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ImmortalRationalist comments on Stupid Questions May 2015 - Less Wrong Discussion
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Using Bayesian reasoning, what is the probability that the sun will rise tomorrow? If we assume that induction works, and that something happening previously, i.e. the sun rising before, increases the posterior probability that it will happen again, wouldn't we ultimately need some kind of "first hyperprior" to base our Bayesian updates on, for when we originally lack any data to conclude that the sun will rise tomorrow?
This is a well-known problem dating back to Laplace (pp 18-19 of the book).
Also, how do we know when the probability surpasses 50%? Couldn't the prior probability of the sun rising tomorrow be astronomically small, and with Bayesian updates using the evidence that the sun will rise tomorrow, merely make the probability slightly less astronomically small?
The prior probability could be anything you want. Laplace advised taking a uniform distribution in the absence of any other data. In other words, unless you have some reason to suppose one outcome is more likely than another, you should weigh them equally. For the sunrise problem, you could invoke the laws of physics and our observations of the solar system to assign a prior probability much greater than 0.5.
Example: If I handed you a biased coin and asked for the prior probability that it comes up heads, it would be reasonable to suppose that there's no reason to suppose it biased to one side over the other and so assign a prior of 0.5. If I asked about the prior probability of three heads in a row, then there's nothing stopping you from saying "Either it happens or it doesn't, so 50/50". However, if your prior is that the coin is fair, then you can compute the prior for three heads in a row as 1/8.