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:

equation for bayesian inference

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(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 evidence".

So;

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 or 0.00000000071427.

 

Applying this to our equation:

Prior probability: 7.1427 X 10-10
New probability: 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 X 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.

 

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