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> But if we already know the generating process, all the hard work is already done. Yes, I agree completely, and I believe that so would most frequentists! The

by _dps 13y ago

> But if we already know the generating process, all the hard work is already done. Yes, I agree completely, and I believe that so would most frequentists! The point, even among the most strident critics, is rarely one of whether Bayesian approaches are useful; it is that Frequentists regard the proper domain of what they call "probability" to be exclusively related to generative processes and relative sampling ratios. From a purely technical point of view I see nothing wrong with that, even if it is a somewhat strict position (having trained primarily in mathematics I'm comfortable allowing people their strict definitions as long as they are recognized as such). As a natural consequence of this divergence, Bayesian modeling allows you to explicitly punt the problem of generative modeling for the tricky bits, and it puts a nice big warning label on it saying "If you botch your prior, you're going to have a bad time." The biased coin is a perfect example. Knowing the bias, the generative process for a sequence of flips is trivial. But how are you going to create a generative process that seriously engages how and where the biases emerge? It is far easier to just say "empirically, biases seem to be distributed like so." At this point the Frequentist says "Well, that assumption is not derived from a formal model and is only loosely falsifiable, but if I accept it as a substitute for a generative model then you and I will reach the same conclusions about posterior probabilities." I know of no frequentist who would disagree that getting good generative models for complex phenomena is often extraordinarily difficult. Where they disagree with Bayesians is whether doing something other than that should, strictly, be called "probability" or, whether it is more appropriate to call it something like "semi-empirical subjective-belief modeling".

pdonis 13y ago
how are you going to create a generative process that seriously engages how and where the biases emerge? E. T. Jaynes would have said that you do this using your knowledge of the physics of coins and coin flipping. One of the examples he uses in his book Probability Theory: The Logic of Science is a robotic coin-flipper that can control the process so as to always make the coin land on the same side, i.e., the "bias" is in the flipping process, not in the coin itself. If you don't know anything about the flipping process or the relevant physics, then you have no way of constructing any hypotheses about what sort of generative process might be involved. It is far easier to just say "empirically, biases seem to be distributed like so." This corresponds to the case where you don't know anything about the underlying physics; in Bayesian terms, you are assuming a maximum entropy prior with a constraint--the constraint being the distribution observed in the flips so far. But if you do know something about the underlying physics--for example, if you know the coin is being flipped by a robotic flipper with such-and-such design--you might be able to come up with a much better prior using that knowledge. I'm not sure whether that sort of thing is included in the frequentist's concept of a "generative model".

by _dps 13y ago

pdonis 13y ago

how are you going to create a generative process that seriously engages how and where the biases emerge? E. T. Jaynes would have said that you do this using your knowledge of the physics of coins and coin flipping. One of the examples he uses in his book Probability Theory: The Logic of Science is a robotic coin-flipper that can control the process so as to always make the coin land on the same side, i.e., the "bias" is in the flipping process, not in the coin itself. If you don't know anything about the flipping process or the relevant physics, then you have no way of constructing any hypotheses about what sort of generative process might be involved. It is far easier to just say "empirically, biases seem to be distributed like so." This corresponds to the case where you don't know anything about the underlying physics; in Bayesian terms, you are assuming a maximum entropy prior with a constraint--the constraint being the distribution observed in the flips so far. But if you do know something about the underlying physics--for example, if you know the coin is being flipped by a robotic flipper with such-and-such design--you might be able to come up with a much better prior using that knowledge. I'm not sure whether that sort of thing is included in the frequentist's concept of a "generative model".