Priors Are Useless

[DragonGod]

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Priors are Useless.

Priors are irrelevant. Given two different prior probabilities $P{r}_{i_1}$, and $P{r}_{i_2}$ for some hypothesis $H_i$.
Let their respective posterior probabilities be $P{r}_{i_{z1}}$ and

.
After sufficient number of experiments, the posterior probability

.
Or More formally:
${lim}_{n\to \infty }\frac{P{r}_{i_{z1}}}{P{r}_{i_{z2}}}=1$.
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 $P{r}_{i_{z{1}_j}}$ gets closer and closer to the true probability of the hypothesis $P{r}_i$. The same holds true for $P{r}_{i_{z{2}_j}}$. 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 $H_i$ $\left(P_i\right)$ is $0.5$ and as we see, after sufficient number of trials, the different $P{r}_{i_{z{1}_j}}$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 ${lim}_{n\to \infty }\frac{P{r}_{i_{z1}}}{P{r}_{i_{z2}}}=1$.