Super interesting post (as always) Professor! I'm curious about the value of P(O | not Theory). How could one begin to conceptualize this value? The set of theories that are not the one being tested in infinitely large — is there an intuitive way to quantify the probability of the observation given that it is in this space?
2. you want to think about the probability of the observation in the absence of theory. so this it's roughly the volume of your predicted interval divided by the volume of the Spielraum.
3. see 1. it's a useful way of thinking, but not one that can be made precise here. Meehl tries to do a more precise version in his paper "Cliometric Metatheory." I may cover this at some point.
Thank you! For point 2, is this basically assuming a uniform distribution in the absence of no other information? Otherwise, how could we approximate the probability in this way?
> Popper didn’t believe in [...] Bayesian thinking. In Popper’s mind, you wanted to test the most improbable theories [...] with the fewest number of parameters
This seems inconsistent. By Kolmogorov, simple theories, with few parameters, are generally each more probable than theories with many parameters.
For example, consider two hypothesis classes for the angle of light refraction: one following Snell's law and another where the refractive index n12 is a property of the pair of media (rather than the ratio of two quantities n2/n1, one for each medium). To be falsifiable, a theory needs to assign values to these quantities. Each assignment to {n_ij} is exponentially less likely than each assignment to {n_i}, unless your prior exponentially prefers the larger class.
So generally, the most improbable theories have the most number of parameters. That's the logic behind things like BIC.
For a Bayesian, this seems straightforwardly correct, though there are doubtless some subtleties to think more about.
It's worth remembering that hardly anyone before about 1950 thought in Bayesian terms, and even now it's not part of most people's mental equipment. For example, Monty Hall problems still fool most, even though they are trivial if you frame them in Bayesian terms. So, it's not surprising that Popper (born 19020 wasn't attracted to this way of thinking.
Super interesting post (as always) Professor! I'm curious about the value of P(O | not Theory). How could one begin to conceptualize this value? The set of theories that are not the one being tested in infinitely large — is there an intuitive way to quantify the probability of the observation given that it is in this space?
as I mention,
1. this should be taken as metaphor, not as math
2. you want to think about the probability of the observation in the absence of theory. so this it's roughly the volume of your predicted interval divided by the volume of the Spielraum.
3. see 1. it's a useful way of thinking, but not one that can be made precise here. Meehl tries to do a more precise version in his paper "Cliometric Metatheory." I may cover this at some point.
Thank you! For point 2, is this basically assuming a uniform distribution in the absence of no other information? Otherwise, how could we approximate the probability in this way?
I’m pretty sure Damn Strange Coincidence refers to getting an outcome in the given narrow range of the Spielraum in a world without the theory T.
Yes. At least, that's what Meehl means in the context of this lecture. Is that not clear from what I wrote? I'll wordsmith it.
Salmon develops a wider definition for a DSC in "Scientific Explanation and the Causal Structure of the World." I'll get to that version eventually.
It was not super clear.
> Popper didn’t believe in [...] Bayesian thinking. In Popper’s mind, you wanted to test the most improbable theories [...] with the fewest number of parameters
This seems inconsistent. By Kolmogorov, simple theories, with few parameters, are generally each more probable than theories with many parameters.
Why? I don't follow.
For example, consider two hypothesis classes for the angle of light refraction: one following Snell's law and another where the refractive index n12 is a property of the pair of media (rather than the ratio of two quantities n2/n1, one for each medium). To be falsifiable, a theory needs to assign values to these quantities. Each assignment to {n_ij} is exponentially less likely than each assignment to {n_i}, unless your prior exponentially prefers the larger class.
So generally, the most improbable theories have the most number of parameters. That's the logic behind things like BIC.
For a Bayesian, this seems straightforwardly correct, though there are doubtless some subtleties to think more about.
It's worth remembering that hardly anyone before about 1950 thought in Bayesian terms, and even now it's not part of most people's mental equipment. For example, Monty Hall problems still fool most, even though they are trivial if you frame them in Bayesian terms. So, it's not surprising that Popper (born 19020 wasn't attracted to this way of thinking.
Perhaps, but Popper was still writing into the 1980s, and he knew about subjective probability arguments in the 1960s and rejected them.
And yet his propensity interpretation isn’t any better.
Heh. You and I are not Popperians. ;)