Let's focus on this part of the question: "On what basis can we ever
say, even if this pattern was detected, that this was generated by
intelligent lifeforms outside earth?"
Bayesian inference framework
One common approach is to apply
Bayesian inference
as a method to compute a new hypothesis credence (posterior probability)
from an old credence (prior probability), a new observation, and the
compatibility of the evidence with the hypothesis. So let's do that to
the radio SETI problem.
Bayes' theorem is
frequently written:
P(E|H) P(H)
P(H|E) = -----------
P(E)
where:
H is the hypothesis of interest, in this case, that extra-terrestrial
technological civilizations currently exist nearby, or existed in the
past at the proper distance for their signal to reach us now.
E is the evidence, in this case the detection of a radio signal with a
particular structure, originating from outside our solar system.
P(H|E), the "probability of H given E", is the updated probability
(credence) that H is true, given we have detected E, and is what we
want to compute.
P(E|H) is the probability that E happens if we assume H is true.
We'll have to make an assumption here, so I'll explore multiple
possible values.
P(H) is the credence we assign H before detecting E. This also
requires an assumption.
P(E) is the probability of detecting E, independent of H.
Let's first deal with P(E). Typically when making scientific
measurements, we don't estimate P(E), but instead P(E|¬H), i.e.,
the probability of observing E assuming that H is false. We can then
compute P(E) by breaking it down into the two cases depending on whether
H is true or false:
P(E) = P(E|H) P(H) + P(E|¬H) P(¬H)
and:
P(¬H) = 1 - P(H)
What value should we use for P(E|¬H)? In physics, it is common to use
a "five sigma" detection
threshold, meaning that the probability that E would be detected if H were
false (that is, due to random chance) is less than 0.00006%.
Astronomers often use three sigma, but for a discovery of this
importance, the more rigorous threshold is appropriate, so I will use
0.00006% as my baseline. You've contemplated 2^-100 so I'll consider
that too.
What about P(E|H)? That is, assuming there is currently at least one
extra-terrestrial technological civilization (ETTC), what is the
probability that we would detect a structured signal from it? There
doesn't seem to be a good way to estimate this since it depends on so
many unknowns, so I'll consider a range of values, with a somewhat
arbitrary 1% as the baseline.
Finally, what about P(H)? That is, prior to any detection, what should
we assume is the probability that there is at least one ETTC? This is
the subject of
the Drake equation, but
in practice that doesn't help much. Again, I'll consider a range of
values, again using 1% as a baseline.
Baseline calculation
With baseline assumptions, we find that a five sigma detection leads
to over 99% updated credence despite only 1% prior credences:
P(H|E) = P(E|H) * P(H) / P(E)
= P(E|H) * P(H) / ( P(E|H) * P(H) + P(E|¬H) * P(¬H) )
= P(E|H) * P(H) / ( P(E|H) * P(H) + P(E|¬H) * (1 - P(H)) )
= 0.01 * 0.01 / ( 0.01 * 0.01 + 0.0000006 * (1 - 0.01) )
= 0.0001 / ( 0.0001 + 0.0000006 * 0.99 )
= 0.0001 / ( 0.0001 + 0.000000594 )
= 0.0001 / ( 0.000100594 )
= 0.994
Moreover, we can see from the structure of this calculation that the
result is primarily determined by how P(E|H)P(H) (the product of our
priors) compares to P(E|¬H) (the probability of detection under the
null hypothesis).
More skeptical priors
With five sigma detection, how low do our priors have to be to
significantly degrade the final credence? Let's try a few values:
P(E|H) P(H) P(H|E)
------ ------ ------
0.01 0.01 0.994 baseline
0.001 0.001 0.625 more skeptical
0.0001 0.0001 0.016 initially highly skeptical, first detection
0.0001 0.016 0.730 initially highly skeptical, second detection
0.0001 0.730 0.998 initially highly skeptical, third detection
So, if you think there's only a 0.01% chance an ETTC exists, and a 0.01%
chance that we would detect one if it did exist, then even with a five
sigma signal detection, your updated credence that an ETTC exists is
only 2%. But then, further detections, if they occur, would quickly
push the credence to near certainty.
Extreme detection
What if P(E|¬H), the probability of observing the evidence due to
random chance, is very low, say, 2^-100 = 7.89e-31? Let's try it:
P(E|¬H) P(E|H) P(H) P(H|E)
---------- ------ ------ ------
0.0000006 0.0001 0.0001 0.016 highly skeptical from above
7.89e-31 0.0001 0.0001 1.000 P(E|¬H) = 2^-100
7.89e-31 1e-10 1e-10 1.000 absurdly skeptical
7.89e-31 1e-15 1e-15 0.559 ... and then some
We find that, if the chance of the detected signal occurring by chance
is 2^-100, then we'd need the product of our priors to be comparable to
that value to have any posterior doubt, which requires at least one to
be, in my opinion, unjustifiably small.
Conclusion
Bayesian inference provides one possible answer to how a rational agent
should update its credences in light of new evidence. Applying it to
the problem of a potential detection of extra-terrestrial technology, we
see that when using reasonably but not excessively skeptical prior
credences (1% each), a five sigma signal detection ought to be regarded
as very convincing, and that with either repeated detection or a signal
with greater structure, even extreme skeptics ought to be persuaded.
But, I can't tell you your priors! Every rational agent arrives at the
inference problem with their own, and for any given detection and
accompanying probability of occurrence by chance, there are priors that
make that (single) detection unpersuasive.
Addendum: Regarding selection of priors, see follow-up question: Can a zero prior probability for some theories be justified?
