The Exclusion Principle of evidence is the concept that in order for a piece of evidence to be useful to support a claim, it must first be excluded, or differentiated, from other possibile explanations.
For example, an individual named Bob claims that Jack smokes. Bob points out that Jack's house burned down as proof for this claim. While it is true that Jack's house burning down is potentially due to a mishap from smoking in bed, Bob is making a fallacy.
He's assuming that smoking in bed is the only legitimate cause for Jack's house burning down. In realty, there's several possibilities.
- Lightning struck the house.
- Bad wiring in the house.
- Mishap while smoking in bed.
- Knocked over scented candle.
- Tissue paper caught fire on stove.
The possibilities are literally endless to what could have started the fire. It's even possible that aliens from a parallel dimension shot a interdimensional heat ray at Jack's house, starting the fire.
Bob, in this case, has seemingly arbitrarily chosen smoking in bed as the cause for the house burning down. The critical thinker would immediately ask, "What makes you think that's the cause, as opposed to bad wiring?" The assertion that he's a smoker because his house burned down makes even less sense if Jack has no history of smoking (Which is analagous to theists asserting supernatural causes when nothing supernatural has been demonstrated yet).
In order to distinguish smoking in bed as the correct cause, and distinguish it from the other possibilities, Bob must provide additional evidence.
For instance, several additional pieces could distinguish smoking in bed as the correct cause.
- Indications the fire started in the bedroom.
- Cigarette butts discovered in the bedroom.
- Disqualify lighting due to clear skies.
- Disqualify scented candle if Jack never uses them.
To simply insist that Jack smokes because his house burned down, with no additional details would be woefully inadequate.
Examples in Religious Claims
In all the different ways that theistic claims can fail the standards of evidence, the most common unmet standard is exclusion. Typically, the theist ignores, or is unaware of, other possibilities. Sometimes, they ignore the null hypothesis and assume that God is the default answer to an asserted piece of evidence, which leads to the God of the gaps phenomenon.
Consider the following examples.
- The proof of God is that you exist.
- Natural causes is also a possible explanation. In fact, we can readily demonstrate that nature exists, and is operational, as opposed to supernature, which is entirely unconfirmed. What's the additional evidence that excludes nature?
- The evidence for a god is answered prayer.
- The supposed evidence for prayer could be a mixture of confirmation bias, a model for prayer that is unfalsifiable, and coincidence. What's the additional evidence that excludes these possibilities?
- The evidence for a god is the apparent design in nature.
- The perception of design in nature is a subjective opinion. What's the evidence that excludes the possibility that it's simply a question of interpretation?
The irony is that they often pick the most ludicrus and far-fetched causes as the proven cause, or simply assume that it's the most reasonable answer, despite it being unproven entirely.
Probabilistic Sorting of Possibilities
The principle of exclusion with evidence is the addressing the problem of determining which, of the infinite possibilities, is the most likely cause for an effect. The first step is to identify reasonable potential causes.
Typically, the reasonable causes are matched up against what is already known. One piece of evidence given for the veracity of the Bible is the discovery of chariot wheels discovered at the bottom of the Red Sea. This apparently supports the story of Moses parting the Red Sea, which came crashing down on the Egyptian army that was following behind him. The evidence here is the discovered chariot wheels, however, there's several possibilities as to how they got there.
- The story of Moses is true, and the wheels are the remains of those drowned soldiers.
- A shipment of chariot parts on a boat was lost during an attempted sailing across the sea, and the boat capsized.
- The account of chariot wheels being found is unreliable and a case of Lying for Jesus. 
- Members of an alien species from another planet abducted some Eqyptian soldiers and discarded the chariots as they flew away, which happened to be over the Red Sea.
Due to a loose application of Occam's Razor, #2 and #3 would seem to be the most reasonable explanations, barring further evidence. It's the one that's not asserting unproven flights of fancy to make the explanation work. What's more, we have evidence that not only is life very robust and versetile, but we're discovering more and more planets that could support life. That means, we have actual evidnece that extraterrestrial life could exist elsewhere.
That means that #4 is actually more plausable than #1, since currently we have no valid evidence of any kind of supernatural realm or being, at all.
Without further evidence to exclude the other possibilities from the Biblican explanation, we must then choose, tentatively, the most reasonable explanation - an accident during the shipment of chariot wheels - as the most likely possibility presented thus far.