This is our monthly thread for collecting these little gems and pearls of wisdom, rationality-related quotes you've seen recently, or had stored in your quotesfile for ages, and which might be handy to link to in one of our discussions.
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Well, suppose I currently assess the odds of God's existence at epsilon. If I encounter evidence with an odds ratio of a million to one, then I update to 1,000,000 * epsilon.
If epsilon is required to be a standard real number, then I am forced to either make epsilon non-zero (but less than 1 ppm), or make it zero and stop calling myself a Bayesian.
But if epsilon is allowed to be a non-standard real - specifically, an infinitesimal - then I think I can have my atheist cake and be a Bayesian too.
Perhaps this example might help. Suppose I tell you that I am thinking of a random point in the closed unit square. You choose a uniform prior. That means you believe that the probability that my point is on the boundary of the square is zero. So, what do you, as a Bayesian, do when I inform you that the point is indeed on the boundary and ask you for the probability that it is on the bottom edge?
Either you had to initially assign a finite probability to the point being on the boundary (and also a finite probability to it having an x coordinate of exactly 0.5, etc.) or else you find some way of claiming that the probability of the point is infinitesimal - that is, if you are forced to pick a real number, you will pick 0, but you refuse to be forced to pick a real number.
For any probablity p strictly between 0 and 1, and any distance r greater than 0, there exists a finite amount of evidence E that would convince a Bayesian tha... (read more)