Bayesian Inference
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:
P(H│E)=(P(E│H)*P(H))/(P(E))
Where:
  means Given
 H any hypothesis whose probability may be affected by data
 E data that were not used in computing the prior probability
 P(H) the prior probability, is the probability of H before E is observed
 P(HE) the posterior probability, is the probability of H given E, i.e., after E is observed
 P(EH) the probability of observing E given H, is also known as the likeP(E) the marginal likelihood or "model evidence".

So;
P(H) is our new probability
P(HE) is our starting point or prior probability
P(EH) is the probability of our evidence or event: (0.5).
P(E) is our “likelihood function” which is: P(EH1) * P(H1) + P(EH2) * 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 1010 or 0.00000000071427.
Applying this to our equation:
Prior probability: 7.1427 X 1010
New probability: 5.36261 X 1018 from the DNA data
0.5 * 7.1427 X 1010 / (7.1427 X 1010 * 0.5) + (5.36261 X 1018 * 0.5) = 0.9999999924292286
Thus the probability of Anthra being ET is 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.