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 are mixing civil and criminal law. Presumption of innocence is only a thing in criminal law as only in criminal law there is a presecution and only in criminal law are they trying to prove guilt.
Balance of probablites is a thing in civil law where one person sues another for breach of contract for example and you are trying to figure out if the contract was fulfilled or not, there is no guilt involved.
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