Priors Are Useless
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Priors are Useless.
Priors are irrelevant. Given two different prior probabilities
Let their respective posterior probabilities be
After sufficient number of experiments, the posterior probability
Or More formally:
Therefore, priors are useless.
The above is true, because as we carry out subsequent experiments, the posterior probability
This is the graph of the above table:
In the example above, the true probability of Hypothesis
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