In statistics, Bayesian inference is a method of inference
in which Bayes' rule is used to update the probability estimate for a
hypothesis as additional evidence is learned. Bayesian updating is an important
technique throughout statistics, and especially in mathematical statistics:
Exhibiting a Bayesian derivation for a statistical method automatically ensures
that the method works as well as any competing method, for some cases. Bayesian
updating is especially important in the dynamic analysis of a sequence of data.
Bayesian inference has found application in a range of fields including science,
engineering, medicine, and law. [wikipedia]
Bayesian inference derives the probability as a result of
two antecedents, a prior probability and a "likelihood function"
derived from a probability model for the data to be observed. Bayesian
inference computes the posterior probability according this equation:
| means Given
H any hypothesis whose probability may be
affected by data
E data that were not used in computing the prior
P(H) the prior probability, is the probability of H before
E is observed
P(H|E) the posterior probability, is the probability of H
given E, i.e., after E is observed
P(E|H) the probability of observing E given H, is also
known as the likelihood. Default: 0.5
P(E) the marginal likelihood or "model
P(H) is our new probability
P(H|E) is our starting point or prior probability
P(E|H) is the probability of our evidence or event (0.5).
P(E) is our “likelihood function” which is: P(E|H1)
* P(H1) + P(E|H2) * P(H2)
To make this work we need a starting point; one that should
give the “anti” hypothesis greater weight. Since the DNA probabilities are the
only indicators available; we shall start there.
In our “Real World Probabilities of Extraterrestrials on
Earth” paper we show the probability of ET being on Earth; at 7.1427 X 10-10
Applying this to our equation:
probability: 7.1427 X 10-10
5.36261 X 10-18 from the DNA data
0.5 * 7.1427 X 10-10 /
(7.1427 X 10-10 * 0.5) + (5.36261
10-18 * 0.5) = 0.9999999924292286
Our DNA data has changed to probability of me being ET to 0.9999999924292286 or 1: 1.00000000757077
making it an almost “sure thing”,
and making the probability
of being Terrestrial: 0.00000000757077 or
1 in 132,086,960.77149.
While this is perhaps not the kind of “proof” many seem to
want; it does serve to illustrate the Mathematical probabilities at play here.