In most countries, the burden of proof is on the accuser and the accused doesn’t need to proven his or her innocence and is treated as innocent until proven otherwise.
We (the rest of the world) can't prove a negative, so it is up to the claimant to supply the proof. Given proof it will be believed, no proof then it's a no-op.
With the murder trial you can assume a-priori that there is a murderer, the question then is 'who', not if a murder is possible or not. These are entirely different problems for a logic perspective.