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Priors are Useless.
Priors are irrelevant. Given two different prior probabilities , and
for some hypothesis
.
Let their respective posterior probabilities be and
.
After sufficient number of experiments, the posterior probability .
Or More formally:.
Where is the number of experiments.
Therefore, priors are useless.
The above is true, because as we carry out subsequent experiments, the posterior probability gets closer and closer to the true probability of the hypothesis
. The same holds true for
. As such, if you have access to a sufficient number of experiments the initial prior hypothesis you assigned the experiment is irrelevant.
To demonstrate.
http://i.prntscr.com/hj56iDxlQSW2x9Jpt4Sxhg.png
This is the graph of the above table:
http://i.prntscr.com/pcXHKqDAS\_C2aInqzqblnA.png
In the example above, the true probability of Hypothesis
is
and as we see, after sufficient number of trials, the different
s get closer to
.
To generalize from my above argument:
If you have enough information, your initial beliefs are irrelevant—you will arrive at the same final beliefs.
Because I can’t resist, a corollary to Aumann’s agreement theorem.
Given sufficient information, two rationalists will always arrive at the same final beliefs irrespective of their initial beliefs.
The above can be generalized to what I call the “Universal Agreement Theorem”:
Given sufficient evidence, all rationalists will arrive at the same set of beliefs regarding a phenomenon irrespective of their initial set of beliefs regarding said phenomenon.
Exercise For the Reader
Prove .