Scott had been to the marina to go fishing. If she was murdered by someone else, then they could have easily taken her body there to make Scott look guilty. Scott's attorney never introduced that possibility to the jury or offered an alternative theory to the prosecution's theory.
Unfortunately, for Scott, his behavior and recorded phone calls, where he made some stupid inferrences, badly hurts his own case, making himself look guilty. All of this weighed on the jury and they found him guilty.
He will now have a much harder time to prove he is not guilty and get his conviction overturned on appeal. What will help him are the weird things that happened on the jury, including the forman who was removed. The former jury forman now states that he was threatened by the other jurers. This and other jury problems will likely cause the conviction to be thrown out. If that happens, Scott can retain new counsel, and get a new trial, where once again, the burden of proof will be on the State and not Scott.