I don’t like Good’s solution. It’s mathematically correct, but, why would our prior look like this? Why would we know there’s b black objects, and more generally, why would we expect there to be (anti)correlation between the color of ravens and the color of non-ravens? (In fact, my prior would be that if all ravens are black, this very slightly increases the chances of non-raven objects being black; objects might mimic ravens (e.g., artwork depicting ravens), and objects might be produced by similar processes as ravens (evolution in some niche) and hence have a similar phenotype.)
More than the prior, I find the sampling process to be nonsensical. Why would we expect to run across the objects in the world uniformly at random? Seeing ravens is highly correlated with being outside, looking at the sky, etc, so why wouldn’t seeing white ravens have similar but much narrower requirements, like being on a specific island? Adding something along these lines to our model should make the bayesian evidence gained by observing either black ravens or non-black non-ravens quickly go to 0.
Many years ago, I was repeatedly amused when reading texts that talked about rare “black swan” events. Where I lived there were hundreds of swans living on the river, and every single one was black.
Getting back to the original point: a uniform distribution is superficially the simplest model, so should be associated with the lowest complexity penalty. However after gathering broader evidence, you see that this doesn’t hold for anything else, so why should you expect it to hold for ravens?
Especially once you get down into gears-level models like inheritance, one should already expect that if there are both black ravens and white ravens, they are more likely to be geographically separated than intermingled.
To address the paradox: If I show you a box and (trustworthily) tell you that there’s one object in the box, and then you look through a tiny hole in the box so all you can see is the color, and you see not-black, that is probably evidence that not all ravens are black (unless we think there’s way more non-black objects in worlds where all ravens are black, enough to overcome the possibility that we’re looking at a non-black raven). Then, if we open the box and see a non-raven, that observation is evidence in favor of “all ravens are black”, because we’ve (mostly) screened off the evidence against—we definitely weren’t looking at a nonblack raven.
I don’t like Good’s solution. It’s mathematically correct, but, why would our prior look like this? Why would we know there’s b black objects, and more generally, why would we expect there to be (anti)correlation between the color of ravens and the color of non-ravens? (In fact, my prior would be that if all ravens are black, this very slightly increases the chances of non-raven objects being black; objects might mimic ravens (e.g., artwork depicting ravens), and objects might be produced by similar processes as ravens (evolution in some niche) and hence have a similar phenotype.)
More than the prior, I find the sampling process to be nonsensical. Why would we expect to run across the objects in the world uniformly at random? Seeing ravens is highly correlated with being outside, looking at the sky, etc, so why wouldn’t seeing white ravens have similar but much narrower requirements, like being on a specific island? Adding something along these lines to our model should make the bayesian evidence gained by observing either black ravens or non-black non-ravens quickly go to 0.
Many years ago, I was repeatedly amused when reading texts that talked about rare “black swan” events. Where I lived there were hundreds of swans living on the river, and every single one was black.
Getting back to the original point: a uniform distribution is superficially the simplest model, so should be associated with the lowest complexity penalty. However after gathering broader evidence, you see that this doesn’t hold for anything else, so why should you expect it to hold for ravens?
Especially once you get down into gears-level models like inheritance, one should already expect that if there are both black ravens and white ravens, they are more likely to be geographically separated than intermingled.
To address the paradox: If I show you a box and (trustworthily) tell you that there’s one object in the box, and then you look through a tiny hole in the box so all you can see is the color, and you see not-black, that is probably evidence that not all ravens are black (unless we think there’s way more non-black objects in worlds where all ravens are black, enough to overcome the possibility that we’re looking at a non-black raven). Then, if we open the box and see a non-raven, that observation is evidence in favor of “all ravens are black”, because we’ve (mostly) screened off the evidence against—we definitely weren’t looking at a nonblack raven.