If you are presumed innocent, it makes sense that OVERWHELMING evidence against you can result in a VERY LOW posterior probability of you being innocent. This sounds reasonable.
However a “balance of probabilities” sounds like something completely different, i.e. “one thing is more likely than the other”. How does this go along with presumption of innocence, where to my understanding the burden of proof beyond reasonable doubt should be on the prosecution rather than defence?
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You’re confusing the burden of proof and the standard of proof.
The presumption of innocence is a burden of proof. It dictates who has to prove their case in court. If both parties show up with no evidence and no arguments, the accused goes free. If the prosecution wants a different outcome, they have the burden of proving it. It doesn’t say anything about what you have to prove or how convincing you have to be, it’s just the default outcome of the trial.
The balance of probabilities is a standard of proof. It dictates how convincing the evidence has to be to satisfy the burden. Also, incidentally, the standard of proof in criminal cases is “beyond a reasonable doubt”, which is much higher than “balance of probabilities”. The balance of probabilities is the typical standard in civil cases, where you want the court to side with whoever seems more likely to be in the right.
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