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In practice, the question of who has burden of proof usually does not change the outcome of a case, because prosecutors rarely bring cases which are marginal enough so that who has burden of proof makes a difference. In any case, most criminal cases in the United States are resolved via plea bargaining in which burden of proof again does not make a significant difference to the outcome of the case.
Outside a legal context, "burden of proof" means that someone suggesting a new theory or stating a claim must provide evidence to support it: it is not sufficient to say "you can't disprove this".
For example, if a person says "The Chinese government is plotting to poison our water supply" it is that person's burden of proof to prove this plot is actually occurring. If he can provide evidence that shows that the plot exists, then it becomes the skeptic's burden of proof to disprove the claim with facts of his own.
A classic example comes from Criswell's final speech at the end of Ed Wood's Plan 9 from Outer Space: "My friends, you have seen this incident, based on sworn testimony. Can you prove that it didn't happen?". Considering that the incident in question involved grave robbers from space, the burden of proof is being incorrectly assigned.
See also presumption of innocence, scientific method, Occam's razor