A defendant who offers evidence of self-defense typically doesn’t have to convince the jury that he or she was justified in using force. In most instances, the burden remains on the prosecution to prove beyond a reasonable doubt that the defendant was not so justified. However, to raise the defense in the first place, the defendant has to produce some evidence that supports the self-defense theory. If there's no evidence whatsoever that the defendant believed the victim posed a threat, there isn't enough to shift the burden to the prosecution.
Example: A jury acquitted neighborhood-watch volunteer George Zimmerman of murdering 17-year-old Trayvon Martin not necessarily because the defense convinced the jury that he acted in self-defense, but because the prosecution didn’t prove that he didn’t. (See The George Zimmerman Verdict: Murder, Manslaughter, and Self-Defense.)