Priors Are Useless

by DragonGod 2y21st Jun 201722 comments

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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.

 

Exercise For the Reader

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

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