In complete absence of any evidence both the statement and its logical negation have equal weight

But there is never "complete absence of any evidence." For example, if I claim to you that I have an invisible flying pig in my backyard, we both have a lifetime of experiences to draw on which are inconsistent with such a claim. e.g. witnessing pigs and similar animals running around but not flying; feeling solid objects which have always been visible in normal light; and so on. So I would bear the burden of proving my claim.