Before conviction, the gov't has the burden of proving guilt beyond reasonable doubt. Once convicted, the burden of proof transfers to the convict making tha appeal. There's nothing wrong with that.
All I can say about the Troy Davis case is that it had been reviewed by numerous courts and other legal entities (ie the Georgia Pardons Board.) None of them, who had the ability to investigate, supoena witnesses, etc., either vacated the conviction or commuted the sentance. To me that says something about the actual strength of the case that the media doesn't want to acknowledge.