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Priors
Personal Blog

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# Priors are Useless.

Priors are irrelevant. Given two different prior probabilities $Pr_{i_1}$, and $Pr_{i_2}$ for some hypothesis $H_i$.
Let their respective posterior probabilities be $Pr_{i_{z1}}$ and $Pr_{i_{z2}$.
After sufficient number of experiments, the posterior probability $Pr_{i_{z1}} \approx [;Pr_{i_{z2}$.
Or More formally:
$\lim_{n \to \infty} \frac{ Pr_{i_{z1}}}{Pr_{i_{z2}}} = 1$.
Where $n$ is the number of experiments.
Therefore, priors are useless.
The above is true, because as we carry out subsequent experiments, the posterior probability $Pr_{i_{z1_j}}$ gets closer and closer to the true probability of the hypothesis $Pr_i$. The same holds true for $Pr_{i_{z2_j}}$. 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 $H_i$ $(P_i)$ is $0.5$ and as we see, after sufficient number of trials, the different $Pr_{i_{z1_j}}$s get closer to $0.5$.

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.

Prove $\lim_{n \to \infty} \frac{ Pr_{i_{z1}}}{Pr_{i_{z2}}} = 1$.