Let A = Hypothesis that aliens are visiting Earth

Let Not[A] = Hypothesis that aliens are not visiting Earth

Let { CE_i } = A set of i independent Close Encounter events

Using Bayes' theorem we find that the (posterior) probability that aliens are visiting Earth given a set of i independent close encounters, P( A | { CE_i } ), is given by

P( A | { CE_i } ) = P( { CE_i } | A ) P( A ) / [ P( { CE_i } | A ) P( A ) + P( { CE_i } | Not[A] ) P( Not[A] ) ]

Now I argue that the probability of a set of i independent close encounters given the hypothesis that aliens are visiting Earth, P( { CE_i } | A ), can be taken to be one by definition. A believer in the alien hypothesis will not be surprised to learn of a set of close encounters with aliens.

I wish to argue backwards to find an expression for the prior probability for aliens P( A ) that gives us an appreciable posterior probability for aliens given reports of i close encounters, P( A | { CE_i } ) = 1/2.

Substituting P( { CE_i } | A ) = 1 and P( A | { CE_i } ) = 1/2 into Bayes' theorem we get

P( A ) / P( Not[A] ) = P( { CE_i } | Not[A] )

P( A ) / P( Not[A] ) = P( CE_1 | Not[A] ) * P( CE_2 | Not[A] ) * P( CE_3 | Not[A] ) * ...

Let us assume that aliens are not visiting Earth and that the witness was awake during the encounter (no bed-time accounts) and that the encounter was faithfully recorded soon after the event. I would say that the witness was either lying, hallucinating or the victim of a hoax.

For the sake of argument let us assume that we have a compelling case so that P( CE | Not[A] ) = 1 / 100

Let us assume that we have 10 good cases of close encounters. In order to end up with an appreciable posterior probability of aliens given those cases, i.e. P( A | { CE_i } ) = 1/2, the prior ratio of our belief to unbelief in aliens, P( A ) / P( Not[A] ) is given by

P( A ) / P( Not[A] ) = (1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100) = 10^-20

Thus if we combine data from compelling close encounter cases even the most hardened skeptic should begin to believe in the alien hypothesis.

Let Not[A] = Hypothesis that aliens are not visiting Earth

Let { CE_i } = A set of i independent Close Encounter events

Using Bayes' theorem we find that the (posterior) probability that aliens are visiting Earth given a set of i independent close encounters, P( A | { CE_i } ), is given by

P( A | { CE_i } ) = P( { CE_i } | A ) P( A ) / [ P( { CE_i } | A ) P( A ) + P( { CE_i } | Not[A] ) P( Not[A] ) ]

Now I argue that the probability of a set of i independent close encounters given the hypothesis that aliens are visiting Earth, P( { CE_i } | A ), can be taken to be one by definition. A believer in the alien hypothesis will not be surprised to learn of a set of close encounters with aliens.

I wish to argue backwards to find an expression for the prior probability for aliens P( A ) that gives us an appreciable posterior probability for aliens given reports of i close encounters, P( A | { CE_i } ) = 1/2.

Substituting P( { CE_i } | A ) = 1 and P( A | { CE_i } ) = 1/2 into Bayes' theorem we get

P( A ) / P( Not[A] ) = P( { CE_i } | Not[A] )

P( A ) / P( Not[A] ) = P( CE_1 | Not[A] ) * P( CE_2 | Not[A] ) * P( CE_3 | Not[A] ) * ...

Let us assume that aliens are not visiting Earth and that the witness was awake during the encounter (no bed-time accounts) and that the encounter was faithfully recorded soon after the event. I would say that the witness was either lying, hallucinating or the victim of a hoax.

For the sake of argument let us assume that we have a compelling case so that P( CE | Not[A] ) = 1 / 100

Let us assume that we have 10 good cases of close encounters. In order to end up with an appreciable posterior probability of aliens given those cases, i.e. P( A | { CE_i } ) = 1/2, the prior ratio of our belief to unbelief in aliens, P( A ) / P( Not[A] ) is given by

P( A ) / P( Not[A] ) = (1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100) = 10^-20

Thus if we combine data from compelling close encounter cases even the most hardened skeptic should begin to believe in the alien hypothesis.

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